Variational Principles with Linear Growth

Abstract

Variational principles which exhibit only linear growth arise in several contexts. As a paradigm for the questions which these principles suggest, we take up here the study of the problem

$$\mathop {\inf }\limits_{v \in {\cal{A}}} \left\{ {\int_\Omega {\phi (\nabla v)dx - \int_\Omega {fvdx} } } \right\}.$$

((0.1))

Here ϕ : R ⁿ → R is a non-negative convex sufficiently differentiable function satisfying

$$\phi (0) = 0,{\phi _p}(0) = 0,\;{\rm{and}}\;|p| - \lambda \le \phi \le |p|$$

((0.2))

for some λ > 0, subject to

$$\mathop {\lim }\limits_{t \to \infty } \phi (tp)/t|p| = 1.$$

((0.3))

The competing v belong to a suitable class A of functions from Ω to R where Ω is a bounded domain with sufficiently smooth boundary and f is given.