Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities

Michał Kowalczyk; Angela Pistoia; Piotr Rybka; Giusi Vaira

doi:10.5802/slsedp.122

Michał Kowalczyk ¹; Angela Pistoia ²; Piotr Rybka ³; Giusi Vaira ⁴

¹ Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS) Universidad de Chile Casilla 170 Correo 3 Santiago Chile
² Dipartimento SBAI Sapienza Universitá di Roma via Antonio Scarpa 16 00161 Roma Italy
³ Institute of Applied Mathematics and Mechanics The University of Warsaw Banacha 2 02-097 Warsaw Poland
⁴ Dipartimento di Matematica e Fisica Universitá della Campania “L. Vanvitelli” viale Lincoln 5 81100 Caserta Italy

Séminaire Laurent Schwartz — EDP et applications (2017-2018), Talk no. 10, 12 p.

Abstract

We consider equations of the form $Δ u + λ^{2} V (x) e^{u} = ρ$ in various two dimensional settings. We assume that $V > 0$ is a given function, $λ > 0$ is a small parameter and $ρ = 𝒪 (1)$ or $ρ \to + \infty$ as $λ \to 0$ . In a recent paper [27] we proved the existence of the maximal solutions for a particular choice $V \equiv 1$ , $ρ = 0$ when the problem is posed in doubly connected domains under Dirichlet boundary conditions. We related the maximal solutions with a novel free boundary problem. The purpose of this note is to derive the corresponding free boundary problems in other settings. Solvability of such problems is, viewed formally, the necessary condition for the existence of the maximal solution.

Published online: 2018-10-29

DOI: 10.5802/slsedp.122

Author's affiliations:

Michał Kowalczyk ¹; Angela Pistoia ²; Piotr Rybka ³; Giusi Vaira ⁴

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     title = {Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
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     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
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Michał Kowalczyk; Angela Pistoia; Piotr Rybka; Giusi Vaira. Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities. Séminaire Laurent Schwartz — EDP et applications (2017-2018), Talk no. 10, 12 p. doi : 10.5802/slsedp.122. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.122/

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