The zero level set for a certain weak solution, with applications to the Bellman equations

Abstract

We will prove a partial regularity result for the zero level set of weak solutions to div(B∇u) = 0, where B = B(u) = I + (A − I)χ {u<0} , where I is the identity matrix and the eigenvalues of A are strictly positive and bounded. We will apply this to describe the regularity of solutions to the Bellman equations.

Boundary regularity for a parabolic obstacle type problem

“…The argument is elementary so we will only give a rough sketch. From Lemma 11 we know that ∂ n u s ≈ (x n ) 1/(p−1) + + o (1). We also know that ∇ ′ u s L 2 (B1) ≤ C s ǫ since ∇ ′ u L 2 (B1(0)) ≤ ǫ.…”

“…Lemma 9. Let u ∈ W 1,p loc (R n ) be a solution to (1) in R n and Ω = R n + . Assume that u satisfies the following growth condition…”

“…In this section we will assume that we have a sequence of solutions u j in B 2 (0) in a normalized coordinate system defined as follows. p (B 1 (0)) be a solution to (1). Then we say that the coordinate system is normalized (with respect to u) if…”

“…where g(x) ≥ 0 is the restriction of a W 1,p (D) function to ∂D. We will denote Ω u = {u > 0} and the free boundary Γ = ∂Ω ∩ D. A solution to (1) can be found by minimizing D 1 p |∇u| p + max(u, 0)dx in K = {u ∈ W 1,p (D); u − g ∈ W 1,p 0 (D)}. It is well known that solutions to (1) are C 1,α loc (D) for some α > 0, see for instance [5].…”

“…The argument is interesting in its own right and based on an idea from [7]. A Lemma from geometric measure theory (see for instance [8]), used in the free boundary context in [1], then implies that a blow-up of u 0 is in fact also a blow-up of u at x 0 -at least for almost every x 0 in the free boundary.…”