Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs

Abstract

Abstract. This paper is concerned with a class of boundary value problems for fully nonlinear elliptic PDEs involving the p-Hessian operator. We first derive a maximum principle for a suitable function involving the solution u (x) and its gradient. This maximum principle is then applied to obtain some sharp estimates for the solution and the magnitude of its gradient. We also investigate some symmetry properties of Ω or u (x) under specific boundary condition or geometry of Ω. Mathematics Subject Classification (2000). Primary 35B50, 35R35; Secondary 35B45.

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

“…Furthermore, G.A. Philippin and A. Safoui [10] (see, also, C. Enache [2] or L. Barbu and C. Enache [1]) have also investigated the general class of Monge-Ampère equations (1.1) and derived a similar maximum principle for a different auxiliary function, namely for:…”

“…on ∂Ω; see G.A. Philippin and A. Safoui [10] or C. Enache [2]). The corresponding P -function P (r, 1) reads in this case:…”

Overdetermined problems for fully nonlinear elliptic equations

“…A very large number of extensions of Serrin's theorem can be found in the literature, and recent years have seen an explosion of works on overdetermined elliptic problems. It is virtually impossible to give a full bibliography, we refer to [1,4,5,6,7,9,8,11,12,13,14,15,16,17,18,19,22,24,26,28,27,31,32] and the references in these papers, for related symmetry results for various degenerate operators, in various geometries, and a variety of methods of proof. Below we discuss in more detail how our results compare to the previous works which deal with fully nonlinear equations.…”

“…To our knowledge, prior to our paper results like Theorem 1.1 for fully nonlinear operators have appeared only for the particular cases when F (M) = S k (M) is a symmetric polynomial of the eigenvalues of M, see [27,5,11], and for equations involving Pucci operators or operators in the form |Du| α M + λ,Λ (D 2 u), with ellipticity constants sufficiently close to each other, see [4]. In Section 4 we will give an extension of the main theorem in [4] to operators satisfying (H2), with a short proof which will also play an important role in the proof of Theorem 1.1.…”