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

2026/07/19 by Amit Kumar Acharya, Ram Baran Verma
#math.AP

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Abstract

In this article, we establish Hölder regularity for viscosity solutions to a class of degenerate fully nonlinear elliptic equations of the form F(D²u,Du)=f(x)~~in~~B₁, 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.

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