On the regularity of the free boundary for quasilinear obstacle problems

Abstract

We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the p(x)-Laplacian. Under the assumption of Lipschitz continuity of the order of the power growth p(x) > 1, we use the growth rate of the solution near the free boundary to obtain its porosity, which implies that the free boundary is of Lebesgue measure zero for p(x)-Laplacian type heterogeneous obstacle problems. Under additional assumptions on the operator heterogeneities and on data we show, in two different cases, that up to a negligible singular set of null perimeter the free boundary is the union of at most a countable family of C 1 hypersurfaces: i) by extending directly the finiteness of the (n − 1)-dimensional Hausdorff measure of the free boundary to the case of heterogeneous p-Laplacian type operators with constant p, 1 < p < ∞; ii) by proving the characteristic function of the coincidence set is of bounded variation in the case of non degenerate or non singular operators with variable power growth p(x) > 1.

Homogenization of obstacle problems in Orlicz–Sobolev spaces

“…Remark 4.1. If ψ 0 = 0 and the right hand side is regular enough, the condition (4.5) holds automatically, since in this particular case one has porosity of the free boundary from [9] (hence, the free boundary has Lebesgue measure zero), which provides (4.5). Proof.…”

“…Step 1 (Apriori estimates). Existence and uniqueness of the solution of (1.8) (and (1.9)) is a classical result (see, for instance, [9,18,19]). As in the proof of [5, Theorem 2.3] (see also [18, page 145]), the coercitivity and boundedness assumptions from (1.2) imply that u ε is bounded in W 1,pε(·) 0…”

Sharp regularity estimates for quasilinear evolution equations

“…Although weak solutions of (1.1) under the compatibility assumptions (C) are known to be locally of the class C 1+α (in the parabolic sense) for some α ∈ (0, 1), the sharp exponent is known only for some specific cases (see [5,6,26,28,33]). This type of quantitative information plays an essential role in the study of blow-up analysis, related geometric and free boundary problems and for proving Liouville type results (see [3,4,12,16,18,40] for some enlightening examples).…”

“…weak solutions of (1.1) are of class C α for some α ∈ (0, 1). Using compactness and geometric tangential methods (see [3,4,11,12,14,19,36,37]) and intrinsic scaling techniques (see [16,20,24,41] In this work we will solve it in the second scenario. More precisely, our main result reveals that bounded week solutions of (1.1) are locally of the class C 1+α (in the parabolic sense) in the critical zone (i.e.…”

Porosity of the free boundary in the degenerate parabolic variational problem

“…Due to the non-invariant property of solutions under the operator div a(x,t, u, ∇u) − ∂ t u, techniques of compactness will be applied to establish the growth when we use intrinsic scaling, which was also applied in [4,5,6,7,10,11,12]. It should be noticed that the finite N − 1-Hausdorff measure of the free boundary in the elliptic obstacle problem was considered in [11,12,13,14,15] based on the property of porosity of the free boundary.…”