Optimal Regularity for the No‐Sign Obstacle Problem

Abstract

In this paper we prove the optimal C 1,1 (B 1 2 )-regularity for a general obstacle type problemunder the assumption that f * N is C 1,1 (B 1 ), where N is the Newtonian potential. This is the weakest assumption for which one can hope to get C 1,1regularity. As a by-product of the C 1,1 -regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point x 0 , the free boundary is locally a C 1 -graph close to x 0 , provided f is Dini. This completely settles the question of the optimal regularity of this problem, that has been under much attention during the last two decades.

Regularity of solutions in semilinear elliptic theory

“…We consider the L 2 projection of D 2 u on the space of Hessians generated by second order homogeneous harmonic polynomials on balls with radius r > 0 and show that the projections stay uniformly bounded as r → 0 + . Although this approach has proven effective in dealing with a variety of free boundary problems [2,6,8,9], Theorem 1.1 illustrates that it is also useful in extending and refining the classical elliptic theory.…”

“…This theorem was proven in [2] (Theorem 1.2) for the case when g(x, t) depends only on x. Under Assumption A, appropriate modifications of the proof in [2] work also for the general case; since the arguments are similar, we provide only a sketch of the proof and highlight the differences. • Interior C 1,1 estimate • Quadratic growth away from the free boundary…”

Optimal regularity in the optimal switching problem

“…Clearly, without the compatibility conditions, there are no solutions to (1) achieving the boundary data. We are interested in deriving C 1,1 -regularity for the solutions to our system, which is the best regularity one can expect.…”

“…By L 0 we denote the interior of the set L , and by pointwise C 2,α regularity we mean uniform approximation with a second order polynomial with the speed r 2+α . The idea of the proof is the same as in deriving the optimal regularity for the no-sign obstacle problem in [1]. The proof is based on the BM O-estimates for In the end we justify our assumption 0 ∈ ∂L 0 with a counterexample: We consider a particular system in R 2 , where the zero loop set L = {0}, then we find an explicit solution, that is not C 1,1 .…”

“…This is a necessary condition for the system to be well-defined. Indeed, let (u 1 , u 2 ) be a solution to (1), then u 1 − u 2 + ψ 1 ≥ 0 and u 2 − u 1 + ψ 2 ≥ 0, which implies…”