An overdetermined problem associated to the Finsler Laplacian

Giulio Ciraolo; Antonio Greco

doi:10.3934/cpaa.2021004

Article Contents

2021, Volume 20, Issue 3: 1025-1038. Doi: 10.3934/cpaa.2021004 open access

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An overdetermined problem associated to the Finsler Laplacian

Giulio Ciraolo^1, , and
Antonio Greco^2,

1.
Department of Mathematics "Federigo Enriques", Università degli Studi di Milano, Italy
2.
Department of Mathematics and Computer Science, Università degli Studi di Cagliari, Italy

^* Corresponding author
^* Corresponding author

Received: October 2020

Revised: November 2020

Early access: January 4, 2021

Published online: January 4, 2021

G. Ciraolo has been partially supported by the PRIN 2017 project "Qualitative and quantitative aspects of nonlinear PDEs" and by GNAMPA of INdAM. A. Greco has been partially supported by the research project Integro-differential Equations and Non-Local Problems, funded by Fondazione di Sardegna (2017)

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Abstract

We prove a rigidity result for the anisotropic Laplacian. More precisely, the domain of the problem is bounded by an unknown surface supporting a Dirichlet condition together with a Neumann-type condition which is not translation-invariant. Using a comparison argument, we show that the domain is in fact a Wulff shape. We also consider the more general case when the unknown surface is required to have its boundary on a given conical surface: in such a case, the domain of the problem is bounded by the unknown surface and by a portion of the given conical surface, which supports a homogeneous Neumann condition. We prove that the unknown surface lies on the boundary of a Wulff shape.

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Mathematics Subject Classification: Primary: 35N25, 35B06; Secondary: 35R35.

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Figure 1. Maximizing the scalar product $ y \cdot {D \! H}_0(x) $ under the constraint $ H_0(y) = R $

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Figure 2. The ball $ B_1(0, H_0) $ (left) is smooth, its dual (right) is not

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Figure 3. Finding the Euclidean norm of $ {D \! H}_0(P_\vartheta) $

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