# zbMATH — the first resource for mathematics

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.

##### MSC:
 35J70 Degenerate elliptic equations 35J60 Nonlinear elliptic equations 35D10 Regularity of generalized solutions of PDE (MSC2000)
Full Text: