A note on the density of the partial regularity result in the class of viscosity solutions

Authors

  • Disson dos Prazeres Department of Mathematics UFS, 49100-000 São Cristóvão-SE, Brazil
  • Edgard A. Pimentel Department of Mathematics, University of Coimbra – CMUC, 3001-501 Coimbra, Portugal and Department of Mathematics, Pontifical Catholic University of Rio de Janeiro – PUC-Rio, 22451-900, Gávea, Rio de Janeiro-RJ, Brazil
  • Giane C. Rampasso Instituto de Matemática e Computação, Universidade Federal de Itajubá – UNIFEI, Campus Prof. José Rodrigues Seabra, Itajubá-MG, Brasil

DOI:

https://doi.org/10.33044/revuma.2358

Abstract

We establish the density of the partial regularity result in the class of continuous viscosity solutions. Given a fully nonlinear equation, we prove the existence of a sequence entitled to the partial regularity result, approximating its solutions. Distinct conditions on the operator driving the equation lead to density in different topologies. Our findings include applications to nonhomogeneous problems, with variable-coefficient models.

Downloads

Download data is not yet available.

References

S. N. Armstrong, L. E. Silvestre and C. K. Smart, Partial regularity of solutions of fully nonlinear, uniformly elliptic equations, Comm. Pure Appl. Math. 65 (2012), no. 8, 1169–1184. MR 2928094.

E. N. Barron, L. C. Evans and R. Jensen, Viscosity solutions of Isaacs' equations and differential games with Lipschitz controls, J. Differential Equations 53 (1984), no. 2, 213–233. MR 0748240.

X. Cabré and L. A. Caffarelli, Interior $C^{2,alpha}$ regularity theory for a class of nonconvex fully nonlinear elliptic equations, J. Math. Pures Appl. (9) 82 (2003), no. 5, 573–612. MR 1995493.

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189–213. MR 1005611.

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. MR 1351007.

L. Caffarelli and L. Silvestre, Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations, in Nonlinear Partial Differential Equations and Related Topics, 67–85, Amer. Math. Soc. Transl. Ser. 2, 229, Amer. Math. Soc., Providence, RI, 2010. MR 2667633.

D. dos Prazeres and E. V. Teixeira, Asymptotics and regularity of flat solutions to fully nonlinear elliptic problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 485–500. MR 3495436.

L. Escauriaza, $W^{2,n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J. 42 (1993), no. 2, 413–423. MR 1237053.

L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363. MR 0649348.

L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton–Jacobi–Isaacs equations, Indiana Univ. Math. J. 33 (1984), no. 5, 773–797. MR 0756158.

W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Univ. Math. J. 38 (1989), no. 2, 293–314. MR 0997385.

R. Hardt, D. Kinderlehrer and F.-H. Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys. 105 (1986), no. 4, 547–570. MR 0852090.

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487–523, 670. MR 0661144.

N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175, 239. MR 0563790.

G. Leoni, A First Course in Sobolev Spaces, Graduate Studies in Mathematics, 105, American Mathematical Society, Providence, RI, 2009. MR 2527916.

N. Nadirashvili and S. Vlăduţ, Nonclassical solutions of fully nonlinear elliptic equations, Geom. Funct. Anal. 17 (2007), no. 4, 1283–1296. MR 2373018.

N. Nadirashvili and S. Vlăduţ, Singular viscosity solutions to fully nonlinear elliptic equations, J. Math. Pures Appl. (9) 89 (2008), no. 2, 107–113. MR 2391642.

N. Nadirashvili and S. Vlăduţ, Octonions and singular solutions of Hessian elliptic equations, Geom. Funct. Anal. 21 (2011), no. 2, 483–498. MR 2795515.

N. Nadirashvili and S. Vlăduţ, Singular solutions of Hessian fully nonlinear elliptic equations, Adv. Math. 228 (2011), no. 3, 1718–1741. MR 2824567.

E. A. Pimentel, Regularity theory for the Isaacs equation through approximation methods, Ann. Inst. H. Poincaré C Anal. Non Linéaire 36 (2019), no. 1, 53–74. MR 3906865.

E. A. Pimentel and M. S. Santos, Asymptotic methods in regularity theory for nonlinear elliptic equations: a survey, in PDE Models for Multi-Agent Phenomena, 167–194, Springer INdAM Ser., 28, Springer, Cham, 2018. MR 3888972.

E. A. Pimentel and A. Święch, Existence of solutions to a fully nonlinear free transmission problem, J. Differential Equations 320 (2022), 49–63. MR 4390963.

E. A. Pimentel and E. V. Teixeira, Sharp Hessian integrability estimates for nonlinear elliptic equations: an asymptotic approach, J. Math. Pures Appl. (9) 106 (2016), no. 4, 744–767. MR 3539473.

O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations 32 (2007), no. 4-6, 557–578. MR 2334822.

L. Silvestre and E. V. Teixeira, Regularity estimates for fully non linear elliptic equations which are asymptotically convex, in Contributions to Nonlinear Elliptic Equations and Systems, 425–438, Progr. Nonlinear Differential Equations Appl., 86, Birkhäuser/Springer, Cham, 2015. MR 3494911.

A. Świech, $W^{1,p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations 2 (1997), no. 6, 1005–1027. MR 1606359.

N. S. Trudinger, On the twice differentiability of viscosity solutions of nonlinear elliptic equations, Bull. Austral. Math. Soc. 39 (1989), no. 3, 443–447. MR 0995142.

N. Winter, $W^{2,p}$ and $W^{1,p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend. 28 (2009), no. 2, 129–164. MR 2486925.

Downloads

Published

2022-10-24