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Quantitative maximal $L^2$-regularity for viscous Hamilton-Jacobi PDEs in 2D and Mean Field Games

Alessandro Goffi

Abstract

We discuss quantitative Calderón-Zygmund estimates in $W^{2,2}$ for 2D viscous Hamilton-Jacobi equations with natural growth in the gradient. We apply the result to obtain the existence of classical solutions for stationary second order Mean Field Games systems in 2D with (defocusing) coupling behaving like $m^α$ for any $α>0$. We also survey on the known results for the regularity of viscous Hamilton-Jacobi equations and second order Mean Field Games and list several open problems.

Quantitative maximal $L^2$-regularity for viscous Hamilton-Jacobi PDEs in 2D and Mean Field Games

Abstract

We discuss quantitative Calderón-Zygmund estimates in for 2D viscous Hamilton-Jacobi equations with natural growth in the gradient. We apply the result to obtain the existence of classical solutions for stationary second order Mean Field Games systems in 2D with (defocusing) coupling behaving like for any . We also survey on the known results for the regularity of viscous Hamilton-Jacobi equations and second order Mean Field Games and list several open problems.
Paper Structure (12 sections, 2 theorems, 38 equations)

This paper contains 12 sections, 2 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Strong maximal regularity for viscous Hamilton-Jacobi equations in 2D
  3. Survey on the classical regularity theory for second order Mean Field Games
  4. Stationary MFGs
  5. Stationary defocusing case
  6. Stationary focusing case
  7. Logarithmic coupling
  8. Parabolic MFGs
  9. Parabolic defocusing case
  10. Parabolic focusing case
  11. Parabolic logarithmic case
  12. Applications to stationary second order Mean Field Games in two independent variables

Key Result

Theorem 2.1

Let $u\in H^1(M)\cap L^\infty(M)$ be a weak solution of HJ2d+ or HJ2d- with $f\in L^2(M)$. Then $u\in W^{2,2}(M)$ and the following quantitative regularity estimate holds

Theorems & Definitions (8)

  • Theorem 2.1
  • Remark 2.2
  • proof
  • Remark 2.3: 1D HJ equations
  • Remark 2.4
  • Theorem 4.1
  • Remark 4.2
  • proof