Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem

Abstract

Abstract. We consider an obstacle-type problemwhere D is a given open set in R n and Ω is an unknown open subset of D. The problem originates in potential theory, in connection with harmonic continuation of potentials. The qualitative difference between this problem and the classical obstacle problem is that the solutions here are allowed to change sign. Using geometric and energetic criteria in delicate combination we show the C 1,1 regularity of the solutions, and the regularity of the free boundary, below the Lipschitz threshold for the right hand side.

Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

“…See the previous works of Caffarelli [6,7,8,9], Weiss [38], and Caffarelli, Karp, Shahgholian [13] for Lipschitz coefficients, and also the work of Caffarelli, Kinderlehrer [14] for some related estimates on the modulus of continuity of the solution or of its gradient. Let us mention that very recently, similar regularity results have been obtained in Petrosyan, Shahgholian [34] for the regular points of the free boundary, for an obstacle problem with no sign assumption on the solution. These results are obtained under geometric and energetic conditions and the assumption…”

“…Let us recall that this result is originally du to Weiss [38] for f ≡ 1 (see also Monneau [32] for a version for some Dini modulus of continuity, and Petrosyan, Shahgholian [34] for a similar monotonicity formula for double Dini modulus of continuity, but for obstacle problems with no sign condition on the solution).…”

The classical obstacle problem with coefficients in fractional Sobolev spaces

“…The Weiss monotonicity formula was proven by Weiss within [45] for the case where A ≡ I n and f ≡ 1; in the same paper he proved the celebrated epiperimetric inequality (see Theorem 8.2) and gave a new way of approaching the problem of the regularity for the free-boundary. In [39] Petrosyan and Shahgholian proved the monotonicity formula for A ≡ I n and f with a double Dini modulus of continuity (but for obstacle problems with no sign condition on the solution). Lederman and Wolanski [31] provided a local monotonicity formula for the perturbated problem to achieve the regularity of Bernoulli and Stefan free-boundary problems, while Ma, Song and Zhao [34] showed the formula for elliptic and parabolic systems in the case in which A ≡ I n and the equations present a first order nonlinear term.…”

Optimal regularity for the parabolic no-sign obstacle type problem

“…The papers we discuss below ( [4], [8], [14] [7]) are technically very sophisticated and we have to refer the reader to the original sources for the full details. It should be mentioned that we will, rather mischievously, slightly change the conceptual framework of the above papers into the BMO framework of this paper in our explanations.…”

“…It is not difficult to prove that the solution is W 2,1 ∞ close to points X 0 where |{u = 0} ∩Q − r (X 0 )| > ǫ|Q − r (X 0 )| for every r > 0. The most sophisticated result of this kind is [14] in the elliptic case and [7] for the parabolic case. The assumptions on the free boundary are, in order to be as week as possible, rather technical so we will have to refer the readers to the original papers for the details.…”