# Regularity Bootstrapping For Fourth Order Non Linear Elliptic Equations

## TL;DR

This paper establishes interior derivative estimates for solutions of certain nonlinear fourth order elliptic equations of double divergence type, where the nonlinearity depends on the Hessian, enhancing understanding of their regularity.

## Contribution

It introduces regularity results for solutions of nonlinear fourth order elliptic equations with Hessian-dependent nonlinearities, expanding the theory of higher-order elliptic PDEs.

## Key findings

- Solutions in C^{2,alpha} have interior estimates on all derivatives.
- The results apply to a specific class of double divergence type equations.
- Regularity theory is extended for Hessian-dependent nonlinearities.

## Abstract

We consider nonlinear fourth order elliptic equations of double divergence type. We show that for a certain class of equations where the nonlinearity is in the Hessian, solutions that are C^{2,alpha} enjoy interior estimates on all derivatives.