Quasiharmonic Fields: a Higher Integrability Result

Di Gironimo, Patrizia

Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 843-851 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

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 $\beta$.

Résumé

In questo lavoro si studia il grado di integrabilità dei campi quasi-armonici. Questi campi sono connessi con lo studio dell'equazione $\operatorname{div}(A(x)\nabla u(x))= 0$, dove la matrice simmetrica $A(x)$ soddisfa la condizione $|\xi|^2+|A(x)\xi|^2 \leq K(x)\langle A(x)\xi,\xi\rangle$. La funzione non negativa $K(x)$ appartiene alla classe esponenziale, cioé esiste $\beta >0$ tale che $\exp(\beta K(x))$ è integrabile. Si dimostra che il gradiente di una soluzione locale dell'equazione appartiene agli spazi di Zygmund $L^2_{\text{loc}} \log^{\alpha - 1}L$, $0 < \alpha = \alpha (\beta)$. Inoltre si prova come il grado di migliore regolarità dipende da $\beta$.

Détail

Publié le : 2007-10-01

MR 2507900 | Zbl 1184.35134

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