The existence of solutions to variational problems of slow growth

Abstract

We consider the existence of solutions, in the space W^{1,1}(Ω), to the problem minimize ∫_{Ω}L(∇v(x))dx on φ+W₀^{1,1}(Ω)

where L is of slow (linear or at most quadratic) growth. We present a necessary and sufficient condition in order that, for any smooth boundary datum φ and for any bounded Ω with smooth boundary, the minimum problem be solvable.