Some properties of the solutions of obstacle problems with measure data (Q1809062)

For a partial differential equation of the form \[ Au = \mu, \] where \(A\) is a linear elliptic operator and \(\mu\) is a bounded Radon measure, the solution to a homogeneous Dirichlet boundary-value problem is sought among functions defined on a domain \(\Omega\subset R^N\) and additionally restricted by an obstacle condition \(u\geq\psi\). Such a problem is called \(OP(\mu, \psi)\). A solution to \(OP(\mu, \psi)\) is defined as the smallest function satisfying the obstacle condition and solving in the sense of \textit{G. Stampacchia} [Ann. Inst. Fourier 15, No. 1, 189-257 (1965; Zbl 0151.15401)] a homogeneous Dirichlet boundary-value problem with no obstacle condition for the equation \[ Au=\mu +\lambda, \] where \(\lambda \) is a nonegative Radon measure. For the solution, the measure \(\lambda\) is called the obstacle reaction. It follows from the linearity of the operator \(A\) that the obstacle reaction is concentrated on the contact set \(\{u=\psi\}\). The main result of the paper characterizes the structure of the obstacle reaction. Namely, it is obtained that if the obstacle measure \(\psi\) satisfies certain two-sided estimates, then \(\lambda\) can be decomposed into \(\lambda = \lambda_0 + \mu_s^- \), where \(\lambda_0\) is a nonnegative Radon measure vanishing on all subsets of zero capacity and the \(\mu_s^- \) is the singular component of the measure \(\mu^-\), i.e., \(\mu_s^- \) is concentrated on sets of capacity zero. From this theorem it is further derived that under the same assumptions, if \(u\) and \(u_0\) are, respectively, the solutions of problems \(OP(\mu,\psi)\) and \(OP(\mu^+-\mu_a^-,\psi)\), then equality \(u=u_0\) holds almost everywhere with respect to the capacity and the corresponding obstacle reaction measures \(\lambda\) and \(\lambda_0\) are related through the equality \(\lambda = \lambda_0 + \mu_s^- \). Here \(\mu_a^-\) is the absolutely continuous component of measure \(\mu^-\) with respect to capacity (i.e., vanishing on all sets of zero capacity).