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May  2020, 40(5): 2945-2962. doi: 10.3934/dcds.2020156

Pointwise properties of $L^p$-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms

 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

* Corresponding author: Andrzej Świȩch

Received  September 2019 Published  March 2020

We prove various pointwise properties of $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations with unbounded measurable terms and possibly quadratically growing gradient terms. These include differentiability properties and "pointwise maximum principle". In particular we discuss an equivalent pointwise definition of $L^p$-viscosity solution which, together with pointwise properties, allows to operate on $L^p$-viscosity solutions almost as if they were strong solutions and move them freely from one equation to another. Such results were proved before in [6,8,30] for equations with linearly growing gradient terms, however it appears that they have not been widely used. Here we generalize pointwise properties of $L^p$-viscosity solutions to a larger class of equations and show that they create a very powerful set of tools which can be used for instance to prove regularity results for equations and differential inequalities.

Citation: Andrzej Świȩch. Pointwise properties of $L^p$-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2945-2962. doi: 10.3934/dcds.2020156
References:
 [1] I. Birindelli, F. Demengel and F. Leoni, $C^{1, \gamma}$ regularity for singular or degenerate fully nonlinear equations and applications, NoDEA Nonlinear Differential Equations Appl., 26 (2019), Art. 40, 13 pp. doi: 10.1007/s00030-019-0586-2. [2] J. E. M. Braga and D. Moreira, Zero Lebesgue measure sets as removable sets for degenerate fully nonlinear elliptic PDEs, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 11, 12 pp. doi: 10.1007/s00030-018-0499-5. [3] J. E. M. Braga, A. Figalli and D. Moreira, Optimal regularity for the convex envelope and semiconvex functions related to supersolutions of fully nonlinear elliptic equations, Comm. Math. Phys., 367 (2019), 1-32.  doi: 10.1007/s00220-019-03370-2. [4] L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math.(2), 130 (1989), 189-213.  doi: 10.2307/1971480. [5] L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043. [6] L. A. Caffarelli, M. G. Crandall, M. Kocan and A. \'Swiȩch, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), 365-397.  doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. [7] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc.(N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [8] M. G. Crandall, M. Kocan, P. Soravia and A. Świȩch, On the equivalence of various weak notions of solutions of elliptic PDEs with measurable ingredients, Progress in Elliptic and Parabolic Partial Differential Equations, Pitman Res. Notes Math. Ser., Longman, Harlow, 350 (1996), 136-162. [9] M. G. Crandall, M. Kocan and A. Świȩch, $L^p$-theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations, 25 (2000), 1997-2053.  doi: 10.1080/03605300008821576. [10] M. G. Crandall and A. Świȩch, A note on generalized maximum principles for elliptic and parabolic PDE, Evolution Equations, Lecture Notes in Pure and Appl. Math., Dekker, New York, 234 (2003), 121-127.  doi: 10.1177/0008068320030111. [11] L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully non-linear equations, Indiana Univ. Math. J., 42 (1993), 413-423.  doi: 10.1512/iumj.1993.42.42019. [12] E. B. Fabes and D. W. Stroock, The $L^p$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J., 51 (1984), 997-1016.  doi: 10.1215/S0012-7094-84-05145-7. [13] P.-K. Fok, Some Maximum Principles and Continuity Estimates for Fully Nonlinear Elliptic Equations of Second Order, Ph.D thesis, University of California, Santa Barbara, 1996, 82 pp. [14] K. Fok, A nonlinear Fabes-Stroock result, Comm. Partial Differential Equations, 23 (1998), 967-983.  doi: 10.1080/03605309808821375. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [16] R. Jensen and A. Świȩch, Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE, Commun. Pure Appl. Anal., 4 (2005), 199-207.  doi: 10.3934/cpaa.2005.4.187. [17] S. Koike and K. Nakagawa, Remarks on the Phragmén-Lindelöf theorem for $L^p$-viscosity solutions of fully nonlinear PDEs with unbounded ingredients, Electron. J. Differential Equations, 2009 (2009), 14 pp. [18] S. Koike and A. Świȩch, Maximum principle and existence of $L^p$-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 491-509.  doi: 10.1007/s00030-004-2001-9. [19] S. Koike and A. Świȩch, Maximum principle for fully nonlinear equations via the iterated comparison function method, Math. Ann., 339 (2007), 461-484.  doi: 10.1007/s00208-007-0125-z. [20] S. Koike and A. Świȩch, Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan, 61 (2009), 723-755.  doi: 10.2969/jmsj/06130723. [21] S. Koike and A. Świȩch, Existence of strong solutions of Pucci extremal equations with superlinear growth in $Du$, J. Fixed Point Theory Appl., 5 (2009), 291-304.  doi: 10.1007/s11784-009-0106-9. [22] N. V. Krylov, Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations, Mathematical Surveys and Monographs, 233, American Mathematical Society, Providence, RI, 2018. [23] P.-L. Lions, A remark on Bony maximum principle, Proc. Amer. Math. Soc., 88 (1983), 503-508.  doi: 10.1090/S0002-9939-1983-0699422-3. [24] K. Nakagawa, Maximum principle for $L^p$-viscosity solutions of fully nonlinear equations with unbounded ingredients and superlinear growth terms, Adv. Math. Sci. Appl., 19 (2009), 89-107. [25] G. Nornberg, $C^{1, \alpha}$ regularity for fully nonlinear elliptic equations with superlinear growth in the gradient, J. Math. Pures Appl.(9), 128 (2019), 297-329.  doi: 10.1016/j.matpur.2019.06.008. [26] G. Nornberg and B. Sirakov, A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient, J. Funct. Anal., 276 (2019), 1806-1852.  doi: 10.1016/j.jfa.2018.06.017. [27] A. Schikorra, A remark on $C^{1, \alpha}$-regularity for differential inequalities in viscosity sense, preprint, (2018), arXiv: 1811.00376v2. [28] B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Ration. Mech. Anal., 195 (2010), 579-607.  doi: 10.1007/s00205-009-0218-9. [29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970. [30] A. Świȩch, $W^{1, p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027. [31] A. Świȩch, A note on the upper perturbation property and removable sets for fully nonlinear degenerate elliptic PDE, NoDEA Nonlinear Differential Equations Appl., 26 (2019), Art. 3, 4 pp. doi: 10.1007/s00030-018-0547-1. [32] N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Rev. Mat. Iberoamericana, 4 (1988), 453-468.  doi: 10.4171/RMI/80. [33] N. S. Trudinger, On the twice differentiability of viscosity solutions of nonlinear elliptic equations, Bull. Austral. Math. Soc., 39 (1989), 443-447.  doi: 10.1017/S0004972700003361. [34] L. H. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.  doi: 10.1002/cpa.3160450103. [35] L. H. Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math., 45 (1992), 141-178.  doi: 10.1002/cpa.3160450202.

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References:
 [1] I. Birindelli, F. Demengel and F. Leoni, $C^{1, \gamma}$ regularity for singular or degenerate fully nonlinear equations and applications, NoDEA Nonlinear Differential Equations Appl., 26 (2019), Art. 40, 13 pp. doi: 10.1007/s00030-019-0586-2. [2] J. E. M. Braga and D. Moreira, Zero Lebesgue measure sets as removable sets for degenerate fully nonlinear elliptic PDEs, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 11, 12 pp. doi: 10.1007/s00030-018-0499-5. [3] J. E. M. Braga, A. Figalli and D. Moreira, Optimal regularity for the convex envelope and semiconvex functions related to supersolutions of fully nonlinear elliptic equations, Comm. Math. Phys., 367 (2019), 1-32.  doi: 10.1007/s00220-019-03370-2. [4] L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math.(2), 130 (1989), 189-213.  doi: 10.2307/1971480. [5] L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. doi: 10.1090/coll/043. [6] L. A. Caffarelli, M. G. Crandall, M. Kocan and A. \'Swiȩch, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), 365-397.  doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. [7] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc.(N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [8] M. G. Crandall, M. Kocan, P. Soravia and A. Świȩch, On the equivalence of various weak notions of solutions of elliptic PDEs with measurable ingredients, Progress in Elliptic and Parabolic Partial Differential Equations, Pitman Res. Notes Math. Ser., Longman, Harlow, 350 (1996), 136-162. [9] M. G. Crandall, M. Kocan and A. Świȩch, $L^p$-theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations, 25 (2000), 1997-2053.  doi: 10.1080/03605300008821576. [10] M. G. Crandall and A. Świȩch, A note on generalized maximum principles for elliptic and parabolic PDE, Evolution Equations, Lecture Notes in Pure and Appl. Math., Dekker, New York, 234 (2003), 121-127.  doi: 10.1177/0008068320030111. [11] L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully non-linear equations, Indiana Univ. Math. J., 42 (1993), 413-423.  doi: 10.1512/iumj.1993.42.42019. [12] E. B. Fabes and D. W. Stroock, The $L^p$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J., 51 (1984), 997-1016.  doi: 10.1215/S0012-7094-84-05145-7. [13] P.-K. Fok, Some Maximum Principles and Continuity Estimates for Fully Nonlinear Elliptic Equations of Second Order, Ph.D thesis, University of California, Santa Barbara, 1996, 82 pp. [14] K. Fok, A nonlinear Fabes-Stroock result, Comm. Partial Differential Equations, 23 (1998), 967-983.  doi: 10.1080/03605309808821375. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [16] R. Jensen and A. Świȩch, Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE, Commun. Pure Appl. Anal., 4 (2005), 199-207.  doi: 10.3934/cpaa.2005.4.187. [17] S. Koike and K. Nakagawa, Remarks on the Phragmén-Lindelöf theorem for $L^p$-viscosity solutions of fully nonlinear PDEs with unbounded ingredients, Electron. J. Differential Equations, 2009 (2009), 14 pp. [18] S. Koike and A. Świȩch, Maximum principle and existence of $L^p$-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 491-509.  doi: 10.1007/s00030-004-2001-9. [19] S. Koike and A. Świȩch, Maximum principle for fully nonlinear equations via the iterated comparison function method, Math. Ann., 339 (2007), 461-484.  doi: 10.1007/s00208-007-0125-z. [20] S. Koike and A. Świȩch, Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan, 61 (2009), 723-755.  doi: 10.2969/jmsj/06130723. [21] S. Koike and A. Świȩch, Existence of strong solutions of Pucci extremal equations with superlinear growth in $Du$, J. Fixed Point Theory Appl., 5 (2009), 291-304.  doi: 10.1007/s11784-009-0106-9. [22] N. V. Krylov, Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations, Mathematical Surveys and Monographs, 233, American Mathematical Society, Providence, RI, 2018. [23] P.-L. Lions, A remark on Bony maximum principle, Proc. Amer. Math. Soc., 88 (1983), 503-508.  doi: 10.1090/S0002-9939-1983-0699422-3. [24] K. Nakagawa, Maximum principle for $L^p$-viscosity solutions of fully nonlinear equations with unbounded ingredients and superlinear growth terms, Adv. Math. Sci. Appl., 19 (2009), 89-107. [25] G. Nornberg, $C^{1, \alpha}$ regularity for fully nonlinear elliptic equations with superlinear growth in the gradient, J. Math. Pures Appl.(9), 128 (2019), 297-329.  doi: 10.1016/j.matpur.2019.06.008. [26] G. Nornberg and B. Sirakov, A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient, J. Funct. Anal., 276 (2019), 1806-1852.  doi: 10.1016/j.jfa.2018.06.017. [27] A. Schikorra, A remark on $C^{1, \alpha}$-regularity for differential inequalities in viscosity sense, preprint, (2018), arXiv: 1811.00376v2. [28] B. Sirakov, Solvability of uniformly elliptic fully nonlinear PDE, Arch. Ration. Mech. Anal., 195 (2010), 579-607.  doi: 10.1007/s00205-009-0218-9. [29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970. [30] A. Świȩch, $W^{1, p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027. [31] A. Świȩch, A note on the upper perturbation property and removable sets for fully nonlinear degenerate elliptic PDE, NoDEA Nonlinear Differential Equations Appl., 26 (2019), Art. 3, 4 pp. doi: 10.1007/s00030-018-0547-1. [32] N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Rev. Mat. Iberoamericana, 4 (1988), 453-468.  doi: 10.4171/RMI/80. [33] N. S. Trudinger, On the twice differentiability of viscosity solutions of nonlinear elliptic equations, Bull. Austral. Math. Soc., 39 (1989), 443-447.  doi: 10.1017/S0004972700003361. [34] L. H. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.  doi: 10.1002/cpa.3160450103. [35] L. H. Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math., 45 (1992), 141-178.  doi: 10.1002/cpa.3160450202.
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