Università degli studi di Modena e Reggio Emilia

Let us consider the autonomous obstacle problemmin(v) integral(Omega) F(Dv(x)) dxon a specific class of admissible functions, where we suppose the Lagrangian satisfies proper hypotheses of convexity and superlinearity at infinity. Our aim is to find a necessary condition for the extremality of the solution, which exists and it is unique, thanks to a primal-dual formulation of the problem. The proof is based on classical arguments of Convex Analysis and on Calculus of Variations' techniques. (c) 2023 Elsevier Ltd. All rights reserved.

We study the asymptotic behavior of a family of functionals which penalize a short-range interaction of convolution type between a finite perimeter set and its complement. We first compute the pointwise limit and we obtain a lower estimate on more regulars sets. Finally, some examples are discussed.

In this paper, we consider a class of obstacle problems of the typemin {integral(Omega) f(x, Dv) dx : v is an element of K-psi(Omega)}where psi is the obstacle, K-psi (Omega) = {v is an element of u(0)+W-0(1,p) (Omega, R) : v >= psi a.e. in Omega}, with v(0) is an element of W-1,W-p (Omega) a fixed boundary datum, the class of the admissible functions and the integrand f (x, Dv) satisfies non standard (p, q)-growth conditions. We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map x bar right arrow A(x, xi) is independent of the dimension n and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one W-1,W-n.