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.