Dini-Campanato Spaces and Applications to Nonlinear Elliptic Equations
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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 Δu = 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.