Scholarlink

Abstract <p>A Steklov-type boundary value problem for the Lamé operator in a half-cylinder containing a small cavity is studied. The case is considered when an elastic homogeneous isotropic medium filling the region with a small cavity is rigidly connected with the lateral boundary of the half-cylinder and the boundary of the small cavity, which corresponds to the homogeneous Dirichlet boundary condition, and a Steklov spectral condition is imposed on the base of the half-cylinder. The main result consists in proving a theorem on the convergence of the eigenelements of such a singularly perturbed boundary value problem to the eigenelements of the limit problem (in a half-cylinder without a cavity) as the small parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2307_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon &gt; 0\)</EquationSource> <!--ComMat2570099Davletov-m1--> </InlineEquation>, which characterizes the diameter of the cavity, tends to zero. To prove the theorem, a Hilbert space of infinitely differentiable vector functions with a finite Dirichlet integral over the half-cylinder was introduced. Unlike the situation with a bounded domain, in the boundary value problem under study, the condition of finiteness of the Dirichlet integral is essential, since it ensures as a whole the finiteness of the norm in the space introduced. The required finiteness of the Dirichlet integral made it possible to establish prior estimates that guarantee the uniqueness of solutions of the limit and perturbed boundary value problems and to establish the equivalence of the norms, necessary to prove the existence of a solution of the singularly perturbed boundary value problem under study.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Convergence of Eigenelements of a Steklov-Type Boundary Value Problem for the Lamé Operator in a Half-Cylinder with a Small Cavity

  • 期刊论文

  • 发表日期：

    2025年9月1日

查看全文
  • D. B. Davletov,
  • O. B. Davletov,
  • R. R. Davletova,
  • A. A. Ershov

摘要

Abstract

A Steklov-type boundary value problem for the Lamé operator in a half-cylinder containing a small cavity is studied. The case is considered when an elastic homogeneous isotropic medium filling the region with a small cavity is rigidly connected with the lateral boundary of the half-cylinder and the boundary of the small cavity, which corresponds to the homogeneous Dirichlet boundary condition, and a Steklov spectral condition is imposed on the base of the half-cylinder. The main result consists in proving a theorem on the convergence of the eigenelements of such a singularly perturbed boundary value problem to the eigenelements of the limit problem (in a half-cylinder without a cavity) as the small parameter \(\varepsilon > 0\) , which characterizes the diameter of the cavity, tends to zero. To prove the theorem, a Hilbert space of infinitely differentiable vector functions with a finite Dirichlet integral over the half-cylinder was introduced. Unlike the situation with a bounded domain, in the boundary value problem under study, the condition of finiteness of the Dirichlet integral is essential, since it ensures as a whole the finiteness of the norm in the space introduced. The required finiteness of the Dirichlet integral made it possible to establish prior estimates that guarantee the uniqueness of solutions of the limit and perturbed boundary value problems and to establish the equivalence of the norms, necessary to prove the existence of a solution of the singularly perturbed boundary value problem under study.