Pointwise properties of <inline-formula><tex-math id="M1">\begin{document} L^p \end{document}</tex-math></inline-formula>-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms

Previous Article
General initial data for a class of parabolic equations including the curve shortening problem
DCDS Home
This Issue
Next Article
Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems

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

Andrzej Świȩch ^,

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.

Keywords: Fully nonlinear uniformly elliptic equations, L^p-viscosity solutions, maximum principle, superlinear gradient terms, differential inequalities.

Mathematics Subject Classification: Primary: 35D40, 35B50, 35B65, 35J15, 35J60.

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.

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.
[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.

[1]	N. V. Krylov. Uniqueness for L_p-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2495-2516. doi: 10.3934/cpaa.2018119
[2]	Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[3]	Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897
[4]	Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[5]	Yuhua Sun. On the uniqueness of nonnegative solutions of differential inequalities with gradient terms on Riemannian manifolds. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1743-1757. doi: 10.3934/cpaa.2015.14.1743
[6]	Xiaohong Li, Fengquan Li. Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities. Communications on Pure and Applied Analysis, 2012, 11 (3) : 935-943. doi: 10.3934/cpaa.2012.11.935
[7]	Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539
[8]	Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897
[9]	Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747
[10]	Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019
[11]	Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771
[12]	Evgeny Galakhov, Olga Salieva. Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets. Conference Publications, 2015, 2015 (special) : 489-494. doi: 10.3934/proc.2015.0489
[13]	Feliz Minhós. Periodic solutions for some fully nonlinear fourth order differential equations. Conference Publications, 2011, 2011 (Special) : 1068-1077. doi: 10.3934/proc.2011.2011.1068
[14]	Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763
[15]	Chuanqiang Chen. On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4761-4811. doi: 10.3934/dcds.2016007
[16]	Chuanqiang Chen. On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3383-3402. doi: 10.3934/dcds.2014.34.3383
[17]	Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 933-982. doi: 10.3934/dcds.2020067
[18]	Ryuji Kajikiya, Daisuke Naimen. Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1593-1612. doi: 10.3934/cpaa.2014.13.1593
[19]	Rainer Buckdahn, Christian Keller, Jin Ma, Jianfeng Zhang. Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 7-. doi: 10.1186/s41546-020-00049-8
[20]	Ibrahim Ekren, Jianfeng Zhang. Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 6-. doi: 10.1186/s41546-016-0010-3

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Andrzej Świȩch
on Google Scholar
Andrzej Świȩch

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Andrzej Świȩch
on Google Scholar
Andrzej Świȩch

[Back to Top]