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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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