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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 & Continuous Dynamical Systems - A, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] K. Fok, A nonlinear Fabes-Stroock result, Comm. Partial Differential Equations, 23 (1998), 967-983.  doi: 10.1080/03605309808821375.  Google Scholar [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [27] A. Schikorra, A remark on $C^{1, \alpha}$-regularity for differential inequalities in viscosity sense, preprint, (2018), arXiv: 1811.00376v2. Google Scholar [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.  Google Scholar [29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.   Google Scholar [30] A. Świȩch, $W^{1, p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] K. Fok, A nonlinear Fabes-Stroock result, Comm. Partial Differential Equations, 23 (1998), 967-983.  doi: 10.1080/03605309808821375.  Google Scholar [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [27] A. Schikorra, A remark on $C^{1, \alpha}$-regularity for differential inequalities in viscosity sense, preprint, (2018), arXiv: 1811.00376v2. Google Scholar [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.  Google Scholar [29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.   Google Scholar [30] A. Świȩch, $W^{1, p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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