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Laplace problem with an exponential nonlinear boundary condition

Jamel Benameur, Chokri Elhechmi, Gmar Benhenda

Abstract

In this paper, we establish a new result for the Laplace problem with exponential Robin boundary conditions posed on the unit disk in $\R^2$. More precisely, we prove the existence and uniqueness of a solution under suitable smallness assumptions on the boundary data. Our approach relies on an iterative method combined with periodic Sobolev embedding results.

Laplace problem with an exponential nonlinear boundary condition

Abstract

In this paper, we establish a new result for the Laplace problem with exponential Robin boundary conditions posed on the unit disk in . More precisely, we prove the existence and uniqueness of a solution under suitable smallness assumptions on the boundary data. Our approach relies on an iterative method combined with periodic Sobolev embedding results.
Paper Structure (10 sections, 9 theorems, 102 equations)

This paper contains 10 sections, 9 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Study of approximation system
  4. Convergence result
  5. Existence of a solution of the problem $(S)$
  6. Uniqueness of solution of the problem $(S)$ in $\overline{B_V(0,M_0)}$
  7. Appendices
  8. Appendix A
  9. Appendix B
  10. Appendix C: Sobolev constants in Fourier series

Key Result

Lemma 2.1

(Jer-Dron) Let $\Omega\subset\mathbb R^2$ be a Lipschitzian domain. Then, for $p\in[2,+\infty)$ we have Precisely, there is a constant $C_{\Omega,p} > 0$, depending only on $p$ and domain $\Omega$, such that and

Theorems & Definitions (17)

  • Remark 1.1
  • Lemma 2.1
  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • Lemma 3.1
  • Proposition 3.1
  • Lemma 4.1: Contraction estimate
  • Remark 4.1
  • ...and 7 more