Electron. J. Diff. Eqns., Vol. 1999(1999), No. 37, pp. 1-20.
### Dini-Campanato spaces and applications to nonlinear elliptic equations

Jay Kovats

**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 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.
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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