On the Lower Semicontinuous Envelope of Functionals Defined on Polyhedral Chains
نویسندگان
چکیده
In this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains. More precisely, denoting by H : R → [0,∞) an even, subadditive, and lower semicontinuous function with H(0) = 0, and by ΦH the functional induced by H on polyhedral m-chains, namely
منابع مشابه
Variational Approximation of Functionals withCurvatures and Related
We consider the problem of approximating via ?-convergence a class of functionals depending on curvatures of smooth compact boundaries. We investigate the connections between the approximation problem and the lower semicontinuous envelope of the original functional. We provide some examples of lower semicontinuous functionals and their variational approximation.
متن کاملRelaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings
In this paper we study the lower semicontinuous envelope with respect to the L-topology of a class of isotropic functionals with linear growth defined on mappings from the n-dimensional ball into RN that are constrained to take values into a smooth submanifold Y of RN . Let B be the unit ball in R and Y a smooth Riemannian manifold of dimension M ≥ 1, isometrically embedded in R for some N ≥ 2....
متن کاملA note on non lower semicontinuous perimeter functionals on partitions
We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in R. Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity.
متن کاملSemicontinuity and relaxation of L∞-functionals
Fixed a bounded open set Ω of R , we completely characterize the weak* lower semicontinuity of functionals of the form F (u,A) = ess sup x∈A f(x, u(x), Du(x)) defined for every u ∈ W 1,∞(Ω) and for every open subset A ⊂ Ω. Without a continuity assumption on f(·, u, ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it can be represented through a level conv...
متن کاملRelaxation of Free-discontinuity Energies with Obstacles
Given a Borel function ψ defined on a bounded open set Ω with Lipschitz boundary and φ ∈ L1(∂Ω,Hn−1), we prove an explicit representation formula for the L lower semicontinuous envelope of Mumford-Shah type functionals with the obstacle constraint u ≥ ψ Hn−1 a.e. on Ω and the Dirichlet boundary condition u = φ on ∂Ω. Mathematics Subject Classification. 49J45, 74R10. Received December 12, 2006. ...
متن کامل