Electronic Journal of Differential Equations,
Vol. 1999(1999), No. 37, pp. 1-20.
Title: Dini-Campanato spaces and applications to nonlinear elliptic equations
Author: Jay Kovats (Florida Institute of Technology, Melbourne, FL, USA)
Abstract:
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 $D^2u$ 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 $F(D^2u,x)=f(x)$ to obtain estimates on the modulus of continuity of
$D^2u$ when the $L^n$ averages of $f$ satisfy the Dini condition.
Submitted January 6, 1999. Revised July 19, 1999. Published September 25, 1999.
Math Subject Classifications: 35B65, 41A10.
Key Words: Fully nonlinear elliptic equations; polynomial approximation;
Dini condition.