The Caffarelli Alternative in Measure for the Nondivergence Form Elliptic Obstacle Problem with Principal Coefficients in VMO

Abstract

We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in [C1].

Reifenberg flatness of free boundaries in obstacle problems with VMO ingredients

“…In the interior of both {w > 0} and {w = 0} it is not hard to show that χ {w k >0} f k converges pointwise almost everywhere to χ {w>0} f (for the interior of {w = 0} one needs to use the nondegeneracy statement), so by Lebesgue's dominated convergence theorem it suffices to prove that ∂{w = 0} has no Lebesgue points. The proof of this fact is very similar to the proof of Lemma 5.1 of [4], but we include it here for the convenience of the reader. Let x 0 ∈ ∂{w = 0} ∩ B 1 , and choose r > 0 such that…”

“…Proof Here again our proof is almost identical to the proof of Theorem 6.3 of [4], so we leave it to the reader.…”

“…Proof The proof of Theorem 5.4 of [4] can be adapted to the current setting without too much difficulty, but we include it for the convenience of the reader. Suppose not.…”

“…For any g ∈ VMO, η g (r ) is referred to as the VMO-modulus. For all conventions regarding VMO we follow [4] which in turn follows [15]. and…”

“…In fact, it is relatively easy to produce nonuniqueness even in the case with a constant right hand side, and it was done in [4] for the nondivergence form case, but that counter-example can be copied almost exactly for the divergence form case. In the case where the coefficients of L are constant, one can use the counter-example in [1] to show nonuniqueness of blowup limits when the right hand side is only assumed to be continuous.…”