The prescribed mean curvature equation in weakly regular domains

Abstract

We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a generalized Gauss-Green theorem based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a weak Young's law for (Λ, r 0 )-minimizers of the perimeter.Other contributions to the theory have been obtained in various directions, see for instance Tam [54,55], Finn [27], Concus-Finn [16], Caffarelli-Friedman [5], as well as more recent works by De Philippis-Maggi [19], Caffarelli-Mellet [6] and Lancaster [35]. However the above list is far from being complete. A necessary condition on the pair (Ω, H) for the existence of a solution to (PMC) can be easily found by integrating (PMC) on any relatively compact set A ⊂ Ω with smooth boundary. Indeed, by applying the divergence theorem we get A

Two examples of minimal Cheeger sets in the plane

“…In [23] we have extended Giusti's results on the existence of solutions to (PMC) and on the characterization of the extremality condition (2)…”

“…At the same time, both solutions will become vertical at the reduced boundary of Ω ε . In conclusion, this example shows that it is not possible to extend the characterization of existence and uniqueness of solutions to (PMC) given in [23,Theorem 4.1] by dropping the assumption of weak regularity of the domain (see the discussion after the proof of Theorem 2.4).…”

“…In particular, the Cheeger problem is closely related to the theory of existence and uniqueness of graph of prescribed mean curvature [12,14,23] which is the cornerstone of the theory of capillary surfaces (a comprehensive treatise is available in [13]). We recall that for an open, bounded and connected set Ω ⊂ R n , a function u : Ω → R is a classical solution to the prescribed mean curvature equation if div ∇u(x)…”

“…Indeed, from this ancestor set, one can build an increasing sequence of minimal Cheeger sets Ω k converging to the unitary disk both in volume and perimeter, in such a way that the stability result [23,Proposition 4.4] holds.…”

Cauchy Fluxes and Gauss–Green Formulas for Divergence-Measure Fields Over General Open Sets

“…The classical Gauss-Green formula for Lipschitz vector fields F over sets of finite perimeter was proved first by De Giorgi [22,23] and Federer [29,30], and by Burago-Maz'ya [8,44] and Vol'pert [62,63] for F in the class of functions of bounded variation (BV ). The Gauss-Green formula for vector fields F P L 8 with divF P M was first investigated by Anzellotti in [4, Theorem 1.9] and [5] on bounded Lipschitz domains, and his methods were then exploited by Ambrosio-Crippa-Maniglia [1], Kawohl-Schuricht [38], Leondardi-Saracco [40], and Scheven-Schmidt [50][51][52]. Independently, motivated by the problems arising from the theory of hyperbolic conservation laws, Chen-Frid [11] first introduced the approach of defining the interior normal traces on the boundary of a Lipschitz deformable set as the limits of the classical normal traces over the boundaries of the interior approximations of the set, in which the Gauss-Green formulas hold.…”