Open Access

Regularity of the Solution of the First Initial-Boundary Value Problem for Hyperbolic Equations in Domains with Cuspidal Points on Boundary

Boundary Value Problems20102009:135730

DOI: 10.1155/2009/135730

Received: 3 July 2009

Accepted: 8 December 2009

Published: 12 January 2010

Abstract

The goal of this paper is to establish the regularity of the solution of the first initial-boundary value problem for general higher-order hyperbolic equations in cylinders with the bases containing cuspidal points.

1. Introduction

Initial boundary-value problems for hyperbolic and parabolic type equations in a cylinder with the base containing conical points have been developed sufficiently by us [14], the main results of which are about the unique existence of the solution and asymptotic expansions of the solution near a neighborhood of a conical point. However, those problems mentioned above in cylinder with base containing cuspidal point, also interesting for applied sciences, have not been studied yet.

In the present paper, we are concerned with the first initial boundary value problems for higher hyperbolic equation in a cylinder, whose base containing cuspidal points.

In [5, 6] we showed the existence of a sequence of smooth domains https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq1_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq3_HTML.gif . Furthermore, we proved the existence, the uniqueness, and the smoothness with respect to time variable of the generalized solution by approximating boundary method, which can be applied for nonlinear equations. With the help of the results in [5, 6] as well as the results for elliptic boundary value problems in [7, 8], we can deal with the regularity with respect to both time variables and spatial ones of the solution.

Our paper is organized as follows: in Section 2, we introduce exterior cusp domain and weight Sobolev spaces. In Section 3, we will state the formulation of the problem. The main results, Theorems 4.3, 4.6, and 4.7, are stated in Section 4, and examples are given in Section 5.

2. Cusp Domain and Weighted Sobolev Spaces

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq4_HTML.gif be an infinitely differentiable positive function on the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq5_HTML.gif satisfying the following conditions:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq6_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq7_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq8_HTML.gif .

These conditions are satisfied; the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq9_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq10_HTML.gif is an example. Obviously, conditions (i) and (ii) imply https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq11_HTML.gif . We assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq12_HTML.gif is a bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq13_HTML.gif is smooth, and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ1_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq14_HTML.gif is a smooth domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq15_HTML.gif . Then the mapping
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ2_HTML.gif
(2.2)

takes the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq16_HTML.gif onto the half-cylinder https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq17_HTML.gif . Moreover, it follows that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ3_HTML.gif
(2.3)

We extend the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq18_HTML.gif to an infinitely differentiable positive function on the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq19_HTML.gif The space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq20_HTML.gif can be defined as the closure of the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq21_HTML.gif with respect to the norm

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ4_HTML.gif
(2.4)

It is known that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq22_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq23_HTML.gif (see [7, Lemma https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq24_HTML.gif ]).

We also denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq25_HTML.gif the Sobolev space of functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq26_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq27_HTML.gif that have generalized derivatives https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq29_HTML.gif . The norm in this space is defined as follows:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ5_HTML.gif
(2.5)

The space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq30_HTML.gif is the completion of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq31_HTML.gif in norm of the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq32_HTML.gif .

Set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq33_HTML.gif ; we proceed to introduce some functional spaces. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq34_HTML.gif be Banach spaces, we denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq35_HTML.gif the spaces consisting of all measurable functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq36_HTML.gif with norm

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ6_HTML.gif
(2.6)

and by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq38_HTML.gif the spaces consisting of all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq39_HTML.gif such that generalized derivatives https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq40_HTML.gif exist and belong to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq41_HTML.gif , (see [9]), with norms

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ7_HTML.gif
(2.7)

For shortness, we set

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ8_HTML.gif
(2.8)
Finally, we define the weighted Sobolev space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq42_HTML.gif as a set of all functions defined in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq43_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ9_HTML.gif
(2.9)

To simplify notation, we continue to write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq44_HTML.gif instead of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq45_HTML.gif .

3. Formulation of the Problem

Let us consider the partial differential operator of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq46_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ10_HTML.gif
(3.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq47_HTML.gif are functions with complex values, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq48_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq49_HTML.gif denotes the transposed conjugate of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq50_HTML.gif ) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq51_HTML.gif are infinitely differentiable in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq52_HTML.gif . Moreover, we assume that the functions
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ11_HTML.gif
(3.2)

satisfy the condition of stabilization for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq53_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq54_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq55_HTML.gif (see [7, Section https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq56_HTML.gif ]). Then the coefficients of the operators https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq57_HTML.gif , which arise from operators https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq58_HTML.gif via the coordinate change https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq59_HTML.gif , stabilize for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq60_HTML.gif . If we replace the coefficients of the differential operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq61_HTML.gif by their limits for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq62_HTML.gif , we get differential operator which has coefficients depending only on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq64_HTML.gif (for the convenience in use, we denote also by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq65_HTML.gif ).

In the paper, we usually use the following Green's formula:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ12_HTML.gif
(3.3)
which is valid for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq66_HTML.gif and a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq67_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ13_HTML.gif
(3.4)

We also suppose that the form https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq68_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq69_HTML.gif -elliptic uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq70_HTML.gif , that is, the inequality

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ14_HTML.gif
(3.5)

is valid for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq71_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq72_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq73_HTML.gif is the positive constant independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq75_HTML.gif . In this paper, we consider the following problem:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ15_HTML.gif
(3.6)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ16_HTML.gif
(3.7)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ17_HTML.gif
(3.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq77_HTML.gif are derivatives with respect to the outer unit normal of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq78_HTML.gif .

Definition 3.1.

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq79_HTML.gif is called a generalized solution of problem (3.6)–(3.8) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq80_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq81_HTML.gif and the equality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ18_HTML.gif
(3.9)

holds for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq82_HTML.gif .

The existence, the uniqueness and the smoothness with respect to the time variable for the generalized solution of problem (3.6)–(3.8) in the Sobolev space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq83_HTML.gif were established in [5, 6] according to following theorems:

Theorem 3.2.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq84_HTML.gif and there exists a positive number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq85_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ19_HTML.gif
(3.10)
Then problem (3.6), (3.8) has the unique generalized solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq86_HTML.gif , and the following estimate holds
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ20_HTML.gif
(3.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq87_HTML.gif is a constant independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq88_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq89_HTML.gif .

Theorem 3.3.

Suppose that the following hypotheses are satisfied:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq90_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq91_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq92_HTML.gif .

Then the generalized solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq93_HTML.gif of problem (3.6), (3.8) has generalized derivatives with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq94_HTML.gif up to order h in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq95_HTML.gif and satisfies the following estimate:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ21_HTML.gif
(3.12)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq96_HTML.gif is a constant independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq97_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq98_HTML.gif .

Owing to the support of the following proposition, we can apply the results of the Dirichlet problem for elliptic equation in domains with exterior cusps.

Proposition 3.4.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq99_HTML.gif is a generalized solution of problem (3.6)–(3.8) and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq100_HTML.gif . Then for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq101_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq102_HTML.gif is a generalized solution in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq103_HTML.gif of the Dirichlet problem for elliptic equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ22_HTML.gif
(3.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq104_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq105_HTML.gif be an orthogonal basis of the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq106_HTML.gif . Setting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq107_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq108_HTML.gif , and substituting the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq109_HTML.gif into (3.9), we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ23_HTML.gif
(3.14)
We will denote by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ24_HTML.gif
(3.15)
that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq110_HTML.gif Noting that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq111_HTML.gif and using Fubini's theorem, we obtain from (3.14) that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq112_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq113_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq114_HTML.gif is a set of measure zero. Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq115_HTML.gif are dense in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq116_HTML.gif , the following equality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ25_HTML.gif
(3.16)

holds for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq117_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq118_HTML.gif . It follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq119_HTML.gif is a generalized solution in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq120_HTML.gif of the Dirichlet problem for elliptic equation (3.13), for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq121_HTML.gif .

4. The Main Results

In this section, we would like to present the main results of the study which is based on our previous results (cf. [5, 6]) and the results of elliptic equations in cusp domains (cf. [7]). For the start of this section, we denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq122_HTML.gif the operator corresponding to the parameter-depending boundary value problem

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ26_HTML.gif
(4.1)

For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq123_HTML.gif we have the operator pencil https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq124_HTML.gif to be Fredholm, and its spectrum consists of a countable number of isolated eigenvalues. Similarly to Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq125_HTML.gif in [7], we have the following lemma.

Lemma 4.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq126_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq127_HTML.gif are real numbers. Additionally, the authors suppose that no eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq128_HTML.gif line in strips https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq129_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq130_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq131_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq132_HTML.gif are eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq133_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq134_HTML.gif . Then the generalized solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq135_HTML.gif of the Dirichlet problem for elliptic equation (3.13), such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq136_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq137_HTML.gif , belongs to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq138_HTML.gif and satisfies the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ27_HTML.gif
(4.2)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq139_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq140_HTML.gif .

Proof.

Setting
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ28_HTML.gif
(4.3)
by the Friederichs inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ29_HTML.gif
(4.4)
therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ30_HTML.gif
(4.5)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq141_HTML.gif . Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ31_HTML.gif
(4.6)

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq142_HTML.gif be the function which arises from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq143_HTML.gif via the coordinate change https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq144_HTML.gif . We set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq145_HTML.gif ; then from the properties of the mapping (2.2) and from inequality (4.6), it follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq146_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq147_HTML.gif is the solution of an elliptic equation in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq148_HTML.gif with coefficients which stabilize for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq149_HTML.gif , that is,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ32_HTML.gif
(4.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq150_HTML.gif , we obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq151_HTML.gif (cf. [7, Lemma https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq152_HTML.gif ]). This implies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq153_HTML.gif . Using the fact that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ33_HTML.gif
(4.8)

as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq154_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq155_HTML.gif , we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq156_HTML.gif From Corollary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq157_HTML.gif in [7] it follows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq158_HTML.gif . Furthermore, (4.2) holds.

Lemma 4.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq159_HTML.gif , and the strip https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq160_HTML.gif does not contain eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq161_HTML.gif . Then the generalized solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq162_HTML.gif of problem (3.6)–(3.8), such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq163_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq164_HTML.gif , belongs to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq165_HTML.gif and satisfies the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ34_HTML.gif
(4.9)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq166_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq167_HTML.gif .

Proof.

Using the smoothness of the generalized solution of problem (3.6)–(3.8) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq168_HTML.gif in Theorem 3.3 and Proposition 3.4, we can see that for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq169_HTML.gif is the generalized solution of Dirichlet problem for (3.13) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq170_HTML.gif . From Lemma 4.1, it implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq171_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq172_HTML.gif and satisfies the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ35_HTML.gif
(4.10)

By integrating the inequality above with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq173_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq174_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq175_HTML.gif , and using the estimates for derivatives of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq176_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq177_HTML.gif again, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq178_HTML.gif , which satisfies inequality (4.9).

Theorem 4.3.

Let the assumptions of Lemma 4.2 be satisfied, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq179_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq180_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq181_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq182_HTML.gif . Then the generalized solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq183_HTML.gif of problem (3.6)–(3.8), such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq184_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq185_HTML.gif , belongs to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq186_HTML.gif and satisfies the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ36_HTML.gif
(4.11)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq187_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq188_HTML.gif .

Proof.

Let us first prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq189_HTML.gif belong to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq190_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq191_HTML.gif and satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ37_HTML.gif
(4.12)
The proof is an induction on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq192_HTML.gif . According to Lemma 4.2, it is valid for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq193_HTML.gif . Now let this assertion be true for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq194_HTML.gif ; we will prove that this also holds for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq195_HTML.gif . Due to Lemma 4.2, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq196_HTML.gif satisfies (3.6). By differentiating both sides of (3.6) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq197_HTML.gif times, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ38_HTML.gif
(4.13)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ39_HTML.gif
(4.14)
By the supposition of the theorem and the inductive assumption, the right-hand side of (4.13) belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq198_HTML.gif . By the arguments analogous to the proof of Lemma 4.2, we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq199_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ40_HTML.gif
(4.15)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq200_HTML.gif is a constant independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq201_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq202_HTML.gif .

By using (4.15) and estimates for derivatives of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq203_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq204_HTML.gif in Theorem 3.3, we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ41_HTML.gif
(4.16)

Remark 4.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq205_HTML.gif be a sufficiently small positive number. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq206_HTML.gif and the strip https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq207_HTML.gif contains no eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq208_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq209_HTML.gif ; then the generalized solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq210_HTML.gif of problem (3.6)–(3.8), such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq211_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq212_HTML.gif , belongs to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq213_HTML.gif . In fact, setting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq214_HTML.gif , we obtain the first initial boundary value problem which differs little from (3.6)–(3.8). Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq215_HTML.gif , and then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq216_HTML.gif . Using the remark above and Lemma 4.1, we obtain the following theorem.

Theorem 4.5.

Let the assumptions of Lemma 4.1 be satisfied. Furthermore, the authors assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq217_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq218_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq219_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq220_HTML.gif . Then the generalized solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq221_HTML.gif of problem (3.6)–(3.8), such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq222_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq223_HTML.gif , belongs to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq224_HTML.gif and satisfies the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ42_HTML.gif
(4.17)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq225_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq226_HTML.gif .

This theorem is proved by arguments analogous to those proofs of Lemma 4.2 and Theorem 4.3. Next, we will prove the well regularity of the generalized solution of problem (3.6)–(3.8).

Theorem 4.6.

Let the assumptions of Lemma 4.1 be satisfied. Furthermore, the authors assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq227_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq228_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq229_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq230_HTML.gif . Then the generalized solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq231_HTML.gif of problem (3.6)–(3.8), such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq232_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq233_HTML.gif , belongs to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq234_HTML.gif and satisfies the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ43_HTML.gif
(4.18)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq235_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq236_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq237_HTML.gif .

Proof.

The theorem is proved by induction on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq238_HTML.gif . Thanks to Theorem 4.5, this theorem is obviously valid for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq239_HTML.gif . Assume that the theorem is true for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq240_HTML.gif , we will prove that it also holds for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq241_HTML.gif . It is only needed to show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ44_HTML.gif
(4.19)
Differentiating both sides of (3.6) again with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq242_HTML.gif times, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ45_HTML.gif
(4.20)
By the supposition of the theorem and the inductive assumption, the right-hand side of (4.20) belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq243_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq244_HTML.gif . Using Lemma 4.1, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq245_HTML.gif . It implies that (4.19) holds for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq246_HTML.gif . Suppose that (4.19) is true for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq247_HTML.gif and set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq248_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ46_HTML.gif
(4.21)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq249_HTML.gif . By the inductive assumption with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq250_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq251_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq252_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq253_HTML.gif . Thus, the right-hand side of (4.21) belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq254_HTML.gif . Applying Lemma 4.1 again for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq255_HTML.gif , we get that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq256_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq257_HTML.gif . It means that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq258_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq259_HTML.gif . Furthermore, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ47_HTML.gif
(4.22)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ48_HTML.gif
(4.23)

It implies that (4.19) holds for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq260_HTML.gif . The proof is complete.

Now we will prove the global regularity of the solution.

Theorem 4.7.

Let the hypotheses of Lemma 4.1 be satisfied. Furthermore, suppose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq261_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq262_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq263_HTML.gif ,for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq264_HTML.gif . Then the generalized solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq265_HTML.gif of problem (3.6)–(3.8) belongs to the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq266_HTML.gif and satisfies the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ49_HTML.gif
(4.24)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq267_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq268_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq269_HTML.gif .

Proof.

We denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq270_HTML.gif the unit ball, and suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq271_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq272_HTML.gif in the neighborhood of the origin https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq273_HTML.gif . It is easy to get that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ50_HTML.gif
(4.25)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq274_HTML.gif is a differential operator, whose coefficients have compact support in a neighborhood of the origin. By arguments analogous to the proof of Theorem 4.6, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ51_HTML.gif
(4.26)
Set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq275_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq276_HTML.gif in a neighborhood of the origin and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq277_HTML.gif , and using the smoothness of the solution of this problem in domain with smooth boundary, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ52_HTML.gif
(4.27)

The proof is complete.

5. Examples

In this section, we apply the results of the previous section to the Cauchy-Dirichlet problem for the wave equation. The assumptions can be described as follows: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq278_HTML.gif is a bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq279_HTML.gif is smooth,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ53_HTML.gif
(5.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq280_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq281_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq282_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq283_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq284_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq285_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq286_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq287_HTML.gif .

We consider the Cauchy-Dirichlet problem for the wave equation in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq288_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ54_HTML.gif
(5.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq289_HTML.gif . It follows the results of Section 4 that if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq290_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq291_HTML.gif is the least positive root of the Bessel function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq292_HTML.gif , then problem (5.2) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq293_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq294_HTML.gif and we have the estimate

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ55_HTML.gif
(5.3)

Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq295_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq296_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq297_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq298_HTML.gif and satisfies

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ56_HTML.gif
(5.4)

For the two-dimensional case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq299_HTML.gif , and letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq300_HTML.gif , we consider problem (5.2) in the cylinder https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq301_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq302_HTML.gif is a bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq303_HTML.gif is smooth, and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ57_HTML.gif
(5.5)

Thus, the change of variables

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ58_HTML.gif
(5.6)

transforms

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ59_HTML.gif
(5.7)

With notations https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq304_HTML.gif , we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ60_HTML.gif
(5.8)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ61_HTML.gif
(5.9)
Hence, the differential operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq305_HTML.gif , which arises from the differential operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq306_HTML.gif via the coordinate change https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq307_HTML.gif , turns out to be
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ62_HTML.gif
(5.10)

Clearly, coefficients of differential operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq308_HTML.gif stabilize for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq309_HTML.gif , and the limit differential operator of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq310_HTML.gif (denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq311_HTML.gif for convenience) is

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ63_HTML.gif
(5.11)

We denote also by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq312_HTML.gif the operator corresponding to the parameter-depending boundary value problem

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ64_HTML.gif
(5.12)

Eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq313_HTML.gif are roots of the Bessel function

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ65_HTML.gif
(5.13)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq314_HTML.gif has only real roots (see [10, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq315_HTML.gif , page 94]). Therefore, they are

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ66_HTML.gif
(5.14)

It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq316_HTML.gif is the least positive root of the Bessel function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq317_HTML.gif . From arguments above in combination with Theorems 4.6 and 4.7, we obtain the following results:

Theorem 5.1.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq318_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq319_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq320_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq321_HTML.gif is a real number and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq322_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq323_HTML.gif . Then problem (5.2) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq324_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq325_HTML.gif and we have the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ67_HTML.gif
(5.15)
Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq326_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq327_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq328_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq329_HTML.gif and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ68_HTML.gif
(5.16)

In case that boundary of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq330_HTML.gif has some cuspidal points, then by arguments analogous to Section 4, we consequently obtain the similar results.

Declarations

Acknowledgment

This work was supported by National Foundation for Science and Technology Development (NAFOSTED), Vietnam.

Authors’ Affiliations

(1)
Department of Mathematics, Hanoi National University of Education
(2)
Department of Mathematics, Taybac University

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