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


In this paper, considered a Borel function g on \mathbf {R}<sup>n</sup> taking its values in [0,+∈fty], verifying some weak hypothesis of continuity, such that (domg)<sup>o</sup> = \emptyset and domg is convex, we obtain an integral representation result for the lower semicontinuous envelope in the L<sup>1</sup>(ω) - topology of the integral functional G<sup>0</sup>(u<sub>0</sub>,ω,u) = ∈top \limit _{ω}g(\nabla u)dx, where(Error rendering LaTeX formula) 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.

DOI Code: 10.1285/i15900932v19n2p231

Classification: 49J45

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