Estimates on elliptic equations that hold only where the Hessian is large

In this article, we establish Hölder regularity for viscosity solutions to a class of degenerate fully nonlinear elliptic equations of the form \[ F(D^2u,Du)=f(x)~~\text{in}~~B_1, \] where the operator is elliptic only in regions where the Hessian is sufficiently large.

Such equations arise naturally in free boundary problems and models with partial ellipticity.

The proof combines a modified cusp function with a decomposition of the contact set in a point-to-measure argument.

As a consequence, interior Hölder continuity follows under natural structural assumptions.