Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities

Abstract

For 1 < p < ∞, we prove radial symmetry for bounded nonnegative solutions ofwhere Ω is a Wulff ball, Σ is a convex cone with vertex at the center of Ω, w is a given weight and f is a possibly discontinuous nonnegative nonlinearity.Given the anisotropic setting that we deal with, the term "radial" is understood in the Finsler framework, that is, the function u is radial if there exists a point x such that u is constant on the Wulff shapes centered at x.When Σ = R N , J. Serra obtained the symmetry result in the isotropic unweighted setting (i.e., when H(ξ) ≡ |ξ| and w ≡ 1). In this case we provide the extension of his result to the anisotropic setting. This provides a generalization to the anisotropic setting of a celebrated result due to Gidas-Ni-Nirenberg and such a generalization is new even for p = 2 whenever N > 2.When Σ R N the results presented are new even in the isotropic and unweighted setting (i.e., when H is the Euclidean norm and w ≡ 1) whenever 2 = p = N . Even for the previously known case of unweighted isotropic setting with p = 2 and Σ R N , the present paper provides an approach to the problem by exploiting integral (in)equalities which is new for N > 2: this complements the corresponding symmetry result obtained via the moving planes method by Berestycki-Pacella.The results obtained in the isotropic and weighted setting (i.e., with w ≡ 1) are new for any p.

An exterior overdetermined problem for Finsler $N$-laplacian in convex cones

“…The physical motivation for studying anisotropic problems comes from wellestablished models of surface energy (see for instance [47]). Moreover, there are many mathematical interesting aspects arising when one considers symmetry problems in an anisotropic setting ( [3,4,6,11,12,15,16,22]).…”

“…It appears in condensed-matter and high-energy physics [31,40], and it is naturally studied in quasiconformal geometry (see for instance [29]). The study of symmetry problems in convex cones has recently attracted the interest of many authors, see for instance [6,12,13,22,25,36,37,42]. As far as the authors know, overdetermined capacity problems in convex cones have not been considered so far, even in the Euclidean case.…”

“…Such an idea is first observed by Figalli and Indrei [25] in order to establish a quantitative version of the isoperimetric inequality in convex cones due to Lions-Pacella [36], corresponding to (1.8) where H is the Euclidean norm. Recently, the ideas in [25] were further adapted in Dipierro-Poggesi-Valdinoci [22] to prove the uniqueness of the minimizers of (1.8) for a general norm H, illustrating that the equality in (1.8) holds if and only if Σ ∩ E = Σ ∩ B H0 R (x 0 ). However, the same argument still works in our case where H is a gauge as required in [6].…”

“…Then the first author and Roncoroni [13] generalized that to more general elliptic operators which are possibly degenerate as well as to space forms. More generally, during the last decade, much interest has been devoted to other parallel problems in convex cones and anisotropic setting (see for instance [6,12,15,22,42]). However, as far as we know, exterior overdetermined problems in unbounded domains contained in cones have not been studied yet even in the isotropic setting.…”

On a Class of Weighted p-Laplace Equation with Singular Nonlinearity

“…Anisotropic p-Laplace equation has been a topic of considerable attention in the recent years. In the nonsingular unweighted case, we refer to Alvino-Ferone-Trombetti-Lions [2], Belloni-Ferone-Kawohl [8], Belloni-Kawohl-Juutinen [10], Bianchini-Giulio [11], Xia [14], Cianchi-Salani [16], Ferone-Kawohl [25], Kawohl-Novaga [35] and with weights, see Dipierro-Poggesi-Valdinoci [19]. When g is singular, for w = 1, anisotropic p-Laplace equations is investigated by Biset-Mebrate-Mohammed [12], Farkas-Winkert [24] and Farkas-Fiscella-Winkert [23], Bal-Garain-Mukherjee [7].…”