Quasiharmonic Fields: a Higher Integrability Result

Di Gironimo, Patrizia

Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 843-851

Résumé

In this paper we study the degree of integrability of quasiharmonic fields. These fields are connected with the study of the equation $\operatorname{div}(A(x)\nabla u(x))= 0$, where the symmetric matrix $A(x)$ satisfies the condition $|\xi|^2+|A(x)\xi|^2 \leq K(x)\langle A(x)\xi,\xi\rangle$.The nonnegative function $K(x)$ belongs to the exponential class, i.e. $\exp(\beta K(x))$ is integrable for some $\beta >0$. We prove that the gradient of a local solution of the equation belongs to the Zygmund spaces $L^2_{\text{loc}} \log^{\alpha - 1}L$, $0 \alpha = \alpha (\beta)$. Moreover we show exactly how the degree of improved regularity depends on $\beta$.