Regularity properties for a class of non-uniformly elliptic Isaacs operators. (English) Zbl 1437.35294
This paper deals with the thorough mathematical analysis of a class of elliptic differential operators defined as the sum of the minimum and the maximum eigenvalue of the Hessian matrix. The operator considered in this paper can be also viewed as a degenerate differential elliptic Isaacs operator, in dimension larger than two. The main results establish the existence and uniqueness for the Dirichlet problem, as well as local and global Hölder estimates for viscosity solutions. The authors also discuss related qualitative properties, including the strong maximum principle and Liouville-type theorems.
Reviewer: Vicenţiu D. Rădulescu (Craiova)
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35J70 | Degenerate elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35B50 | Maximum principles in context of PDEs |
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35D40 | Viscosity solutions to PDEs |
Keywords:
weighted partial trace operators; Bellman-Isaacs equations; viscosity solutions; global Hölder estimatesReferences:
| [1] | A. D. Aleksandrov, Certain estimates for the Dirichlet problem (in Russian), Dokl. Akad. Nauk. SSSR 134 (1960), 1001-1004; translation in Soviet Math. Dokl. 1 (1960), 1151-1154. · Zbl 0100.09303 |
| [2] | A. D. Aleksandrov, Uniqueness conditions and bounds for the solution of the Dirichlet problem (in Russian), Vestnik Leningrad. Univ. Ser. Mat. Meh. Astronom. 18 (1963), no. 3, 5-29; translation in Amer. Math. Soc. Transl. (2) 68 (1968), 89-119. · Zbl 0139.05702 |
| [3] | M. E. Amendola, G. Galise and A. Vitolo, Riesz capacity, maximum principle, and removable sets of fully nonlinear second-order elliptic operators, Differential Integral Equations 26 (2013), no. 7-8, 845-866. · Zbl 1299.35120 |
| [4] | M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second-order elliptic equations in unbounded domains, Abstr. Appl. Anal. 2008 (2008), Article ID 178534. · Zbl 1187.35065 |
| [5] | I. J. Bakel’man, On the theory of quasilinear elliptic equations (in Russian), Sibirsk. Mat. Z. 2 (1961), 179-186. · Zbl 0100.30503 |
| [6] | M. Bardi and F. Da Lio, Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. I. Convex operators, Nonlinear Anal. 44 (2001), no. 8, 991-1006. · Zbl 1135.35342 |
| [7] | M. Bardi and F. Da Lio, Propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations. II. Concave operators, Indiana Univ. Math. J. 52 (2003), no. 3, 607-627. · Zbl 1135.35343 |
| [8] | M. Bardi and P. Mannucci, On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations, Commun. Pure Appl. Anal. 5 (2006), no. 4, 709-731. · Zbl 1142.35041 |
| [9] | H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994), no. 1, 47-92. · Zbl 0806.35129 |
| [10] | I. Birindelli, I. Capuzzo Dolcetta and A. Vitolo, ABP and global Hölder estimates for fully nonlinear elliptic equations in unbounded domains, Commun. Contemp. Math. 18 (2016), no. 4, Article ID 1550075. · Zbl 1344.35036 |
| [11] | I. Birindelli and F. Demengel, \mathscr{C}^{1,\beta} regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations, ESAIM Control Optim. Calc. Var. 20 (2014), no. 4, 1009-1024. · Zbl 1315.35108 |
| [12] | I. Birindelli, G. Galise and H. Ishii, A family of degenerate elliptic operators: maximum principle and its consequences, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 2, 417-441. · Zbl 1390.35079 |
| [13] | I. Birindelli, G. Galise and F. Leoni, Liouville theorems for a family of very degenerate elliptic nonlinear operators, Nonlinear Anal. 161 (2017), 198-211. · Zbl 1375.35171 |
| [14] | P. Blanc, C. Esteve and J. D. Rossi, The evolution problem associated with eigenvalues of the Hessian, preprint (2019), . |
| [15] | P. Blanc and J. D. Rossi, Games for eigenvalues of the Hessian and concave/convex envelopes, J. Math. Pures Appl. (9) 127 (2019), 192-215. · Zbl 1423.35118 |
| [16] | R. Buckdahn, P. Cardaliaguet and M. Quincampoix, Some recent aspects of differential game theory, Dyn. Games Appl. 1 (2011), no. 1, 74-114. · Zbl 1214.91013 |
| [17] | R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations, SIAM J. Control Optim. 47 (2008), no. 1, 444-475. · Zbl 1157.93040 |
| [18] | X. Cabré, On the Alexandroff-Bakel’man-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 48 (1995), no. 5, 539-570. · Zbl 0828.35017 |
| [19] | X. Cabré and L. A. Caffarelli, Regularity for viscosity solutions of fully nonlinear equations F(D^2u)=0, Topol. Methods Nonlinear Anal. 6 (1995), no. 1, 31-48. · Zbl 0866.35025 |
| [20] | V. Cafagna and A. Vitolo, On the maximum principle for second-order elliptic operators in unbounded domains, C. R. Math. Acad. Sci. Paris 334 (2002), no. 5, 359-363. · Zbl 1004.35039 |
| [21] | L. Caffarelli, M. G. Crandall, M. Kocan and A. Świȩch, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 (1996), no. 4, 365-397. · Zbl 0854.35032 |
| [22] | L. Caffarelli, Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations III: viscosity solutions including parabolic operators, Comm. Pure Appl. Math. 66 (2013), no. 1, 109-143. · Zbl 1279.35044 |
| [23] | L. Caffarelli, Y. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. I, J. Fixed Point Theory Appl. 5 (2009), no. 2, 353-395. · Zbl 1215.35068 |
| [24] | L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189-213. · Zbl 0692.35017 |
| [25] | L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ. 43, American Mathematical Society, Providence, 1995. · Zbl 0834.35002 |
| [26] | I. Capuzzo Dolcetta, F. Leoni and A. Vitolo, On the inequality F(x,D^2u)\geq f(u)+g(u)|Du|^q, Math. Ann. 365 (2016), no. 1-2, 423-448. · Zbl 1342.35116 |
| [27] | I. Capuzzo Dolcetta and A. Vitolo, C^{1,\alpha} and Glaeser type estimates, Rend. Mat. Appl. (7) 29 (2009), no. 1, 17-27. · Zbl 1196.35061 |
| [28] | I. Capuzzo Dolcetta and A. Vitolo, Glaeser’s type gradient estimates for non-negative solutions of fully nonlinear elliptic equations, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 539-557. · Zbl 1193.35043 |
| [29] | I. Capuzzo Dolcetta and A. Vitolo, The weak maximum principle for degenerate elliptic operators in unbounded domains, Int. Math. Res. Not. IMRN 2018 (2018), no. 2, 412-431. · Zbl 1407.35038 |
| [30] | I. Capuzzo Dolcetta and A. Vitolo, Directional ellipticity on special domains: Weak maximum and Phragmén-Lindelöf principles, Nonlinear Anal. 184 (2019), 69-82. · Zbl 1421.35105 |
| [31] | S. Cho and M. Safonov, Hölder regularity of solutions to second-order elliptic equations in nonsmooth domains, Bound. Value Probl. 2007 (2007), Article ID 57928. · Zbl 1159.35019 |
| [32] | M. G. Crandall, Viscosity solutions: A primer, Viscosity Solutions and Applications, Lecture Notes in Math. 1660, Springer, Berlin (1997), 1-47. · Zbl 0901.49026 |
| [33] | 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), no. 1, 1-67. · Zbl 0755.35015 |
| [34] | M. G. Crandall and A. Świȩch, A note on generalized maximum principles for elliptic and parabolic PDE, Evolution Equations, Lecture Notes Pure Appl. Math. 234, Dekker, New York (2003), 121-127. · Zbl 1039.35024 |
| [35] | A. Cutrì and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 2, 219-245. · Zbl 0956.35035 |
| [36] | F. Da Lio and B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 2, 317-330. · Zbl 1176.35068 |
| [37] | F. Ferrari, An application of the theorem on Sums to viscosity solutions of degenerate fully nonlinear equations, Proc. Roy. Soc. Edinburgh Sect. A (2019), 10.1017/prm.2018.147. · Zbl 1451.35059 · doi:10.1017/prm.2018.147 |
| [38] | F. Ferrari, Q. Liu and J. Manfredi, On the characterization of p-harmonic functions on the Heisenberg group by mean value properties, Discrete Contin. Dyn. Syst. 34 (2014), no. 7, 2779-2793. · Zbl 1293.35350 |
| [39] | F. Ferrari and A. Pinamonti, Characterization by asymptotic mean formulas of q-harmonic functions in Carnot groups, Potential Anal. 42 (2015), no. 1, 203-227. · Zbl 1325.35049 |
| [40] | F. Ferrari and E. Vecchi, Hölder behavior of viscosity solutions of some fully nonlinear equations in the Heisenberg group, Topol. Methods Nonlinear Anal., to appear. · Zbl 1439.35124 |
| [41] | W. H. Fleming and M. Nisio, Differential games for stochastic partial differential equations, Nagoya Math. J. 131 (1993), 75-107. · Zbl 0787.60071 |
| [42] | W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Appl. Math. 1, Springer, Berlin, 1975. · Zbl 0323.49001 |
| [43] | W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Appl. Math. (New York) 25, Springer, New York, 1993. · Zbl 0773.60070 |
| [44] | W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games, Indiana Univ. Math. J. 38 (1989), no. 2, 293-314. · Zbl 0686.90049 |
| [45] | K. Fok, A nonlinear Fabes-Stroock result, Comm. Partial Differential Equations 23 (1998), no. 5-6, 967-983. · Zbl 0911.35026 |
| [46] | G. Galise and A. Vitolo, Removable singularities for degenerate elliptic Pucci operators, Adv. Differential Equations 22 (2017), no. 1-2, 77-100. · Zbl 1366.35048 |
| [47] | D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer, Berlin, 2001. · Zbl 1042.35002 |
| [48] | F. R. Harvey and H. B. Lawson, Jr., Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math. 62 (2009), no. 3, 396-443. · Zbl 1173.35062 |
| [49] | F. R. Harvey and H. B. Lawson, Jr., Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry, Surveys in Differential Geometry. Geometry and Topology, Surv. Differ. Geom. 18, International Press, Somerville (2013), 103-156. · Zbl 1323.35036 |
| [50] | F. R. Harvey and H. B. Lawson, Jr., Removable singularities for nonlinear subequations, Indiana Univ. Math. J. 63 (2014), no. 5, 1525-1552. · Zbl 1308.35110 |
| [51] | E. Hopf, Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Sitzungsberichte Akad. Berlin 1927 (1927), 147-152. · JFM 53.0454.02 |
| [52] | C. Imbert, Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations, J. Differential Equations 250 (2011), no. 3, 1553-1574. · Zbl 1205.35124 |
| [53] | C. Imbert and L. Silvestre, C^{1,\alpha} regularity of solutions of some degenerate fully non-linear elliptic equations, Adv. Math. 233 (2013), 196-206. · Zbl 1262.35065 |
| [54] | H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations 83 (1990), no. 1, 26-78. · Zbl 0708.35031 |
| [55] | S. Koike and A. Świȩch, Maximum principle for fully nonlinear equations via the iterated comparison function method, Math. Ann. 339 (2007), no. 2, 461-484. · Zbl 1387.35243 |
| [56] | J. Kovats, The minmax principle and W^{2,p} regularity for solutions of the simplest Isaacs equations, Proc. Amer. Math. Soc. 140 (2012), no. 8, 2803-2815. · Zbl 1275.35062 |
| [57] | J. Kovats, On the second order derivatives of solutions of a special Isaacs equation, Proc. Amer. Math. Soc. 144 (2016), no. 4, 1523-1533. · Zbl 1344.35037 |
| [58] | N. V. Krylov, Controlled Diffusion Processes, Springer, New York, 1977. · Zbl 0513.60043 |
| [59] | J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for p-harmonic functions, Proc. Amer. Math. Soc. 138 (2010), no. 3, 881-889. · Zbl 1187.35115 |
| [60] | K. Miller, Extremal barriers on cones with Phragmén-Lindelöf theorems and other applications, Ann. Mat. Pura Appl. (4) 90 (1971), 297-329. · Zbl 0231.35004 |
| [61] | A. Mohammed, G. Porru and A. Vitolo, Harnack inequality for nonlinear elliptic equations with strong absorption, J. Differential Equations 263 (2017), no. 10, 6821-6843. · Zbl 1378.35126 |
| [62] | A. Mohammed and A. Vitolo, On the strong maximum principle, Complex Var. Elliptic Equ. (2019), 10.1080/17476933.2019.1594207. · Zbl 1466.35167 · doi:10.1080/17476933.2019.1594207 |
| [63] | I. Netuka and J. Veselý, Mean value property and harmonic functions, Classical and Modern Potential Theory and Applications (Chateau de Bonas 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 430, Kluwer Academic, Dordrecht (1994), 359-398. · Zbl 0863.31012 |
| [64] | M. Nisio, Stochastic differential games and viscosity solutions of Isaacs equations, Nagoya Math. J. 110 (1988), 163-184. · Zbl 0850.93741 |
| [65] | A. M. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5871-5886. · Zbl 1244.35056 |
| [66] | Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167-210. · Zbl 1206.91002 |
| [67] | M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, New York, 1984. · Zbl 0549.35002 |
| [68] | C. Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. (4) 74 (1966), 15-30. · Zbl 0144.35801 |
| [69] | P. Pucci and V. D. Rădulescu, The maximum principle with lack of monotonicity, Electron. J. Qual. Theory Differ. Equ. 2018 (2018), Paper No. 58. · Zbl 1413.35198 |
| [70] | P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations 196 (2004), no. 1, 1-66. · Zbl 1109.35022 |
| [71] | J.-P. Sha, p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437-447. · Zbl 0563.53032 |
| [72] | J.-P. Sha, Handlebodies and p-convexity, J. Differential Geom. 25 (1987), no. 3, 353-361. · Zbl 0661.53028 |
| [73] | A. Świech, W^{1,p}-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations 2 JYea1997, no. 6, 1005-1027. · Zbl 1023.35509 |
| [74] | N. S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Invent. Math. 61 (1980), no. 1, 67-79. · Zbl 0453.35028 |
| [75] | N. S. Trudinger, Hölder gradient estimates for fully nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), no. 1-2, 57-65. · Zbl 0653.35026 |
| [76] | J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191-202. · Zbl 0561.35003 |
| [77] | A. Vitolo, On the maximum principle for complete second-order elliptic operators in general domains, J. Differential Equations 194 (2003), no. 1, 166-184. · Zbl 1160.35362 |
| [78] | A. Vitolo, A note on the maximum principle for second-order elliptic equations in general domains, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 11, 1955-1966. · Zbl 1137.35323 |
| [79] | A. Vitolo, Removable singularities for degenerate elliptic equations without conditions on the growth of the solution, Trans. Amer. Math. Soc. 370 (2018), no. 4, 2679-2705. · Zbl 1386.35095 |
| [80] | H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525-548. · Zbl 0639.53050 |
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