Energy stability for a class of semilinear elliptic problems

In this paper, we consider semilinear elliptic problems in a bounded domain \(\Omega\) contained in a given unbounded Lipschitz domain \(\mathcal C \subset \mathbb R^N\). Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain \(\Omega\) in...

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Bibliographic Details
Published in:arXiv.org 2023-07
Main Authors: Danilo Gregorin Afonso, Iacopetti, Alessandro, Pacella, Filomena
Format: Article
Language:English
Subjects:
Stability
Online Access:Get full text
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Summary:In this paper, we consider semilinear elliptic problems in a bounded domain \(\Omega\) contained in a given unbounded Lipschitz domain \(\mathcal C \subset \mathbb R^N\). Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain \(\Omega\) inside \(\mathcal C\). Once a rigorous variational approach to this question is set, we focus on the cases when \(\mathcal C\) is a cone or a cylinder and we consider spherical sectors and radial solutions or bounded cylinders and special one-dimensional solutions, respectively. In these cases, we show both stability and instability results, which have connections with related overdetermined problems.
ISSN:2331-8422