Gradient potential estimates for elliptic obstacle problems
Section snippets
Introduction and main results
The obstacle problems discussed in this paper are related to the following quasilinear equation: where is a bounded open set and μ is a Radon measure defined on Ω with finite total mass. Moreover we assume that and satisfies The obstacle condition that we impose on the solution is of the form a.e. on Ω, where is a given function with and G is
Preliminaries
In this section, we introduce some notions and results which will be used in this paper. Firstly, we denote by m any number in the natural number set , it is easily verified that for any measurable set and every function such that . If , we have Definition 2.1 A Young function B is called an N-function if and The Young conjugate of a Young function B is defined as
Existence result
In this section we shall prove the existence result for the obstacle problems with measure data. Let's start with the following lemma, which will be useful for Lemma 3.2. Lemma 3.1 Let be a bounded domain. Assume that , satisfies for , and fixed constants . Then we have where .
Proof We first assume that , the inequality (3.1)
Comparison estimates
This section is devoted to compare the solutions of obstacle problems to the solutions of elliptic equations. Therefore we can obtain a similar excess decay estimate for solutions of obstacle problems with measure data.
Firstly, we shall prove the comparison estimate between the solutions of elliptic obstacle problems with measure data and the solutions of the corresponding homogeneous obstacle problems. Lemma 4.1 Under the assumption (1.2), let and the map
Gradient estimates by potentials
In this section, we first prove a pointwise estimate of fractional operator by precise iteration methods, and therefore allow to get -bounds. We thus obtain pointwise and oscillation estimates for the gradients of solutions by means of the sharp maximal function estimates. Proof of Theorem 1.7 We choose a geometric sequence whose spread will be a certain function of the fixed quantities , and will be determined later. More precisely we define
Acknowledgments
The authors are supported by the National Natural Science Foundation of China (NNSF Grant No.12071229 and No.11671414). The authors would like to express their gratitude to the anonymous reviewers for their constructive comments and suggestions that improved the last version of the manuscript.
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