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Nonlinear elliptic systems with measure-valued right hand side. (English) Zbl 0895.35029
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.
Reviewer: V.Moroz (Minsk)

MSC:
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35R05 PDEs with low regular coefficients and/or low regular data
35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
28A33 Spaces of measures, convergence of measures
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