Geometric constraints on the domain for a class of minimum problems

Abstract

Abstract.We consider minimization problems of the form Mathematics Subject Classification. 49J10, 49L25.

A boundary value problem for a PDE model in mass transfer theory: Representation of solutions and applications

“…(We recall that r is the supremum of the radii of all balls contained in .) This result has been extended to convex domains in R n and to more general functionals in subsequent works (see [6,7,10,23,24]). One common point of all these results is that the set is always a convex subset of R n .…”

“…Proof Let v f be the continuous function defined in (10). We claim that the following bound on v f holds true:…”

A sharp uniqueness result for a class of variational problems solved by a distance function

“…In fact, the hard part of the proof of the existence result in [11] is the construction of such a function v. Moreover it is shown, by examples, that the growth condition Λ r Ω cannot be improved. The result in [11] has been extended to convex domains in R n and to more general functionals in subsequent works (see [9,10,16,21,22]). Recently in [6,7] it was proved that for every given nonnegative continuous function f there exists a unique nonnegative continuous function v f solving…”

Variational Problems for a Class of Functionals on Convex Domains

“…Another class of functionals with this property was exhibited in [4,18] (see also [3,10,17]). More precisely, if f is a nonnegative lower semicontinuous function such that f(R)=0 for some R \ 0 and f(t) It remains an open problem to establish for which functionals other than those described above there exists a minimizer in W 1, 1 0 (W) belonging to the class K.…”