Potential estimates for fully nonlinear elliptic equations with bounded ingredients

Edgard A. Pimentel; Miguel Walker; Edgard A. Pimentel; Miguel Walker

doi:10.3934/mine.2023063

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-16. doi: 10.3934/mine.2023063

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Potential estimates for fully nonlinear elliptic equations with bounded ingredients

Edgard A. Pimentel ^{1,2
,
,},
Miguel Walker ²

1.
Centre for Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal
2.
Pontifical Catholic University of Rio de Janeiro – PUC-Rio, 22451-900, Gávea, Rio de Janeiro-RJ, Brazil

Received: 02 September 2022 Revised: 04 November 2022 Accepted: 04 November 2022 Published: 24 November 2022

We examine $ L^p $-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $ p_0 < p < d $, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We survey recent breakthroughs in regularity theory arising from (nonlinear) potential estimates. Our findings follow from – and are inspired by – fundamental facts in the theory of $ L^p $-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi and Giuseppe Mingione ^[10].
- fully nonlinear equations,
- $ L^p $-viscosity solutions,
- potential estimates,
- gradient-regularity estimates
Citation: Edgard A. Pimentel, Miguel Walker. Potential estimates for fully nonlinear elliptic equations with bounded ingredients[J]. Mathematics in Engineering, 2023, 5(3): 1-16. doi: 10.3934/mine.2023063

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Abstract

We examine $ L^p $-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $ p_0 < p < d $, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We survey recent breakthroughs in regularity theory arising from (nonlinear) potential estimates. Our findings follow from – and are inspired by – fundamental facts in the theory of $ L^p $-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi and Giuseppe Mingione ^[10].

References

[1]	L. Boccardo, T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149–169. http://doi.org/10.1016/0022-1236(89)90005-0 doi: 10.1016/0022-1236(89)90005-0
[2]	L. Boccardo, T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Commun. Part. Diff. Eq., 17 (1992), 641–655. http://doi.org/10.1080/03605309208820857 doi: 10.1080/03605309208820857
[3]	L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. Math., 130 (1989), 189–213. http://doi.org/10.2307/1971480 doi: 10.2307/1971480
[4]	L. A. Caffarelli, X. Cabré, Fully nonlinear elliptic equations, Providence, RI: American Mathematical Society, 1995. http://doi.org/10.1090/coll/043
[5]	L. A. Caffarelli, M. G. Crandall, M. Kocan, A. Święch, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365–397. http://doi.org/10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A
[6]	A. Cianchi, Nonlinear potentials, local solutions to elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 10 (2011), 335–361. http://doi.org/10.2422/2036-2145.2011.2.04 doi: 10.2422/2036-2145.2011.2.04
[7]	M. G. Crandall, L. C. Evans, P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487–502. http://doi.org/10.1090/S0002-9947-1984-0732102-X doi: 10.1090/S0002-9947-1984-0732102-X
[8]	M. G. Crandall, H. Ishii, P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1–67. http://doi.org/10.1090/S0273-0979-1992-00266-5 doi: 10.1090/S0273-0979-1992-00266-5
[9]	M. G. Crandall, P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1–42. http://doi.org/10.1090/S0002-9947-1983-0690039-8 doi: 10.1090/S0002-9947-1983-0690039-8
[10]	P. Daskalopoulos, T. Kuusi, G. Mingione, Borderline estimates for fully nonlinear elliptic equations, Commun. Part. Diff. Eq., 39 (2014), 574–590. http://doi.org/10.1080/03605302.2013.866959 doi: 10.1080/03605302.2013.866959
[11]	C. De Filippis, Quasiconvexity and partial regularity via nonlinear potentials, J. Math. Pure. Appl., 163 (2022), 11–82. http://doi.org/10.1016/j.matpur.2022.05.001 doi: 10.1016/j.matpur.2022.05.001
[12]	C. De Filippis, G. Mingione, Lipschitz bounds and nonautonomous integrals, Arch. Rational Mech. Anal., 242 (2021), 973–1057. http://doi.org/10.1007/s00205-021-01698-5 doi: 10.1007/s00205-021-01698-5
[13]	C. De Filippis, G. Mingione, Nonuniformly elliptic schauder theory, arXiv: 2201.07369.
[14]	C. De Filippis, M. Piccinini, Borderline global regularity for nonuniformly elliptic systems, arXiv: 2206.15330.
[15]	C. De Filippis, B. Stroffolini, Singular multiple integrals and nonlinear potentials, arXiv: 2203.05519.
[16]	F. Duzaar, G. Mingione, Partial differential equations–gradient estimates in non-linear potential theory, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 20 (2009), 179–190.
[17]	F. Duzaar, G. Mingione, Gradient continuity estimates, Calc. Var., 39 (2010), 379–418. http://doi.org/10.1007/s00526-010-0314-6 doi: 10.1007/s00526-010-0314-6
[18]	F. Duzaar, G. Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal., 259 (2010), 2961–2998. http://doi.org/10.1016/j.jfa.2010.08.006 doi: 10.1016/j.jfa.2010.08.006
[19]	F. Duzaar, G. Mingione, Gradient estimates via non-linear potentials, Amer. J. Math., 133 (2011), 1093–1149.
[20]	L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413–423.
[21]	L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Commun. Pure Appl. Math., 35 (1982), 333–363. http://doi.org/10.1002/cpa.3160350303 doi: 10.1002/cpa.3160350303
[22]	E. B. Fabes, 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. http://doi.org/10.1215/S0012-7094-84-05145-7 doi: 10.1215/S0012-7094-84-05145-7
[23]	T. Kilpeläinen, J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), 591–613.
[24]	N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), 487–523.
[25]	N. V. Krylov, M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161–175.
[26]	T. Kuusi, G. Mingione, A surprising linear type estimate for nonlinear elliptic equations, C. R. Math., 349 (2011), 889–892. http://doi.org/10.1016/j.crma.2011.07.025 doi: 10.1016/j.crma.2011.07.025
[27]	T. Kuusi, G. Mingione, Pointwise gradient estimates, Nonlinear Anal. Theor., 75 (2012), 4650–4663. http://doi.org/10.1016/j.na.2011.11.021
[28]	T. Kuusi, G. Mingione, Potential estimates and gradient boundedness for nonlinear parabolic systems, Rev. Mat. Iberoam., 28 (2012), 535–576. http://doi.org/10.4171/RMI/684 doi: 10.4171/RMI/684
[29]	T. Kuusi, G. Mingione, Gradient regularity for nonlinear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 12 (2013), 755–822. http://doi.org/10.2422/2036-2145.201103_006 doi: 10.2422/2036-2145.201103_006
[30]	T. Kuusi, G. Mingione, Linear potentials in nonlinear potential theory, Arch. Rational Mech. Anal., 207 (2013), 215–246. http://doi.org/10.1007/s00205-012-0562-z doi: 10.1007/s00205-012-0562-z
[31]	T. Kuusi, G. Mingione, Borderline gradient continuity for nonlinear parabolic systems, Math. Ann., 360 (2014), 937–993. http://doi.org/10.1007/s00208-014-1055-1 doi: 10.1007/s00208-014-1055-1
[32]	T. Kuusi, G. Mingione, Guide to nonlinear potential estimates, Bull. Math. Sci., 4 (2014), 1–82. http://doi.org/10.1007/s13373-013-0048-9 doi: 10.1007/s13373-013-0048-9
[33]	T. Kuusi, G. Mingione, Riesz potentials and nonlinear parabolic equations, Arch. Rational Mech. Anal., 212 (2014), 727–780. http://doi.org/10.1007/s00205-013-0695-8 doi: 10.1007/s00205-013-0695-8
[34]	T. Kuusi, G. Mingione, The Wolff gradient bound for degenerate parabolic equations, J. Eur. Math. Soc., 16 (2014), 835–892. http://doi.org/10.4171/JEMS/449 doi: 10.4171/JEMS/449
[35]	T. Kuusi, G. Mingione, Nonlinear potential theory of elliptic systems, Nonlinear Anal., 138 (2016), 277–299. http://doi.org/10.1016/j.na.2015.12.022 doi: 10.1016/j.na.2015.12.022
[36]	T. Kuusi, G. Mingione, Partial regularity and potentials, Journal de l'École polytechnique – Mathématiques, 3 (2016), 309–363. http://doi.org/10.5802/jep.35 doi: 10.5802/jep.35
[37]	T. Kuusi, G. Mingione, Vectorial nonlinear potential theory, J. Eur. Math. Soc., 20 (2018), 929–1004. http://doi.org/10.4171/JEMS/780 doi: 10.4171/JEMS/780
[38]	G. Mingione, Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459–486. http://doi.org/10.4171/JEMS/258
[39]	G. Mingione, Nonlinear measure data problems, Milan J. Math., 79 (2011), 429–496. http://doi.org/10.1007/s00032-011-0168-1 doi: 10.1007/s00032-011-0168-1
[40]	G. Mingione, Recent advances in nonlinear potential theory, In: Trends in contemporary mathematics, Cham: Springer, 2014,277–292. http://doi.org/10.1007/978-3-319-05254-0_20
[41]	G. Mingione, Recent progress in nonlinear potential theory, In: European Congress of Mathematics, Zürich: Eur. Math. Soc., 2018,501–524.
[42]	A. Święch, $W^{1, p}$-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005–1027.
[43]	N. S. Trudinger, Hölder gradient estimates for fully nonlinear elliptic equations, P. Roy. Soc. Edinb. A, 108 (1988), 57–65. http://doi.org/10.1017/S0308210500026512 doi: 10.1017/S0308210500026512
[44]	N. S. Trudinger, X.-J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math., 124 (2002), 369–410. http://doi.org/10.1353/ajm.2002.0012 doi: 10.1353/ajm.2002.0012
[45]	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), 129–164. http://doi.org/10.4171/ZAA/1377 doi: 10.4171/ZAA/1377

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