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

Andrzej Świȩch

doi:10.3934/dcds.2020156

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 & 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

[1]	Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[2]	Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020469
[3]	Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050
[4]	Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020110
[5]	Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137
[6]	Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020048
[7]	Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227
[8]	Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439
[9]	Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561
[10]	Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1579-1613. doi: 10.3934/dcdsb.2020174
[11]	Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180
[12]	Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115
[13]	Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020436
[14]	Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128
[15]	Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229
[16]	Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318
[17]	Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280
[18]	Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020462
[19]	Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020398
[20]	Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors