Lipschitz regularity for local minimizers of some widely degenerate problems (original) (raw)
Existence and Lipschitz Regularity for Minima
Proceedings of the American Mathematical Society
We prove the existence, uniqueness and Lipschitz regularity of the minima of the integral functional I(u )= Z L(x;u;ru)dx on u + W 1;q 0 () (1 q +1) for a class of integrands L(x;z;p )= f(p )+ g(x;z) that are convex in (z;p) and for boundary data satisfying some barrier conditions. We do not impose regularity or growth assumptions on L.
arXiv: Analysis of PDEs, 2018
We prove an a priori estimate for the second derivatives of local minimizers of integral functionals of calculus of variation with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.
Lipschitz regularity of almost minimizers in one-phase problems driven by the ppp-Laplace operator
arXiv (Cornell University), 2022
We prove that, given p > max 2n n+2 , 1 , the nonnegative almost minimizers of the nonlinear free boundary functional J p (u, Ω) :=ˆΩ |∇u(x)| p + χ {u>0} (x) dx are Lipschitz continuous. The setting of almost minimizers is also classical in the calculus of variations (dating back, at least up to a certain extent, to a famous sentence in Leibniz's Specimen Geometriae Luciferae, probably written in the mid-1690s, "pro minimis adhiberi possunt quasi minima", that is "the almost minimizers can be exploited in place of minimizers"). More specifically, the mathematical setting that we consider here goes as follows. Let Ω ⊂ R n be a given domain and p > max 2n n+2 , 1. We consider the energy functional (1.1) J p (u, Ω) :=ˆΩ |∇u(x)| p + χ {u>0} (x) dx for all u ∈ W 1,p (Ω) with u 0. The condition that u is nonnegative corresponds, in the framework of free boundary problems, to considering "one-phase" solutions (solutions which may change sign being related to "twophase" problems). The precise notion of almost minimizers that we use in this paper is the following one: Definition 1.1. Let κ 0 and β > 0. We say that u ∈ W 1,p (Ω) is an almost minimizer for J p in Ω, with constant κ and exponent β, if u 0 a.e. in Ω and (1.2) J p (u, B ̺ (x)) (1 + κ̺ β)J p (v, B ̺ (x)), for every ball B ̺ (x) such that B ̺ (x) ⊂ Ω and for every v ∈ W 1,p (B ̺ (x)) such that v = u on ∂B ̺ (x) in the sense of the trace. In some sense, Definition 1.1 is one of the possible modern formalizations of Leibniz's initial intuition reported at the beginning of this paper: namely, almost minimizers are natural objects to look at, for instance, to deal with minimizers of "perturbed" functionals. As a concrete example, if we consider J p (u, Ω) := J p (u, Ω) +¨Ω ×Ω Φ(u(y)) Φ(u(z)) Φ(u(y) − u(z)) dy dz for a function Φ : R → [0, 1] with Φ = 0 in (−∞, 0], we readily see that J p (u, B r (x)) J p (u, B r (x)) and J p (u, B r (x)) J p (u, B r (x)) +¨B r (x)×Br(x) χ {u>0} (y) χ {u>0} (z) dy dz J p (u, B r (x)) + |B r |ˆB r (x) χ {u>0} (y) dy (1 + |B 1 | r n)J p (u, B r (x)). Accordingly, a minimizer for the "complicated" functional J p turns out to be an almost minimizer for the "simpler" functional J p. As usual, the constants depending only on n and p are called universal. If u is an almost minimizer, the structural constants may depend on κ and β as well.
Existence of Minimizers for NonLevel Convex Supremal Functionals
Siam Journal on Control and Optimization, 2014
The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem inf ess sup x∈Ω f (∇u(x)) : u ∈ u0 + W 1,∞ 0 (Ω) , when the supremand f is not necessarily level convex. These conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum. Several conditions of convexity for the supremand f are also investigated. Résumé Dans cet article onétudie des conditions nécessaires et suffisantes pour l'existence de solutions pour le problème de minimisation inf ess sup x∈Ω f (∇u(x)) : u ∈ u0 + W 1,∞ 0 (Ω) lorsque la fonction f n'est pas une fonction convexe par niveaux. La stratégie utilisée pour obtenir ces conditions est celle de comparer ce problème avec son problème relaxé. On obtient comme condition nécessaire et suffisante une inclusion différentielle sur la donnée au bord. Onétudie aussi plusieurs conditions de convexité.
Lipschitz regularity for some asymptotically convex problems
ESAIM: Control, Optimisation and Calculus of Variations, 2011
We establish a local Lipschitz regularity result for local minimizers of variational integrals under the assumption that the integrand becomes appropriately elliptic at infinity. The exponent that measures the ellipticity of the integrand is assumed to be less than two.