Quasi-Monotonicity Formulas for Classical Obstacle Problems with Sobolev Coefficients and Applications

Abstract

We establish Weiss' and Monneau's type quasi-monotonicity formulas for quadratic energies having matrix of coefficients in a Sobolev space W 1,p , p > n, and provide an application to the corresponding free boundary analysis for the related classical obstacle problems.

Lipschitz Bounds and Nonautonomous Integrals

“…As for the nonuniformly elliptic setting, this approach has been used in [37]; see also Marcellini's survey [63] for a general overview. We also mention that, over the last several years, Sobolev coefficients have been systematically considered as a replacement of usual Lipschitz ones to find optimal conditions in several other fields of analysis and PDE; see for instance [8,26,40].…”

How a minimal surface leaves a thin obstacle

“…The basic idea of exploiting the regularity of u itself to reduce the problem to quadratic energies with Lipschitz coefficients has been recently considered in the literature for the classical obstacle problem (see, e.g., [17,43,18]). 6.2.…”

The classical obstacle problem with Hölder continuous coefficients

“…In the last years several contributions have been devoted to the extension of the regularity theory for obstacle type problems to the case in which the involved linear elliptic operator in divergence form has coefficients with low regularity [1,17,2,3,12,22,23,30,18,14,20]. The aim of this note is to make another step in that direction for the classical obstacle problem by establishing Weiss' and Monneau's quasi-monotonicity formulas for quadratic forms having matrix of coefficients which are Dini, double-Dini continuous, respectively (for the sake of simplicity the lower order terms have the same regularity).…”

“…In the last years Theorem 1.1 has been extended to the case in which A either is Lipschitz continuous in [12] or belongs to a fractional Sobolev space W 1+s,p in [18], with sp > 1 and p ≥ n ∧ n 2 n(1+s)−1 , or belongs to the Sobolev space W 1,p with p > n in [14]. We point out that in all those cases the involved matrix fields A turns out to be Hölder continuous in view of Sobolev type embeddings.…”

“…We point out that in all those cases the involved matrix fields A turns out to be Hölder continuous in view of Sobolev type embeddings. Applications of the results in [12,14] to the study of the obstacle problem for a large class of nonlinear energies were then given in [13]. In addition, counterexamples to smoothness of the free boundary are shown in case of the Laplacian if f is not Dini continuous in [1], and in case f is constant and A is VMO in [3,Remark 4.2].…”