The overdetermined problem for an anisotropic infinity Laplacian

In this paper, we study the Serrin-type overdetermined problem for the following anisotropic infinity Laplacian {−ΔF;∞Nu=1inΩ,u=0on∂Ω,F(Du)=con∂Ω, where ΔF;∞Nu stands for the Finsler infinity Laplacian (i.e., an anisotropic infinity Laplacian), F is a Finsler norm on Rn and c>0 is a constant. We introduce the anisotropic gradient flow, combine the convex envelope method and Weinberger’s technique to prove that the existence of a C1-viscosity solution to this problem is equivalent to Ω being an anisotropic stadium-like domain when Ω is bounded and convex. The result is an anisotropic version of Crasta and Fragalà (Arch. Ration. Mech. Anal., pp 1577–1607, 2015).