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.

The classical obstacle problem with Hölder continuous coefficients

“…Higher order regularity of solutions, namely W 2,p loc , in combination with their well-known quadratic growth from free boundary points then allow to conclude. In this note we push forward the ideas in [14] and extend the conclusions there to Hölder continuous matrix fields A for which the above mentioned W 2,p loc regularity of solutions is not guaranteed.…”

“…In particular, in this section we improve upon [12, Theorems 4.12 and 4.14], [18, Theorem 1.3] and [14,Theorem 2.1], since in all those cases the matrix field A turns out to be in particular Hölder continuous due to Sobolev type embeddings.…”

“…In this section we establish Weiss' and Monneau's type quasi-monotonicity formulas. The monotone quantities we consider in Section 3.1 are modeled upon the classical Dirichlet energy as in [14] under a pointwise normalization condition on the coefficients. We shall show in the subsequent Section 3.2 how to reduce to that formulation in a pointwise way via a suitable change of variables following [12].…”

“…In those paper, instead, the natural quadratic energy E associated to the obstacle problem under study had been considered. The W 2,p loc regularity of solutions and the quadratic growth from the free boundary were key properties to establish the quasi-monotonicity formulas in [14] for the Dirichlet based quantities. The new contribution of the current paper is to avoid the use of the former piece of information, which is not guaranteed in our setting, thanks to an elementary energy comparison argument to prove Weiss' formula (cf.…”

The classical obstacle problem with Hölder continuous coefficients

“…Higher order regularity of solutions, namely W 2,p loc , in combination with their well-known quadratic growth from free boundary points then allow to conclude. In this note we push forward the ideas in [14] and extend the conclusions there to Hölder continuous matrix fields A for which the above mentioned W 2,p loc regularity of solutions is not guaranteed.…”

“…In particular, in this section we improve upon [12, Theorems 4.12 and 4.14], [18, Theorem 1.3] and [14,Theorem 2.1], since in all those cases the matrix field A turns out to be in particular Hölder continuous due to Sobolev type embeddings.…”

“…In this section we establish Weiss' and Monneau's type quasi-monotonicity formulas. The monotone quantities we consider in Section 3.1 are modeled upon the classical Dirichlet energy as in [14] under a pointwise normalization condition on the coefficients. We shall show in the subsequent Section 3.2 how to reduce to that formulation in a pointwise way via a suitable change of variables following [12].…”

“…In those paper, instead, the natural quadratic energy E associated to the obstacle problem under study had been considered. The W 2,p loc regularity of solutions and the quadratic growth from the free boundary were key properties to establish the quasi-monotonicity formulas in [14] for the Dirichlet based quantities. The new contribution of the current paper is to avoid the use of the former piece of information, which is not guaranteed in our setting, thanks to an elementary energy comparison argument to prove Weiss' formula (cf.…”

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].…”

The classical obstacle problem with Hölder continuous coefficients

“…Higher order regularity of solutions, namely W 2,p loc , in combination with their well-known quadratic growth from free boundary points then allow to conclude. In this note we push forward the ideas in [14] and extend the conclusions there to Hölder continuous matrix fields A for which the above mentioned W 2,p loc regularity of solutions is not guaranteed.…”

“…In particular, in this section we improve upon [12, Theorems 4.12 and 4.14], [18, Theorem 1.3] and [14,Theorem 2.1], since in all those cases the matrix field A turns out to be in particular Hölder continuous due to Sobolev type embeddings.…”

“…In this section we establish Weiss' and Monneau's type quasi-monotonicity formulas. The monotone quantities we consider in Section 3.1 are modeled upon the classical Dirichlet energy as in [14] under a pointwise normalization condition on the coefficients. We shall show in the subsequent Section 3.2 how to reduce to that formulation in a pointwise way via a suitable change of variables following [12].…”

“…In those paper, instead, the natural quadratic energy E associated to the obstacle problem under study had been considered. The W 2,p loc regularity of solutions and the quadratic growth from the free boundary were key properties to establish the quasi-monotonicity formulas in [14] for the Dirichlet based quantities. The new contribution of the current paper is to avoid the use of the former piece of information, which is not guaranteed in our setting, thanks to an elementary energy comparison argument to prove Weiss' formula (cf.…”

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].…”

The classical obstacle problem with Hölder continuous coefficients

“…Higher order regularity of solutions, namely W 2,p loc , in combination with their well-known quadratic growth from free boundary points then allow to conclude. In this note we push forward the ideas in [14] and extend the conclusions there to Hölder continuous matrix fields A for which the above mentioned W 2,p loc regularity of solutions is not guaranteed.…”

“…In particular, in this section we improve upon [12, Theorems 4.12 and 4.14], [18, Theorem 1.3] and [14,Theorem 2.1], since in all those cases the matrix field A turns out to be in particular Hölder continuous due to Sobolev type embeddings.…”

“…In this section we establish Weiss' and Monneau's type quasi-monotonicity formulas. The monotone quantities we consider in Section 3.1 are modeled upon the classical Dirichlet energy as in [14] under a pointwise normalization condition on the coefficients. We shall show in the subsequent Section 3.2 how to reduce to that formulation in a pointwise way via a suitable change of variables following [12].…”

“…In those paper, instead, the natural quadratic energy E associated to the obstacle problem under study had been considered. The W 2,p loc regularity of solutions and the quadratic growth from the free boundary were key properties to establish the quasi-monotonicity formulas in [14] for the Dirichlet based quantities. The new contribution of the current paper is to avoid the use of the former piece of information, which is not guaranteed in our setting, thanks to an elementary energy comparison argument to prove Weiss' formula (cf.…”

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].…”