Some fully nonlinear elliptic boundary value problems with ellipsoidal free boundaries

problem (¢u = 1 in ›, and u = 0 on @›) has a constant outward normal derivative on @›, is the N-dimensional ball. Serrin’s proof is based on the moving plane method and Hopf’s maximum principles. Under weaker assumptions on the regularity of @›, H.F. Weinberger presented in a subsequent paper [20] a simpler proof of Serrin’s basic result. His method of proof combines a maximum principle for a suitable P-function and a Rellich type identity resulting from Green’s theorem. However, Weinberger’s proof relies on the linearity of the Laplacian, while Serrin’s method could be extended to more general nonlinear elliptic equations.

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