A new symmetry criterion based on the distance function and applications to PDEʼs

Abstract

We prove that, if Ω ⊂ R n is an open bounded starshaped domain of class C 2 , the constancy over ∂Ω of the functionimplies that Ω is a ball. Here kj(y) and λ(y) denote respectively the principal curvatures and the cut value of a boundary point y ∈ ∂Ω. We apply this geometric result to different symmetry questions for PDE's: an overdetermined system of Monge-Kantorovich type equations (which can be viewed as the limit as p → +∞ of Serrin's symmetry problem for the p-Laplacian), and equations in divergence form whose solutions depend only on the distance from the boundary in some subset of their domain.

On the characterization of some classes of proximally smooth sets

“…On the other hand, if p = q, then d S (q) = |q − p| ∈ (0, r), so that q must have a unique projection onto S, in contradiction with the fact that, by construction, both p and η(t) are projections of q onto S. Hence, in our coordinate system, the point γ(t) must lie on the right side of the line through γ(t) and γ(s); hence, in view of (20), we infer that γ(t) belongs to the set E ∩ {x 1 > 0} (corresponding to the shaded region in Figure 3). We conclude that (21) γ(t) − γ(s), γ (s) > 0 .…”

“…This notion has been considered from different points of views: in [10,36,37] the regularity of the normal distance under different requirements on the boundary has been investigated, along with some applications to Hamilton-Jacobi equations and to PDEs related with granular matter theory; in [13,23,24,25] the normal distance has been exploited in order to study the minimizing properties of the so-called web functions. Let us also mention that, in a previous paper, we proved a roundedness criterion based on the constancy along the boundary of a C 2 domain of a certain function, depending on the normal distance and on the principal curvatures, see [21,Thm. 1].…”