Regularity in free boundary problems

26/04/2018 · The main mathematical challenge is understanding the regularity of free boundaries. The obstacle problem is the most classical and motivating example in the study of free boundary problems. REGULARITY OF THE FREE BOUNDARY FOR THE VECTORIAL BERNOULLI PROBLEM DARIO MAZZOLENI, SUSANNA TERRACINI, BOZHIDAR VELICHKOV Abstract. In this paper we study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of nite measure D ˆRd

This book is concerned with several elliptic and parabolic obstacle-type problems with a focus on the cases where the free and fixed boundaries meet. The results presented complement those found in existing books in the subject, which mainly treat regularity properties away from the fixed boundary. We provide a rather complete description of the sharp regularity theory to a family of heterogeneous, two-phase free boundary problems, J γ → min, ruled by nonlinear, p-degenerate elliptic operators.Included in such family are heterogeneous cavitation problems of Prandtl–Batchelor type, singular degenerate elliptic equations; and obstacle type systems.

FREE BOUNDARY REGULARITY PROBLEMS T. TORO In the 1980’s Alt-Ca arelli and Alt-Ca arilly-Friedman initiated the study of minimizing problems with free boundary, since then this eld as boomed. In this situation the minimizer satis es a PDE, and the central question … Title: An epiperimetric inequality for the regularity of some free boundary problems: the $2$-dimensional case Authors: Luca Spolaor , Bozhidar Velichkov (Submitted on 6 Dec 2016 ( …

Problem (1) can be regarded as a simpli ed version of the classical one-phase free boundary problem: ˆ u= 0 in := fu>0g; jruj= 1 on @: (3) The regularity of the free boundary problems actually has been a subject arXiv:1702.00465v1 [math.AP] 1 Feb 2017 A GEOMETRIC APPROACH TO REGULARITY FOR NONLINEAR FREE BOUNDARY PROBLEMS ARAM L. KARAKHANYAN Abstract. Let u be a weak solutions of the free boundary problem Lu = λ0H1v∂red{u > 0},u ≥ 0 where Lu = div(g(∇u)∇u) and g(ξ) is a given function of ξ satisfying some standard structural conditions.

arXiv:1702.00465v1 [math.AP] 1 Feb 2017 A GEOMETRIC APPROACH TO REGULARITY FOR NONLINEAR FREE BOUNDARY PROBLEMS ARAM L. KARAKHANYAN Abstract. Let u be a weak solutions of the free boundary problem Lu = λ0H1v∂red{u > 0},u ≥ 0 where Lu = div(g(∇u)∇u) and g(ξ) is a given function of ξ satisfying some standard structural conditions. We study the free boundary of solutions to a class of nonlinear obstacle problems. This class of problems contains a particular model derived from the Ginzburg-Landau equation of superconductivity. We consider solutions in a Lipschitz bounded open set and prove the regularity of the free boundary when it is close enough to the xed boundary @