Solutions to the obstacle problem as Green potentials
An obstacle problem is discussed by using a new type of the ''sweeping-out principle'' of measures and the theory due to Cartan and Deny. The first result is a fundamental inequality between measures. The balayage measure \(\mu\) of a solution \(G\mu\) to the obstacle problem with obstacle \(\psi =G\nu_ 1-G\nu_ 2\) satisfies \(\mu \leq \nu_ 1\), where \(G\mu\) is the Green potential of \(\mu\) and \(\nu_ 1\), \(\nu_ 2\) are positive measures with finite Green energy. The second consequence is that quadrature domains with weight are closely related to the complements of the coincident sets in the obstacle problem.
scientific article; zbMATH DE number 3923521