On the Relaxation of Some Types of Dirichlet Minimum Problems for Unbounded Functionals

Abstract

In this paper, considered a Borel function g on $\mathbf {R}ⁿ$ taking its values in $[0,+∈fty]$, verifying some weak hypothesis of continuity, such that $(domg)^o = \emptyset$ and $domg$ is convex, we obtain an integral representation result for the lower semicontinuous envelope in the $L¹(ω)$ - topology of the integral functional $G⁰(u₀,ω,u) = ∈top \limit _{ω}g(\nabla u)dx$, where $u ∈ W_{(loc)}^(1,∈fty)(\mathbf R ⁿ), u = u_{0}$ only on suitable pin is of the boundary of $ω$ that lie, for example, on affine spaces orthogonal to $aff(domg)$, for boundary values $u{0}$ satisfying suitable compatibility conditions and $ω$ is geometrically well situated respect to $domg$. Then we apply this result to Dirichlet nunimum problems.