Dini-Campanato spaces and applications to nonlinear elliptic equations

Kovats, Jay

Electronic Journal of Differential Equations, Tome 1999 (1999).

Résumé

Summary: We generalize a result due to Campanato [C] and use this to obtain regularity results for classical solutions of fully nonlinear elliptic equations. We demonstrate this technique in two settings. First, in the simplest setting of Poisson's equation $\Delta u=f$ in $B$, where $f$ is Dini continuous in $B$, we obtain known estimates on the modulus of continuity of second derivatives $D^{2}$u in a way that does not depend on either differentiating the equation or appealing to integral representations of solutions. Second, we use this result in the concave, fully nonlinear setting $F(D^{2}$u,x)=$f(x)$ to obtain estimates on the modulus of continuity of $D^{2}$u when the $L^{n}$ averages of $f$ satisfy the Dini condition.

Détail

Classification : 35B65, 41A10
Keywords: fully nonlinear elliptic equations, polynomial approximation, dini condition