On the optimal $L^q$-regularity for viscous Hamilton-Jacobi equations with subquadratic growth in the gradient

TL;DR

This paper establishes optimal $L^q$-regularity results for nonlinear elliptic equations with gradient nonlinearities exhibiting sub-quadratic growth, using duality and interpolation techniques.

Contribution

It introduces a novel approach combining linear elliptic regularity and duality methods to analyze maximal $L^q$-regularity for equations with zero-th order terms and sub-quadratic gradient growth.

Findings

Derived optimal $L^q$-regularity estimates for the class of equations.

Analyzed both global and local regularity properties of solutions.

Identified the critical summability threshold $q=d(eta-1)/eta$.

Abstract

This paper studies a maximal $L^q$-regularity property for nonlinear elliptic equations of second order with a zero-th order term and gradient nonlinearities having superlinear and sub-quadratic growth, complemented with Dirichlet boundary conditions. The approach is based on the combination of linear elliptic regularity theory and interpolation inequalities, so that the analysis of the maximal regularity estimates boils down to determine lower order integral bounds. The latter are achieved via a $L^p$ duality method, which exploits the regularity properties of solutions to stationary Fokker-Planck equations. For the latter problems, we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic boundary-value problems are the treatment of equations with a zero-th order term together with the analysis of the end-point summability threshold $q=d(\gamma-1)/\gamma$, $d$ being the dimension of the ambient space and $\gamma>1$ the growth of the first-order term in the gradient variable.