Nonlinear elliptic systems with measure-valued right hand side
The authors consider existence and regularity questions for elliptic system \[ -\text{div} \sigma \bigl(x,u(x), Du(x)\bigr) =\mu \quad\text{in } \Omega, \] with measure-valued right hand side \(\mu\) and Dirichlet boundary conditions on a bounded domain \(\Omega\subset \mathbb{R}^N\). A serious technical obstacle is that the solutions of the system in general do not belong to the Sobolev space \(W^{1,1}\). This fact has led to the use of the weakest notion of solution where the weak derivative \(Du\) is replaced by the approximate derivative \(ap\) \(Du\). Under certain growth, monotonicity and structure conditions on \(\sigma\) the authors prove the existence of a solution satisfying a weak Lebesgue space a priori estimate. The BMO-regularity of the solution is also discussed. The key point in the proof of the theorem is a variant of the ``div-curl inequality'' for Young measures which is given in the paper.
scientific article; zbMATH DE number 1121732