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An existence theorem for elliptic equations with nonlocal boundary conditions

Chiun-Chang Lee

Abstract

The focus of this study is on exploring some qualitative properties of solutions to a class of semilinear elliptic problems in bounded domains, where the boundary conditions depend non-locally on the unknown solution at specified interior points and its integral. The primary approach integrates a fixed-point argument with refined asymptotic estimates to establish the existence and structure of solutions. Furthermore, the maximum principles are established under practical nonlocal-type boundary conditions.

An existence theorem for elliptic equations with nonlocal boundary conditions

Abstract

The focus of this study is on exploring some qualitative properties of solutions to a class of semilinear elliptic problems in bounded domains, where the boundary conditions depend non-locally on the unknown solution at specified interior points and its integral. The primary approach integrates a fixed-point argument with refined asymptotic estimates to establish the existence and structure of solutions. Furthermore, the maximum principles are established under practical nonlocal-type boundary conditions.
Paper Structure (7 sections, 5 theorems, 70 equations)

This paper contains 7 sections, 5 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Heuristics and Preliminaries
  3. Examples for the non-existence, uniqueness and multiplicity
  4. Overview and strategy
  5. The main result: existence, uniqueness and maximum principles
  6. Estimates of $\boldsymbol{v_{\lambda,\mu}}$ and $\boldsymbol{\mathsf{T}}_{\lambda}$
  7. Proof of Theorem \ref{['m-thm']} and Corollary \ref{['cor1']}

Key Result

Theorem 3.1

For given $\xi_j\in\Omega$, $j=1,...,m$, we define $\mathfrak{m}(\boldsymbol{\xi})$ in mxi, and assume that $f(x,s)$ satisfies (f1)--(f2). Then, for $D\in\mathrm{C}^{1,\tau}(\overline{\Omega};(0,\infty))$, $g\in\mathrm{C}^{0,\tau}(\partial\Omega;\mathbb{R})$ and $w\in\text{L}^1(\Omega)$, we consider

Theorems & Definitions (15)

  • Example 2.1
  • Example 2.2
  • Theorem 3.1
  • Corollary 3.2: Maximum principle
  • Proposition 3.3
  • Remark 3.4
  • proof : Proof of Proposition \ref{['prop-v']}
  • Remark 3.5
  • Proposition 3.6
  • Remark 3.7
  • ...and 5 more