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On very weak solutions of certain elliptic systems. (English) Zbl 0796.35061
Consider a second order degenerate elliptic system whose principal part is given by the \(p\)-Laplacian \(\partial_ \alpha (| Du |^{p-2} \partial_ \alpha u)\). Problems of this type are usually studied in the Sobolev-space \(H^{1,p} (\Omega,\mathbb{R}^ N)\). The author shows that it is sufficient to assume \(u \in H^{1,p - \delta} (\Omega,\mathbb{R}^ N)\) for some \(\delta>0\) depending on the data in order to deduce \(u \in H^{1,p + \delta} (\Omega,\mathbb{R}^ N)\) which means that \(u\) is a classical weak solution. A similar result is obtained for systems of higher order. The proof makes use of Whitney extension theorem and Gehring’s reverse Hölder inequality.

35J70 Degenerate elliptic equations
35J60 Nonlinear elliptic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
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