Elliptic obstacle problems with measure data: potentials and low order regularity
The author treats measure data problems of the type \[ -\text{div} a(x,Du) = \mu \qquad \text{on } \Omega \] coupled with an obstacle constraint of the form \(u \geq \psi\), where \(\mu\) a Radon measure and the left-hand side of the equation an elliptic differential operator of \(p\)-Laplace type for \(p > \frac{2n}{n+1}\). Here it is only assumed that \(a: \Omega \times \mathbb{R}^n \to \mathbb{R}^n\) is a Carathéodory function. As the measure data \(\mu\) is only assumed to be a Radon measure, one cannot work with the usual weak formulation of the problem. Instead, the author discusses the concept of a limit \(u\) of (more regular) approximating solutions. In this setting, the author then proves pointwise estimates for the solutions in terms of the Wolff potential. As applications, criteria for Lorentz space regularity, for the continuity and the Hölder continuity of the solutions are derived.
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