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.

Energy stability for a class of semilinear elliptic problems / GREGORIN AFONSO, Danilo; Iacopetti, Alessandro; Pacella, Filomena. - In: JOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1559-002X. - 34:(2024). [10.1007/s12220-023-01525-1]

Energy stability for a class of semilinear elliptic problems

Danilo Gregorin Afonso;Filomena Pacella
2024

Abstract

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.
2024
Mathematics - Analysis of PDEs; Mathematics - Analysis of PDEs; 35J61, 35B35, 35B38, 49Q10
01 Pubblicazione su rivista::01a Articolo in rivista
Energy stability for a class of semilinear elliptic problems / GREGORIN AFONSO, Danilo; Iacopetti, Alessandro; Pacella, Filomena. - In: JOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1559-002X. - 34:(2024). [10.1007/s12220-023-01525-1]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1700176
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