On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem

Abstract

In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ballWe prove that the free boundary touches the fixed boundary (uniformly) tangentially if the boundary data f and its first and second derivatives vanish at the touch-point.

On the non-tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem

“…Proof of step 3 We may conclude that u 0 = x 1 n i=2 a i x i for some constants a i . We want to show that a i = 0 for i = 3, .…”

“…We classify the angles of touch between the fixed and free boundaries at 0. It should be mentioned also that in [3] the authors and N. Matevosyan proved that if a ± = 0 then the free boundary of u approaches the fixed one at 0 tangentially. Under some growth assumptions they proved that this approach is uniform.…”

“…Proof of Claim 1 This can be done in a similar way as in [3], where it was proven that if a + = a − = 0 along the touching set then u is quadratically bounded. We will show that (0, 0) ∈ Q.…”

“…Under some growth assumptions they proved that this approach is uniform. Several papers have been investigating similar problems using similar techniques (see for instance [2,3,8]). The main new contribution is in the proof of Lemma 13.…”

“…If f grows slower than |x| 2 then we end up in a situation similar to [3] and if f grows faster, say f (x ) = x 2 √ |x 2 |, then the boundary values will be of higher order than the right hand side of (1) and the asymptotic development of u at the origin will be dominated by its Harmonic part and not by the right hand side of (1). We will therefore assume that f has quadratic growth away from the origin.…”