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

  • NguyenManh Hung1 and

    Affiliated with

    • VuTrong Luong2Email author

      Affiliated with

      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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq1_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq2_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq5_HTML.gif satisfying the following conditions:

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

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

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

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

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

      We extend the functions http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq19_HTML.gif The space http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq21_HTML.gif with respect to the norm

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

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

      We also denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq25_HTML.gif the Sobolev space of functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq26_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq27_HTML.gif that have generalized derivatives http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq28_HTML.gif , http://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:

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

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

      Set http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq34_HTML.gif be Banach spaces, we denote by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq36_HTML.gif with norm

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

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

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

      For shortness, we set

      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq43_HTML.gif such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq44_HTML.gif instead of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq46_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ10_HTML.gif
      (3.1)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq47_HTML.gif are functions with complex values, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq48_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq49_HTML.gif denotes the transposed conjugate of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq50_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq51_HTML.gif are infinitely differentiable in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq52_HTML.gif . Moreover, we assume that the functions
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq53_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq54_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq55_HTML.gif (see [7, Section http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq56_HTML.gif ]). Then the coefficients of the operators http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq57_HTML.gif , which arise from operators http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq58_HTML.gif via the coordinate change http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq59_HTML.gif , stabilize for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq61_HTML.gif by their limits for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq63_HTML.gif and http://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 http://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:

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq66_HTML.gif and a.e. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq67_HTML.gif , where
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq68_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq69_HTML.gif -elliptic uniformly with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq70_HTML.gif , that is, the inequality

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

      is valid for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq71_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq72_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq73_HTML.gif is the positive constant independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq74_HTML.gif and http://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:

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq76_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq78_HTML.gif .

      Definition 3.1.

      A function http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq80_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq81_HTML.gif and the equality
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ18_HTML.gif
      (3.9)

      holds for all http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq84_HTML.gif and there exists a positive number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq85_HTML.gif such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq86_HTML.gif , and the following estimate holds
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ20_HTML.gif
      (3.11)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq87_HTML.gif is a constant independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq88_HTML.gif and http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq90_HTML.gif

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

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

      Then the generalized solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq94_HTML.gif up to order h in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq95_HTML.gif and satisfies the following estimate:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ21_HTML.gif
      (3.12)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq96_HTML.gif is a constant independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq97_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq100_HTML.gif . Then for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq101_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq102_HTML.gif is a generalized solution in http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ22_HTML.gif
      (3.13)

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

      Proof.

      Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq106_HTML.gif . Setting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq107_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq108_HTML.gif , and substituting the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq109_HTML.gif into (3.9), we conclude that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ23_HTML.gif
      (3.14)
      We will denote by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ24_HTML.gif
      (3.15)
      that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq110_HTML.gif Noting that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq112_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq113_HTML.gif , where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq115_HTML.gif are dense in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq116_HTML.gif , the following equality
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ25_HTML.gif
      (3.16)

      holds for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq117_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq118_HTML.gif . It follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq119_HTML.gif is a generalized solution in http://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. http://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 http://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

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

      For each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq123_HTML.gif we have the operator pencil http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq126_HTML.gif , where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq128_HTML.gif line in strips http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq129_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq130_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq131_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq132_HTML.gif are eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq133_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq134_HTML.gif . Then the generalized solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq136_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq137_HTML.gif , belongs to the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq138_HTML.gif and satisfies the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ27_HTML.gif
      (4.2)

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

      Proof.

      Setting
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ29_HTML.gif
      (4.4)
      therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ30_HTML.gif
      (4.5)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq141_HTML.gif . Hence,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ31_HTML.gif
      (4.6)

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq142_HTML.gif be the function which arises from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq143_HTML.gif via the coordinate change http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq144_HTML.gif . We set http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq146_HTML.gif . Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq148_HTML.gif with coefficients which stabilize for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq149_HTML.gif , that is,

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

      as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq154_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq155_HTML.gif , we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq156_HTML.gif From Corollary http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq157_HTML.gif in [7] it follows that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq159_HTML.gif , and the strip http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq160_HTML.gif does not contain eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq161_HTML.gif . Then the generalized solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq163_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq164_HTML.gif , belongs to the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq165_HTML.gif and satisfies the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ34_HTML.gif
      (4.9)

      where the constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq166_HTML.gif is independent of http://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 http://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. http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq171_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq172_HTML.gif and satisfies the inequality
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq173_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq174_HTML.gif to http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq176_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq177_HTML.gif again, we obtain http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq179_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq180_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq181_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq182_HTML.gif . Then the generalized solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq184_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq185_HTML.gif , belongs to the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq186_HTML.gif and satisfies the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ36_HTML.gif
      (4.11)

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

      Proof.

      Let us first prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq189_HTML.gif belong to the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq190_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq191_HTML.gif and satisfy
      http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq195_HTML.gif . Due to Lemma 4.2, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq197_HTML.gif times, we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ38_HTML.gif
      (4.13)
      where
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq199_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ40_HTML.gif
      (4.15)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq200_HTML.gif is a constant independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq201_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq203_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq204_HTML.gif in Theorem 3.3, we have

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

      Remark 4.4.

      Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq206_HTML.gif and the strip http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq207_HTML.gif contains no eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq208_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq209_HTML.gif ; then the generalized solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq211_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq212_HTML.gif , belongs to the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq213_HTML.gif . In fact, setting http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq215_HTML.gif , and then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq217_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq218_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq219_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq220_HTML.gif . Then the generalized solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq222_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq223_HTML.gif , belongs to the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq224_HTML.gif and satisfies the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ42_HTML.gif
      (4.17)

      where the constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq225_HTML.gif is independent of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq227_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq228_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq229_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq230_HTML.gif . Then the generalized solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq232_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq233_HTML.gif , belongs to the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq234_HTML.gif and satisfies the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ43_HTML.gif
      (4.18)

      where the constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq235_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq236_HTML.gif and http://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 http://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 http://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 http://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 http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq242_HTML.gif times, we obtain
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq243_HTML.gif for a.e. http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq247_HTML.gif and set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq248_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ46_HTML.gif
      (4.21)
      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq250_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq251_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq252_HTML.gif for a.e. http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq254_HTML.gif . Applying Lemma 4.1 again for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq255_HTML.gif , we get that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq256_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq257_HTML.gif . It means that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq258_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq259_HTML.gif . Furthermore, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ47_HTML.gif
      (4.22)
      Therefore,
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq261_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq262_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq263_HTML.gif ,for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq264_HTML.gif . Then the generalized solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq266_HTML.gif and satisfies the inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ49_HTML.gif
      (4.24)

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

      Proof.

      We denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq270_HTML.gif the unit ball, and suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq271_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq272_HTML.gif in the neighborhood of the origin http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq273_HTML.gif . It is easy to get that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ50_HTML.gif
      (4.25)
      where http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ51_HTML.gif
      (4.26)
      Set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq275_HTML.gif , then http://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 http://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
      http://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: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq278_HTML.gif is a bounded domain in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq279_HTML.gif is smooth,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ53_HTML.gif
      (5.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq280_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq281_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq282_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq283_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq284_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq285_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq286_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq288_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ54_HTML.gif
      (5.2)

      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq290_HTML.gif , where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq293_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq294_HTML.gif and we have the estimate

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

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq299_HTML.gif , and letting http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq301_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq302_HTML.gif is a bounded domain in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq303_HTML.gif is smooth, and

      http://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

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

      transforms

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

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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ60_HTML.gif
      (5.8)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ61_HTML.gif
      (5.9)
      Hence, the differential operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq305_HTML.gif , which arises from the differential operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq306_HTML.gif via the coordinate change http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq307_HTML.gif , turns out to be
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq308_HTML.gif stabilize for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq309_HTML.gif , and the limit differential operator of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq310_HTML.gif (denoted by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq311_HTML.gif for convenience) is

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

      We denote also by http://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

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

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

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq315_HTML.gif , page 94]). Therefore, they are

      http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq318_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq319_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq320_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq321_HTML.gif is a real number and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq322_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq324_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq325_HTML.gif and we have the estimate
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_Equ67_HTML.gif
      (5.15)
      Moreover, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq326_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq327_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq328_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F135730/MediaObjects/13661_2009_Article_827_IEq329_HTML.gif and satisfies
      http://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 http://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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