A short proof of the $C^{0,\alpha}$--regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications

Abstract Recently I. Capuzzo Dolcetta, F. Leoni and A. Porretta obtained a very surprising regularity result for fully nonlinear, superquadratic, elliptic equations by showing that viscosity subsolutions of such equations are locally Holder continuous, and even globally if the boundary of the domain is regular enough. The aim of this paper is to provide a simplified proof of their results, together with an interpretation of the regularity phenomena, some extensions and various applications.

[1]  Z. Artstein,et al.  The Value Function of Singularly Perturbed Control Systems , 2000 .

[2]  Francesca Da Lio,et al.  REMARKS ON THE STRONG MAXIMUM PRINCIPLE FOR VISCOSITY SOLUTIONS TO FULLY NONLINEAR PARABOLIC EQUATIONS , 2004 .

[3]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[4]  Guy Barles,et al.  Space-Time Periodic Solutions and Long-Time Behavior of Solutions to Quasi-linear Parabolic Equations , 2001, SIAM J. Math. Anal..

[5]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[6]  P. Lions,et al.  Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints , 1989 .

[7]  Régis Monneau,et al.  Homogenization of First-Order Equations with $$(u/\varepsilon)$$ -Periodic Hamiltonians. Part I: Local Equations , 2007 .

[8]  P. Souganidis.  Stochastic homogenization of Hamilton–Jacobi equations and some applications , 1999 .

[9]  Thierry Tabet Tchamba.  Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian , 2009, Asymptot. Anal..

[10]  H. Ishii.  Almost periodic homogenization of Hamilton-Jacobi equations , 2000 .

[11]  Guy Barles,et al.  REMARKS ON THE DIRICHLET AND STATE-CONSTRAINT PROBLEMS FOR QUASILINEAR PARABOLIC EQUATIONS , 2003 .

[12]  M. Arisawa.  Ergodic problem for the Hamilton-Jacobi-Bellman equation. I. Existence of the ergodic attractor , 1997 .

[13]  H. Ishii.  Homogenization of the Cauchy Problem for Hamilton-Jacobi Equations , 1999 .

[14]  H. Ishii,et al.  Homogenization of Hamilton-Jacobi Equations on Domains with Small Scale Periodic Structure , 1998 .

[15]  Guy Barles,et al.  A Geometrical Approach to the Study¶of Unbounded Solutions¶of Quasilinear Parabolic Equations , 2002 .

[16]  M. Bardi.  On Differential Games with Long-Time-Average Cost , 2009 .

[17]  G. Barles,et al.  Exit Time Problems in Optimal Control and Vanishing Viscosity Method , 1988 .

[18]  Martino Bardi,et al.  Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations.: I: Convex operators , 2001 .

[19]  H. Ishii.  Perron’s method for Hamilton-Jacobi equations , 1987 .

[20]  L. Evans.  Periodic homogenisation of certain fully nonlinear partial differential equations , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[21]  G. Barles,et al.  Remarks on the Dirichlet Problem for Quasilinear Elliptic and Parabolic Equations , 1999 .

[22]  H. Ishii,et al.  HAMILTON-JACOBI EQUATIONS WITH PARTIAL GRADIENT AND APPLICATION TO HOMOGENIZATION , 2001 .

[23]  O. Alvarez.  Homogenization of Hamilton-Jacobi Equations in Perforated Sets , 1999 .

[24]  L. Evans.  The perturbed test function method for viscosity solutions of nonlinear PDE , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[25]  Some homogenization results for non-coercive Hamilton–Jacobi equations , 2006, math/0605619.

[26]  Alessio Porretta,et al.  Hölder estimates for degenerate elliptic equations with coercive Hamiltonians , 2010 .

[27]  P. Lions,et al.  ON ERGODIC STOCHASTIC CONTROL , 1998 .

[28]  Guy Barles,et al.  On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations , 2004 .

[29]  E. Barron,et al.  Homogenization in L , 2002 .

[30]  P. Lions.  Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[31]  M. Arisawa.  Ergodic problem for the Hamilton-Jacobi-Bellman equation. II , 1998 .

[32]  F. D. Lio,et al.  COMPARISON RESULTS FOR QUASILINEAR EQUATIONS IN ANNULAR DOMAINS AND APPLICATIONS1* , 2002 .

[33]  G. Barles,et al.  Comparison principle for dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations , 1990 .

[34]  Martino Bardi,et al.  Propagation of maxima and Strong Maximum Principle for viscosity solutions of degenerate elliptic equations. II: Concave operators , 2003 .

[35]  Martino Bardi,et al.  Viscosity Solutions Methods for Singular Perturbations in Deterministic and Stochastic Control , 2001, SIAM J. Control. Optim..

[36]  Guy Barles,et al.  Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions , 2008 .

[37]  M. Bardi,et al.  Singular Perturbations of Nonlinear Degenerate Parabolic PDEs: a General Convergence Result , 2003 .

[38]  Martino Bardi,et al.  Propagation of Maxima and Strong Maximum Principle for Viscosity Solutions of Degenerate Elliptic Equations , 2000 .

[39]  M. Bardi,et al.  Ergodic Problems in Differential Games , 2007 .