Symmetry for a general class of overdetermined elliptic problems

Abstract

Abstract. Let Ω be a bounded domain in R N , and let u ∈ C 1 (Ω) be a weak solution of the following overdetermined BVP:and λ is positive and nondecreasing. We show that Ω is a ball and u satisfies some "local" kind of symmetry. The proof is based on the method of continuous Steiner symmetrization.

A symmetry result for an elliptic problem arising from the 2-D thin film equation

“…Let Ω be a bounded domain in R 2 and f ∈ C([0, ∞)). In this note we are concerned with the symmetry of positive functions which satisfy the equation (1) Δu + f (u) = c , for some constant c in Ω.…”

“…In this celebrated work C 2 -regularity of the domain and the solution being in C 2 (Ω) are assumed. Recently, in [1], a radially symmetry result was obtained without the regularity of the boundary based on continuous Steiner symmetrization, where it is required that the modulus of the gradient of the solution is close to being radially symmetric near the boundary in a certain way. Here we work on a bounded domain in R 2 without any regularity assumption, and, to compensate, the energy stability condition is used in an essential way.…”

“…Taking V = f (u) in Lemma 3, as we have just shown that there are at least three connected components in the non-zero set of w, w cannot be a minimizer of (1). In view of ( 2) and ( 3), one must have μ 0 < 1.…”

“…Nevertheless, non-negative, weak global solutions have been constructed in many cases for the one dimensional thin film equation ([3], [2], [5], [6]). Although these solutions still constitute flows in S, due to the possibility of touch-down, the steady state is defined as a nonnegative function in H 1 (Ω 0 ) which satisfies (1) in each connected component of {u > 0} where the constant c may vary on the component. Steady states not only describe the ultimate pattern of (4) but also are important in the study of the dynamical behavior of the flow.…”

The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem

“…Garofalo and Lewis [10] proved this result via Weinberger's approach; Brock and Henrot [5] proposed a different proof via Steiner symmetrization for p ≥ 2; Damascelli and Pacella [7] succeeded in adapting the moving plane method to the case 1 < p < 2. Later, many other refinements and generalizations to more general operators have been proposed, we refer for instance to [9,8,4] and the references therein.…”