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 $$F(D^{2}u,Du)=f(x)\text{ in }{B}_1,$$ 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.