A minimum problem with free boundary for a degenerate quasilinear operator

Abstract

In this paper we prove C 1,α regularity (near flat points) of the free boundary ∂{u > 0} ∩ Ω in the Alt-Caffarelli type minimum problem for the p-Laplace operator: (2000): 35R35, 35J60 Mathematics Subject Classification

Full regularity of the free boundary in a Bernoulli-type problem in two dimensions

“…Moreover, the conditions (1.2)-(1.3) are satisfied on Γ red in the classical sense. This result has been established by Alt and Caffarelli in their fundamental work [AC81] when p = 2 and later generalized for all 1 < p < ∞ by the authors in [DP05].…”

“…These three theorems correspond to Theorem 3.3, Lemma 4.2, and Theorem 4.4 in [DP05], respectively. The only difference is that we now allow p to change in I μ ⊂⊂ (1, ∞).…”

“…In the next two sections we recall some known properties of minimizers of the functional J p taken from [DP05]. In fact, we state the results in a slightly more general form, namely with constants depending uniformly on…”

An isoperimetric inequality related to a Bernoulli problem

“…Since the energy minimising solutions have attracted quite some interest with the work in [1] in case p = 2 and [4] for general p ∈ (1, ∞), our result should still be of interest. We give a proof of (1.6) in Section 3 and compute the optimal values in Section 4.…”

“…Such free boundary value problems originally arose from two dimensional flows (see [2,7]), but also have applications to heat flows or electro-chemical machining (see the references in [4]). …”