Isovolumetric and Isoperimetric Problems for a Class of Capillarity Functionals

Abstract

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in R 3 as the sum of the area integral and an anisotropic term of suitable form. In the class of parametric surfaces with the topological type of S 2 and with fixed volume, extremals of capillarity functionals are surfaces whose mean curvature is prescribed up to a constant. For a certain class of anisotropies vanishing at infinity, we prove existence and nonexistence of volumeconstrained, S 2 -type, minimal surfaces for the corresponding capillarity functionals. Moreover, in some cases, we show existence of extremals for the full isoperimetric inequality.

Existence of isovolumetric $\mathbb S^2$-type stationary surfaces for capillarity functionals

“…In particular t + = ∞ if K ≤ 0 everywhere (but also if K ≤ 0 on the tail of an open cone). Other conditions on K, different from (1.6) and regarding the radial oscillation of K are also displayed in [5]. Moreover in [5] it is proved that…”

“…Some of them are well known and classical. Others, more related to our problem, are discussed in [5]. We refer to that paper for the proofs or for additional, useful bibliography.…”

“…The present paper is a continuation and a completion of [5]. Here we focus on the issue of existence of critical points in the case of nonexistence of minima.…”

“…In general, even if the non homogeneous term vanishes at infinity, its presence in the capillarity functional has important consequences on the issue of the existence of extremals for the corresponding isoperimetric inequality. In [5] one can find some results concerning both existence and non-existence of critical points corresponding to minima for the isoperimetric problems (1 + Q K (u) · ν) dΣ (1.4) and V(u) is the algebraic volume functional, defined as the unique continuous extension to H 1 (S 2 , R 3 ) of the integral functional…”

“…A great part of the tools we use in the present paper is contained in [5] and for the sake of convenience we recall them in Section 2. Sections 3, 4 and 5 are devoted, respectively, to the construction of the minimax scheme, to the existence of constrained Palais-Smale sequences and to the proof of Theorem 1.3.…”