Open Access

Cauchy-Neumann Problem for Second-Order General Schrödinger Equations in Cylinders with Nonsmooth Bases

Boundary Value Problems20092009:231802

DOI: 10.1155/2009/231802

Received: 26 February 2009

Accepted: 18 June 2009

Published: 14 July 2009

Abstract

The main goal of this paper is to obtain the regularity of weak solutions of Cauchy-Neumann problems for the second-order general Schrödinger equations in domains with conical points on the boundary of the bases.

1. Introduction and Notations

Cauchy-Dirichlet problem for general Schrödinger systems in domains containing conical points has been investigated in [1, 2]. Cauchy-Neumann problems have been dealt with for hyperbolic systems in [3] and for parabolic equations in [46]. In this paper we consider the Cauchy-Neumann problem for the second-order general Schrödinger equations in infinite cylinders with nonsmooth bases. The solvability of this problem has been considered in [7]. Our main purpose here is to study the regularity of weak solution of the mentioned problem.

The paper consists of six sections. In Section 1, we introduce some notations and functional spaces used throughout the text. A weak solution of the problem is defined in Section 2 together with some results of its unique existence and smoothness with the time variable. Our main result, the regularity with respect to both of time and spatial variables of the weak solution of the problem, is stated in Section 3. The proof of this result is given in Section 4 with some auxiliary lemmas. In Section 5 we specify that result for the classical Schrödinger equations in quantum mechanics. Finally, some conclusions of our results are given in Section 6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq1_HTML.gif be a bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq3_HTML.gif denote the closure and the boundary of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq4_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq5_HTML.gif . We suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq6_HTML.gif is an infinitely differentiable surface everywhere except the coordinate origin and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq7_HTML.gif coincides with the cone https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq8_HTML.gif in a neighborhood of the origin point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq9_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq10_HTML.gif is a smooth domain on the unit sphere https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq11_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq12_HTML.gif We begin by introducing some notations and functional spaces which are used fluently in the rest.

Denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq14_HTML.gif is the closure of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq15_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq16_HTML.gif . For each multi-index https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq17_HTML.gif , set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq19_HTML.gif .

In this paper we will use usual functional spaces: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq20_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq21_HTML.gif (see [1, 2] for the precise definitions).

Denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq22_HTML.gif is a space of all measurable complex functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq23_HTML.gif that satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq24_HTML.gif —a space of all measurable complex functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq25_HTML.gif that have generalized derivatives up to order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq26_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq27_HTML.gif and up to order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq28_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq29_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ2_HTML.gif
(1.2)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq30_HTML.gif —a space of all measurable complex functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq31_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ3_HTML.gif
(1.3)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq32_HTML.gif —a weighted space with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ4_HTML.gif
(1.4)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq33_HTML.gif be a Banach space. Denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq34_HTML.gif a space of all measurable functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq35_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ5_HTML.gif
(1.5)

2. Formulation of the Problem and Obvious Results

In this paper we consider following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ6_HTML.gif
(2.1)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ7_HTML.gif
(2.2)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ8_HTML.gif
(2.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq36_HTML.gif is a formal self-adjoint differential operator of second-order defined in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq37_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ9_HTML.gif
(2.4)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq38_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ10_HTML.gif
(2.5)

is the conormal derivative on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq39_HTML.gif is the unit exterior normal to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq40_HTML.gif is a given function.

Set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ11_HTML.gif
(2.6)
Throughout this paper, we assume that the coefficients of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq41_HTML.gif are infinitely differentiable and bounded in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq42_HTML.gif together with all their derivatives. Moreover, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq43_HTML.gif are continuous in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq44_HTML.gif uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq45_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq46_HTML.gif In addition, assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq47_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq48_HTML.gif —coercive uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq49_HTML.gif that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ12_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq50_HTML.gif is a positive constant independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq52_HTML.gif

The function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq53_HTML.gif is called a weak solution in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq54_HTML.gif of the problem (2.1)–(2.3) if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq55_HTML.gif , satisfying for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq56_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ13_HTML.gif
(2.8)

for all test functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq58_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq59_HTML.gif .

Now we derive here some our obvious results of the unique existence and smoothness with respect to time variable of the weak solution of the problem (2.1)–(2.3) as lemmas of main results.

Lemma 2.1.

The solvability of the problem, (see [7, Theorems 3.1, 3.2]). There exists a positive number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq60_HTML.gif such that if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq61_HTML.gif then for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq62_HTML.gif the Cauchy-Neumann problem (2.1)–(2.3) has exactly one weak solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq63_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq64_HTML.gif , that satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ14_HTML.gif
(2.9)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq65_HTML.gif does not depend on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq66_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq67_HTML.gif .

The constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq68_HTML.gif depends only on the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq69_HTML.gif and the dimension of the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq70_HTML.gif

Lemma 2.2.

The regularity with respect to time variable of the weak solution (see [7, Theorem 4.1]).Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq71_HTML.gif be a nonnegative integer. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq72_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq73_HTML.gif and if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq74_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq75_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq76_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq77_HTML.gif Then for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq78_HTML.gif , the weak solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq79_HTML.gif of the problem (2.1)–(2.3) has generalized derivatives with respect to time variable up to order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq80_HTML.gif , which belong to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq81_HTML.gif moreover
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ15_HTML.gif
(2.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq82_HTML.gif is a constant independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq83_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq84_HTML.gif .

3. Formulation of the Main Result

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq85_HTML.gif be the principal homogenous part of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq86_HTML.gif We can write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq87_HTML.gif in the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ16_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq88_HTML.gif is an arbitrary local coordinate system on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq89_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq90_HTML.gif is a linear operator with smooth coefficients.

Denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq91_HTML.gif is an eigenvalue of Neumann problem for following equation:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ17_HTML.gif
(3.2)

It is well known in [8] that for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq92_HTML.gif the spectrum of this problem is an enumerable set of eigenvalues.

Recall that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq93_HTML.gif is the positive real number in Lemma 2.1. Now, let us give the main result of the present paper.

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq94_HTML.gif be a nonnegative integer. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq95_HTML.gif is a weak solution in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq96_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq97_HTML.gif of the problem (2.1)–(2.3) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq98_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq99_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq100_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq101_HTML.gif . In addition, suppose that in the strip
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ18_HTML.gif
(3.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq102_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq103_HTML.gif according to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq104_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq105_HTML.gif there is no point from the spectrum of the Neumann problem for the equation (3.2) for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq106_HTML.gif . Then we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq107_HTML.gif and the following estimate holds
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ19_HTML.gif
(3.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq108_HTML.gif is a constant independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq109_HTML.gif .

4. Proof of Theorem  3.1

By using the same arguments as in [1, 2] and Lemmas 2.1, 2.2, we can prove following lemma.

Lemma 4.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq110_HTML.gif arbitrary. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq111_HTML.gif is a weak solution of the problem (2.1)–(2.3) in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq112_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq113_HTML.gif . Then for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq114_HTML.gif the equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ20_HTML.gif
(4.1)

holds for all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq115_HTML.gif .

Now we surround the origin by a neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq116_HTML.gif with a sufficiently small diameter such that the intersection of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq118_HTML.gif coincides with the cone https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq119_HTML.gif We begin by proving some auxiliary lemmas.

Lemma 4.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq120_HTML.gif be a weak solution in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq121_HTML.gif of the problem (2.1)–(2.3) such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq122_HTML.gif outside https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq123_HTML.gif . Moreover, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq124_HTML.gif . Then for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq125_HTML.gif one has

(i)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq126_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq127_HTML.gif ,

(ii)if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq128_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq129_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq130_HTML.gif arbitrary.

Proof.

Because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq131_HTML.gif from Lemma 2.2 we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq132_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq133_HTML.gif for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq134_HTML.gif . Following Lemma 4.1, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq135_HTML.gif is a solution of the Neumann problem for elliptic equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ21_HTML.gif
(4.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq136_HTML.gif for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq137_HTML.gif .Denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq138_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq139_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq140_HTML.gif be large enough such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq141_HTML.gif . By choosing a smooth domain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq142_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq143_HTML.gif , from the theory of the regular of solutions of the boundary value problem for elliptic systems in smooth domains and near the piece smooth boundary of domain (see [9] for reference), we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq144_HTML.gif for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq145_HTML.gif and the following inequality holds
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ22_HTML.gif
(4.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq146_HTML.gif is a positive constant independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq147_HTML.gif . It follows
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ23_HTML.gif
(4.4)
By choosing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq148_HTML.gif and setting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq149_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ24_HTML.gif
(4.5)
Return to the variable https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq150_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ25_HTML.gif
(4.6)

where the positive constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq151_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq152_HTML.gif

Case 1 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq153_HTML.gif ).

Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ26_HTML.gif
(4.7)
It follows from (4.6) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ27_HTML.gif
(4.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq154_HTML.gif does not depend on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq155_HTML.gif Taking sum with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq156_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ28_HTML.gif
(4.9)
This implies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ29_HTML.gif
(4.10)
Because in out of a neighborhood of conical point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq157_HTML.gif is a smooth domain, so we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ30_HTML.gif
(4.11)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq158_HTML.gif almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq159_HTML.gif From (4.7), (4.11) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq160_HTML.gif we receive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq161_HTML.gif for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq162_HTML.gif

Case 2 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq163_HTML.gif ).

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq164_HTML.gif so for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq165_HTML.gif one has https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq166_HTML.gif This implies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq167_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq168_HTML.gif arbitrary, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq169_HTML.gif is a positive constant. Because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq170_HTML.gif outside https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq171_HTML.gif , so we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ31_HTML.gif
(4.12)
For all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq172_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq173_HTML.gif , so it follows from [8, Lemma  7.1.1, page 268] that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ32_HTML.gif
(4.13)
From the inequality (4.6), for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq174_HTML.gif one gets
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ33_HTML.gif
(4.14)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq175_HTML.gif does not depend on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq176_HTML.gif By using analogous arguments used in Case 1, from (4.13), (4.14) we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ34_HTML.gif
(4.15)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq177_HTML.gif almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq178_HTML.gif That is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq179_HTML.gif . The lemma is proved.

Lemma 4.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq180_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq181_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq182_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq183_HTML.gif is a weak solution in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq184_HTML.gif of the problem (2.1)–(2.3) such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq185_HTML.gif outside https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq186_HTML.gif . In addition, suppose that the strip
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ35_HTML.gif
(4.16)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq187_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq188_HTML.gif according to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq189_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq190_HTML.gif , does not contain any point of the spectrum of the Neumann problem for the equation (3.2) for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq191_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq192_HTML.gif .

Proof.

We can rewrite (2.1) in the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ36_HTML.gif
(4.17)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq193_HTML.gif then by applying Lemma 4.2 we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq194_HTML.gif . In another way, because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq195_HTML.gif are continuous in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq196_HTML.gif uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq197_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq198_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq199_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq200_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq201_HTML.gif is a constant independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq202_HTML.gif . Therefore, from the hypotheses of this lemma one gets https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq203_HTML.gif for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq204_HTML.gif . Since in the strip https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq205_HTML.gif there is no spectral point of the Neumann problem for the equation (3.2) for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq206_HTML.gif , then following results of the work [9], one gets https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq207_HTML.gif and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ37_HTML.gif
(4.18)
for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq208_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq209_HTML.gif is a positive constant. Using the same arguments in the proof of Lemma 4.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ38_HTML.gif
(4.19)
for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq210_HTML.gif . Multiplying this inequality with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq211_HTML.gif , then integrating with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq212_HTML.gif from 0 to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq213_HTML.gif , from Lemma 2.2 one gets
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ39_HTML.gif
(4.20)

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq214_HTML.gif is a function in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq215_HTML.gif

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq216_HTML.gif then following Lemma 4.2 we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq217_HTML.gif for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq218_HTML.gif . This and the property of the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq219_HTML.gif continuous in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq220_HTML.gif uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq221_HTML.gif follows https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq222_HTML.gif . Because the strip https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq223_HTML.gif does not contain any spectral point of the Neumann problem for (3.2), so from results of the work [9] we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq224_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ40_HTML.gif
(4.21)

Repeating the proof in the case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq225_HTML.gif we achieve https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq226_HTML.gif , too.

Now differentiating (2.1) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq227_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ41_HTML.gif
(4.22)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq228_HTML.gif . From the hypotheses of the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq229_HTML.gif and Lemma 2.2 we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq230_HTML.gif for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq231_HTML.gif . Repeating arguments used for function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq232_HTML.gif we receive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq233_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq234_HTML.gif .

In another way, it follows from Lemma 2.2 that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ42_HTML.gif
(4.23)

From (4.23) and the assertion that both https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq235_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq236_HTML.gif are in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq237_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq238_HTML.gif . This lemma is proved.

Lemma 4.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq239_HTML.gif be a nonnegative integer number, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq240_HTML.gif be a real number satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq241_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq242_HTML.gif be a weak solution in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq243_HTML.gif of the problem (2.1)–(2.3) such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq244_HTML.gif outside https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq245_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq246_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq247_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq248_HTML.gif . Moreover, suppose that the strip
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ43_HTML.gif
(4.24)
does not contain any point of the spectrum of the Neumann problem for the equation (3.2) for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq249_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq250_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq251_HTML.gif according to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq252_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq253_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq254_HTML.gif , satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ44_HTML.gif
(4.25)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq255_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq256_HTML.gif

Proof.

We use the induction by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq257_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq258_HTML.gif then we had Lemma 4.3 with noting that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq259_HTML.gif . Assume that lemma's assertion holds up to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq260_HTML.gif , we need to prove this holds up to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq261_HTML.gif . It means that we have to prove following inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ45_HTML.gif
(4.26)

for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq262_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq263_HTML.gif is a positive constant.

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq264_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq265_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq266_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq267_HTML.gif . In another way, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq268_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq269_HTML.gif . Then from Lemma 2.2 we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq270_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq271_HTML.gif Hence, by using similar arguments in the proof of Lemma 4.3 we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq272_HTML.gif . This means that (4.26) holds for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq273_HTML.gif .

Assume that (4.26) holds for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq274_HTML.gif . By putting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq275_HTML.gif (by inductive hypothesis) and differentiating (2.1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq276_HTML.gif -times with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq277_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ46_HTML.gif
(4.27)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq278_HTML.gif Following the assumptions of the induction of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq279_HTML.gif and the hypotheses of the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq280_HTML.gif one has https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq281_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq282_HTML.gif . It follows https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq283_HTML.gif . In another way since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq284_HTML.gif so we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq285_HTML.gif for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq286_HTML.gif . Because the strip https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq287_HTML.gif does not contain any point of the spectrum of the Neumann problem for (3.2) for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq288_HTML.gif , then following results of the work [9], one gets https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq289_HTML.gif . This implies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq290_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq291_HTML.gif then by applying [8, Theorem 7.3.2] one gets https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq292_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ47_HTML.gif
(4.28)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq293_HTML.gif is a positive constant. In another way, it is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ48_HTML.gif
(4.29)
Hence from the inductive assumptions we receive
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ49_HTML.gif
(4.30)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq294_HTML.gif is a constant independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq295_HTML.gif . It means that (4.26) is proved. Finally we only need to fix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq296_HTML.gif in (4.26) to complete the proof of this lemma.

Now let us prove Theorem 3.1.

Proof.

Denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq297_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq298_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq299_HTML.gif in a neighborhood of coordinate origin. The function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq300_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ50_HTML.gif
(4.31)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq301_HTML.gif is a linear differential operator order 1. Coefficients of this operator depend on the choice of the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq302_HTML.gif and equal to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq303_HTML.gif outside https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq304_HTML.gif Denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq305_HTML.gif . It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq306_HTML.gif is equal to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq307_HTML.gif in a neighborhood of conical point. Therefore we can apply the theorem on the smoothness of a solution of elliptic problem in a smooth domain to this function to conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq308_HTML.gif for almost all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq309_HTML.gif By applying Lemma 2.2 we receive https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq310_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ51_HTML.gif
(4.32)

Now, let us prove Theorem 3.1 by induction by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq311_HTML.gif When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq312_HTML.gif then functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq313_HTML.gif satisfy the hypotheses of Lemma 4.3. So https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq314_HTML.gif It follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq315_HTML.gif is in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq316_HTML.gif Assume that the theorem holds up to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq317_HTML.gif then we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq318_HTML.gif By using analogous arguments in the proof of Lemma 4.4, with note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq319_HTML.gif (from the hypothesis of induction), we can prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq320_HTML.gif . So https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq321_HTML.gif The inequality in Theorem 3.1 can derive from inequality (4.25) (for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq322_HTML.gif ) and inequality (4.32). The theorem is proved completely.

5. Cauchy-Neumann Problem for Classical Schrödinger Equation in Quantum Mechanics

In this section we apply the previous result to the Cauchy-Neumann problem for classical Schrödinger equations in quantum mechanics. It is shown that the smoothness of the weak solution of this problem depends on the structure of the boundary of the domain, the right hand side and the dimension https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq323_HTML.gif of the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq324_HTML.gif

The classical Schrödinger equation in quantum mechanics has the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ52_HTML.gif
(5.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq325_HTML.gif is the Laplace operator. Now we consider the Cauchy-Neumann problem for (5.1) in infinite cylinder https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq326_HTML.gif with the initial condition
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ53_HTML.gif
(5.2)
and the boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ54_HTML.gif
(5.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq327_HTML.gif is the unit exterior normal to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq328_HTML.gif

The Laplace operator in polar coordinate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq329_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq330_HTML.gif can be written in the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ55_HTML.gif
(5.4)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq331_HTML.gif is the Laplace-Beltrami operator on the unit sphere https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq332_HTML.gif Therefore, the corresponding spectral problem for (3.2) is the Neumann problem for following equation:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ56_HTML.gif
(5.5)

The regularity of the weak solution of the problem (5.1)–(5.3) can be stayed as follows.

Theorem 5.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq333_HTML.gif be a weak solution in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq334_HTML.gif of the Cauchy-Neumann problem (5.1)–(5.3) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq335_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq336_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq337_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq338_HTML.gif .

Proof.

Note https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq339_HTML.gif be nonnegative eigenvalues of the Neumann problem for equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ57_HTML.gif
(5.6)
Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq340_HTML.gif are eigenvalues of the Neumann problem for (5.5). It is easy to see that when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq341_HTML.gif the strip
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_Equ58_HTML.gif
(5.7)

does not contain any eigenvalue of the Neumann problem for (5.5). By applying Theorem  3.1 we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq342_HTML.gif . The theorem is proved.

6. Conclusions

The Schrödinger equation has received a great deal of attention from mathematicians, in particular because of its application to quantum mechanics and optics. It is therefore important to research boundary value problems for it. Such problems have been previously proposed and analyzed for Schrödinger equations whose coefficients are independent of the time variable and in finite cylinders https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq343_HTML.gif (see, e.g., [10]). In infinite cylinder https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq344_HTML.gif , the first initial boundary value problem for this kind of equation with coefficients depend on both of time and spatial variables has been considered (see [1, 2]). In this paper, for a general Schrödinger equation in infinite cylinder https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq345_HTML.gif with conical points in the boundary of base, we proved regularity property of solution of second initial boundary value problem. As a special application of these new results, we received the regularity of solution of a classical Schrödinger equation in quantum mechanics when the dimension of space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq346_HTML.gif . The similar questions for the case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq347_HTML.gif can be answered after researching the asymptotic of solution in the case the strip https://static-content.springer.com/image/art%3A10.1155%2F2009%2F231802/MediaObjects/13661_2009_Article_834_IEq348_HTML.gif contains eigenvalues of the associated spectral problem. This is also the aim of our future research.

Authors’ Affiliations

(1)
Department of Mathematics, Hanoi University of Education
(2)
Faculty of Applied Mathematics and Informatics, Hanoi University of Technology

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© Nguyen Manh Hung et al. 2009

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