Electron. J. Diff. Eqns., Vol. 1999(1999), No. 37, pp. 1-20.

Dini-Campanato spaces and applications to nonlinear elliptic equations

Jay Kovats

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 D2u 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(D2u,x)=f(x) to obtain estimates on the modulus of continuity of D2u when the Ln 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.

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Jay Kovats
Department of Mathematical Sciences
Florida Institute of Technology
Melbourne, FL 32901, USA
e-mail address: jkovats@zach.fit.edu
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