Isoperimetric cones and minimal solutions of partial overdetermined problems

Abstract

In this paper we consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that, in cones having an isoperimetric property, the only domains which admit a solution and which minimize a torsional energy functional are spherical sectors centered at the vertex of the cone. We also show that cones close in the C 1,1 -metric to an isoperimetric one are also isoperimetric, generalizing so a result of [1]. This is achieved by using a characterization of constant mean curvature polar graphs in cones which improves a result of [18].

Existence of nonradial domains for overdetermined and isoperimetric problems in nonconvex cones

“…It was proved in [18], and later in [25,13,10], that if Σ D is a convex cone then the only minimizer of P(E; Σ D ) with a fixed volume are the spherical sectors Ω D,R . This holds also in "almost" convex cones as shown in [2] (see also [23]). If the cone is not convex, a counterexample is given in [18].…”

“…is the diffeomorphism given by (3.4). From the proof of [23,Proposition 4.3] we know that Φ is differentiable and thus we infer that u Ωϕ+tv is differentiable with respect to t. Hence, by the Leibniz integral rule for differentiation of integral functions we get that…”

“…in the space H 1 0 (Ω ∪ Γ 1,Ω ) and we remark that u Ω > 0 a.e. in Ω, by the maximum principle (we refer to [22,23] for more details).…”

“…Using domain derivative techniques, as for other similar problems in shape-optimization theory it can be proved that the critical points of the functional E(Ω; Σ D ) with respect to volume-preserving deformations which leave the cone invariant, correspond to domains Ω for which ∂uΩ ∂ν is constant on Γ Ω , i.e. u Ω satisfies the overdetermined problem (1.2) (see in [23,Proposition 4.3] if Γ Ω is smooth and u Ω has some Sobolev regularity, or Proposition 7.4 in the present paper in the nonsmooth case).…”

“…In particular any local minimizer of E(Ω; Σ D ) with a volume constraint is a spherical sector. Actually, using symmetrization methods in cones [19,24] it can be proved (see [23]) that this holds in a more general class of cones which are the ones having an isoperimetric property.…”