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Exact solution structures on some nonlocal overdetermined problems

Kazuki Sato, Futoshi Takahashi

TL;DR

The paper studies Serrin-type overdetermined problems incorporating a Kirchhoff-type nonlocal term, for both interior and exterior domains. It shows that the number of solutions is governed by the roots of a transcendental equation defined by the nonlocal term, and provides explicit solution forms via a base radially symmetric solution $U_{R,x_0}$. A scaling argument reduces interior problems to a single equation in a scalar parameter, while an exterior-domain Kelvin-transform approach yields a parallel reduction, together enabling precise solution counting and explicit representations. The results extend prior work on nonlocal and Hessian-type overdetermined problems, establishing a unified framework for exact solution structures and their enumeration through simple transcendental equations.

Abstract

In this paper, we study the solution structures of Serrin-type overdetermined problems with Kirchhoff-type nonlocal terms. We prove that the exact number of solutions is the same as those of some transcendental equations defined by the nonlocal terms. We also obtain the explicit form of solutions by using the unique solutions of the overdetermined problems without the nonlocal terms.

Exact solution structures on some nonlocal overdetermined problems

TL;DR

The paper studies Serrin-type overdetermined problems incorporating a Kirchhoff-type nonlocal term, for both interior and exterior domains. It shows that the number of solutions is governed by the roots of a transcendental equation defined by the nonlocal term, and provides explicit solution forms via a base radially symmetric solution . A scaling argument reduces interior problems to a single equation in a scalar parameter, while an exterior-domain Kelvin-transform approach yields a parallel reduction, together enabling precise solution counting and explicit representations. The results extend prior work on nonlocal and Hessian-type overdetermined problems, establishing a unified framework for exact solution structures and their enumeration through simple transcendental equations.

Abstract

In this paper, we study the solution structures of Serrin-type overdetermined problems with Kirchhoff-type nonlocal terms. We prove that the exact number of solutions is the same as those of some transcendental equations defined by the nonlocal terms. We also obtain the explicit form of solutions by using the unique solutions of the overdetermined problems without the nonlocal terms.

Paper Structure

This paper contains 3 sections, 3 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Theorem:H']}.
  3. Exterior overdetermined problem.

Key Result

Theorem 1

Let $M : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$, $0 < p, q \le \infty$, $\lambda > 0$, and $k \in \{ 1, 2, \dots, N \}$. Then if OD_HM admits a solution $u \in C^2(\overline{\Omega})$ for an unknown constant $c > 0$, then $\Omega$ must be a ball and $u$ must be radially symmetric with re where $U_{R, x_0}$ defined in U_R is the unique solution of OD_H for $\Omega = B_R(x_0)$. Then for

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • proof
  • Proposition 1
  • proof