Partial regularity of solutions of fully nonlinear, uniformly elliptic equations (Q2903844)

The main result in the present paper establishes that a viscosity solution of a uniformly elliptic, fully nonlinear equation is of class \(C^{2,\alpha}\) on the compliment of a closed set of Hausdorff dimension at most \(\varepsilon\) less than the dimension. The equation is assumed to be \(C^1\) and the constant \(\varepsilon > 0\) depending only on the dimension and the ellipticity constants. The first key argument in the proof asserts that any viscosity solution of that is sufficiently close to a quadratic polynomial must be of class \(C^{2,\alpha}\). Second, the \(L^\varepsilon\) integrability of the modulus of the quadratic expansion then restricts the Hausdorff dimension of the singular set.