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An Overdetermined Neumann boundary value problem with a general driving force

Ignace Aristide Minlend, Jing Wu

TL;DR

The paper proves the existence of nontrivial subdomains \(\Omega_h\) in the manifold \(\mathcal{M}=\mathbb{R}^N\times\mathbb{R}/2\pi\mathbb{Z}\) that admit solutions to the overdetermined Neumann problem with a general driving force \(g\). A pull-back transformation to a fixed cylinder \(\Omega_* = B_1\times\mathbb{R}\) coupled with a tailored Banach-space framework reduces the problem to a single nonlinear equation \(G_\lambda(u)=0\) and enables the use of Crandall–Rabinowitz bifurcation. The linearization at the trivial branch is shown to be Fredholm of index zero with a one-dimensional kernel, leading to a bifurcation parameter \(\lambda_*= -\gamma_{1,*}\); a local bifurcating branch of solutions is then constructed. The main result provides explicit first-order corrections: for small parameter \(s\), the domain boundary is perturbed as \(h^*_s(x)=\frac{1}{\sqrt{\lambda_*(s)}}+s\beta_*\cos x+o(s)\) and the solution obeys \(w_s(\frac{t}{h^*_s(x)},x)=U_*(|t|)+s\{U_*(|t|)+\delta_*|t|U_*'( |t|)\}\cos x+o(s)\), with these constructions valid for a broad class of nonlinearities \(g\) under Assumption (A). The work extends prior linear-case results to nonlinear driving forces and illustrates that the resulting boundaries are not isoparametric or homogeneous. The techniques provide a robust mechanism for generating nontrivial, nonisoparametric solution domains in a Riemannian setting. All results are framed within a rigorous functional-analytic and bifurcation-theoretic context.

Abstract

In this paper, we prove the existence of a family of non trivial compact subdomains $Ø$ in the manifold $\mathcal{M}=\R^N\times \R/2π\Z, N\geq 2$ for which the overdetermined Neumann boundary value problem \begin{align}\label{Neumann1} \left \{ \begin{aligned} $-\D w&=μg(w) && \qquad \text{in $ Ω$,}$ \frac{\partial w}{\partialη} &=0 &&\qquad \text{on $\partial Ω,$} w&=c\ne 0 &&\qquad \text{on $\partial Ω$,} \end{aligned} \right. \end{align} admits solutions for some $μ> 0$ and a $C^{1, α}$ function $g:\R \rightarrow \R.$ The domains we construct have nonconstant principal curvature, and therefore are not isoparametric nor homogeneous. The argument we develop applies for both linear and non-linear functions $g$. By this, we generalise a recent result obtained by Fall, Weth and the first named author in \cite{Fall-MinlendI-Weth4}, where the overdetermined Neumann eigenvalue problem for the Laplacian was considered.

An Overdetermined Neumann boundary value problem with a general driving force

TL;DR

The paper proves the existence of nontrivial subdomains in the manifold that admit solutions to the overdetermined Neumann problem with a general driving force . A pull-back transformation to a fixed cylinder coupled with a tailored Banach-space framework reduces the problem to a single nonlinear equation \(G_\lambda(u)=0\) and enables the use of Crandall–Rabinowitz bifurcation. The linearization at the trivial branch is shown to be Fredholm of index zero with a one-dimensional kernel, leading to a bifurcation parameter ; a local bifurcating branch of solutions is then constructed. The main result provides explicit first-order corrections: for small parameter , the domain boundary is perturbed as \(h^*_s(x)=\frac{1}{\sqrt{\lambda_*(s)}}+s\beta_*\cos x+o(s)\) and the solution obeys \(w_s(\frac{t}{h^*_s(x)},x)=U_*(|t|)+s\{U_*(|t|)+\delta_*|t|U_*'( |t|)\}\cos x+o(s)\), with these constructions valid for a broad class of nonlinearities under Assumption (A). The work extends prior linear-case results to nonlinear driving forces and illustrates that the resulting boundaries are not isoparametric or homogeneous. The techniques provide a robust mechanism for generating nontrivial, nonisoparametric solution domains in a Riemannian setting. All results are framed within a rigorous functional-analytic and bifurcation-theoretic context.

Abstract

In this paper, we prove the existence of a family of non trivial compact subdomains in the manifold for which the overdetermined Neumann boundary value problem \begin{align}\label{Neumann1} \left \{ \begin{aligned} Ω \frac{\partial w}{\partialη} &=0 &&\qquad \text{on } w&=c\ne 0 &&\qquad \text{on ,} \end{aligned} \right. \end{align} admits solutions for some and a function The domains we construct have nonconstant principal curvature, and therefore are not isoparametric nor homogeneous. The argument we develop applies for both linear and non-linear functions . By this, we generalise a recent result obtained by Fall, Weth and the first named author in \cite{Fall-MinlendI-Weth4}, where the overdetermined Neumann eigenvalue problem for the Laplacian was considered.
Paper Structure (5 sections, 7 theorems, 103 equations)

This paper contains 5 sections, 7 theorems, 103 equations.

Table of Contents

  1. Introduction and main result
  2. The pull back problem
  3. Functional setting
  4. Study of the linearised operator $DG_\lambda (0)$
  5. Proof of Theorem \ref{['Theo1-ND']}

Key Result

Theorem 1.2

Let $N\geq2$ and $g: \mathbb{R}\rightarrow\mathbb{R}$ is a $C^{1}$ function such that Assumptions (A) holds. Then there exist some constants $\mu_*,\varepsilon _*, \lambda _* >0$, $c_*,\beta_*,\delta_* \in \mathbb{R} \setminus \{0\}$, depending only on $N$, and a smooth curve with $\lambda _*(s) |_{s=0}= \lambda_*$, and the property that the overdetermined boundary value problem admits a class

Theorems & Definitions (16)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 3.1
  • Lemma 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 6 more