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On the solvability of parameter-dependent elliptic functional BVPs on annular-like domains

Alessandro Calamai, Gennaro Infante

TL;DR

This paper addresses the solvability of parameter-dependent elliptic BVPs with deviated arguments on annular-like domains under functional boundary conditions. It recasts the PDE into a fixed-point problem via the Nemytskii operator $\mathcal{F}$ and the elliptic solution operator $\mathcal{G}$, and applies a variant of the Birkhoff–Kellogg theorem in affine cones to obtain nontrivial solutions in a translated cone $K_\phi$. A constructive 2D example demonstrates the method when the nonlinearity is not radially symmetric, using an ODE reduction to verify hypotheses and obtaining uncountably many pairs $(u_\rho,\lambda_\rho)$. These results extend existence theory for PDEs with spatial delays and nonlocal boundary conditions and provide a principled approach for heat-flow models with global feedback.

Abstract

We investigate the existence of nontrivial solutions of parameter-dependent elliptic equations with deviated argument in annular-like domains in $\mathbb{R}^{n}$, with $n\geq 2$, subject to functional boundary conditions. In particular we consider a boundary value problem that may be used to model heat-flow problems. We obtain an existence result by means of topological methods; in particular, we make use of a recent variant in affine cones of the celebrated Birkhoff--Kellogg theorem. Using an ODE argument, we illustrate in an example the applicability of our theoretical result.

On the solvability of parameter-dependent elliptic functional BVPs on annular-like domains

TL;DR

This paper addresses the solvability of parameter-dependent elliptic BVPs with deviated arguments on annular-like domains under functional boundary conditions. It recasts the PDE into a fixed-point problem via the Nemytskii operator and the elliptic solution operator , and applies a variant of the Birkhoff–Kellogg theorem in affine cones to obtain nontrivial solutions in a translated cone . A constructive 2D example demonstrates the method when the nonlinearity is not radially symmetric, using an ODE reduction to verify hypotheses and obtaining uncountably many pairs . These results extend existence theory for PDEs with spatial delays and nonlocal boundary conditions and provide a principled approach for heat-flow models with global feedback.

Abstract

We investigate the existence of nontrivial solutions of parameter-dependent elliptic equations with deviated argument in annular-like domains in , with , subject to functional boundary conditions. In particular we consider a boundary value problem that may be used to model heat-flow problems. We obtain an existence result by means of topological methods; in particular, we make use of a recent variant in affine cones of the celebrated Birkhoff--Kellogg theorem. Using an ODE argument, we illustrate in an example the applicability of our theoretical result.

Paper Structure

This paper contains 3 sections, 2 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some known facts and setting of the problem
  3. Main result

Key Result

Theorem 2.2

Let $(X,\| \, \|)$ be a real Banach space, $K\subset X$ be a cone and $D\subset X$ be an open bounded set with $y \in D_{K_y}$ and $\overline{D}_{K_y}\ne K_y$. Assume that $\mathcal{F}:\overline{D}_{K_y}\to K$ is a compact map and assume that Then there exist $x^*\in \partial D_{K_y}$ and $\lambda^*\in (0,+\infty)$ such that $x^*= y+\lambda^* \mathcal{F} (x^*)$.

Figures (1)

  • Figure 1:

Theorems & Definitions (6)

  • Definition 2.1
  • Theorem 2.2: acgi2, Corollary 2.4
  • Definition 2.3
  • Theorem 3.1
  • proof
  • Example 3.2