Overdetermined boundary value problems with strongly nonlinear elliptic PDE

Wulff shape characterizations in overdetermined anisotropic elliptic problems

“…For p = 2, Theorem 1.1 has been proved in [12] under the weaker assumption that the boundary of Ω is of class C 1 (see also [54] for H ∈ C 4 (R N \ {0})). The case p = 2 has been investigated in [37] for H ∈ C 4 (R N \ {0}) following the ideas in [20,22,54]; however, in [37] it is not clear how the approximation argument used for the P -function in [37] can exclude that the P -function attains the maximum at critical points of u. Now we describe the results regarding exterior domains.…”

On a free boundary problem for a class of anisotropic equations

“…More precisely, the main ingredients of our proof are a maximum principle for an appropriate functional combination of and , that is, a P ‐function in the sense of L. E. Payne (see the book of R. Sperb ), a Rellich‐type integral identity, some properties of the given P ‐function, and geometric arguments involving the anisotropic curvature of the free boundary. Let us finally remark that the techniques from the aforementioned papers have been previously employed to deal with Serrin‐type symmetry results for anisotropic equations (see , respectively, ).…”