Globally Lipschitz minimizers for variational problems with linear growth

Abstract

We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space W 1,1 with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately elliptic Euler-Lagrange equation). Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and in fact might fail, as it is well-known already for the nonparametric minimal surface problem. Assuming radial structure, we establish a necessary and sufficient condition on the integrand such that the Dirichlet problem is in general solvable, in the sense that a Lipschitz solution exists for any regular domain and all prescribed regular boundary values, via the construction of appropriate barrier functions in the tradition of Serrin's paper [19]. p

Integrability of the derivative of solutions to a singular one-dimensional parabolic problem

“…In the setting of [4] the smooth dependence of W on p is important for the argument. In [5], in a similar setting Lipschitz continuity of minimizers is shown.…”

Some Geometric Properties of Nonparametric $$\mu $$-Surfaces in $$\pmb {{\mathbb {R}}}^3$$

“…The limit case μ = 2 also plays an important role in studying the regularity of solutions. While our geometric considerations are based on the finiteness in condition (1.8), we note that, e.g., in (1.9) of [14] the condition…”

“…characterizes the existence of regular solutions assuming prescribed boundary values. We also like to refer to the introductory remarks of [14] and to the classical paper [15], where related conditions can be found. Here we let for g: [0, ∞) → R and all t ≥ 0 (g ∈ C 2 [0, ∞) , g (0) = 0, g (t) > 0 for all t > 0)…”

Some geometric properties of nonparametric $μ$-surfaces in $\mathbb{R}^3$

“…The limit case µ = 2 also plays an important role in studying the regularity of solutions. While our geometric considerations are based on the finiteness in condition (1.8), we note that, e.g., in (1.9) of [14] the condition…”