Comparison principles for infinity-Laplace equations in Finsler metrics. (English) Zbl 1433.35100
Summary: In this paper we study comparison principles for normalized Finsler infinity-Laplace operators with nonhomogeneous terms that depend on solutions and their gradients. This is achieved by using a combination of sup-convolution methods to approximate viscosity solutions by semiconvex functions and finite difference approximation schemes. Characterizations of viscosity subsolutions and supersolutions of equations with constant nonhomogeneous terms through comparison with quadratic Finsler cone are also discussed.
References:
| [1] | Armstrong, Scott N.; Smart, Charles K., An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions, Calc. Var. Partial Differential Equations, 37, 3-4, 381-384 (2010) · Zbl 1187.35104 |
| [2] | Armstrong, Scott N.; Smart, Charles K., A finite difference approach to the infinity Laplace equation and tug-of-war games, Trans. Amer. Math. Soc., 364, 2, 595-636 (2012) · Zbl 1239.91011 |
| [3] | Aronsson, G.; Crandall, M. G.; Juutinen, P., A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41, 439-505 (2004) · Zbl 1150.35047 |
| [4] | Bao, D.; Chern, S.-S.; Shen, Z., An Introduction to Riemann-Finsler Geometry (2000), Springer-Verlag: Springer-Verlag New York · Zbl 0954.53001 |
| [5] | Barbu, Luminita; Enache, Cristian, Maximum principles, Liouville-type theorems and symmetry results for a general class of quasilinear anisotropic equations, Adv. Nonlinear Anal., 5, 4, 395-405 (2016) · Zbl 1357.35068 |
| [6] | Barles, G.; Busca, J., Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Differential Equations, 26, 2323-2337 (2001) · Zbl 0997.35023 |
| [7] | Bellettini, G.; Novaga, M.; Paolini, M., On a crystalline variational problem, Part I: First variation and global \(L^\infty\) regularity, Arch. Ration. Mech. Anal., 157, 165-191 (2001) · Zbl 0976.58016 |
| [8] | Bellettini, G.; Novaga, M.; Paolini, M., On a crystalline variational problem, Part II: BV regularity and structure of minimizers on facets, Arch. Ration. Mech. Anal., 157, 193-217 (2001) · Zbl 0976.58017 |
| [9] | Belloni, M.; Ferone, V.; Kawohl, B., Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Z. Angew. Math. Phys., 54, 5, 771-783 (2003), Special issue dedicated to Lawrence E. Payne · Zbl 1099.35509 |
| [10] | Belloni, M.; Kawohl, B.; Juutinen, P., The p-Laplace eigenvalue problem as \(p \to \infty\) in a Finsler metric, J. Eur. Math. Soc., 8, 1, 123-138 (2006) · Zbl 1245.35078 |
| [11] | Bhattacharya, Tilak, An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions, Electron. J. Differential Equations, 44, 8 (2001), (electronic) · Zbl 0966.35052 |
| [12] | Bhattacharya, Tilak; Mohammed, Ahmed, On solutions to Dirichlet problems involving the infinity-Laplacian, Adv. Calc. Var., 4, 4, 445-487 (2011) · Zbl 1232.35056 |
| [13] | Bhattacharya, Tilak; Mohammed, Ahmed, Inhomogeneous Dirichlet problems involving the infinity-Laplacian, Adv. Differential Equations, 17, 3-4, 225-266 (2012) · Zbl 1258.35094 |
| [14] | Birindelli, I.; Capuzzo Dolcetta, I.; Vitolo, A., ABP and global Hölder estimates for fully nonlinear elliptic equations in unbounded domains, Commun. Contemp. Math., 18, 04, Article 1550075 pp. (2016), 16 pp · Zbl 1344.35036 |
| [15] | Brandolini, B.; Nitsch, C.; Salani, P.; Trombetti, C., Serrin type overdetermined problems: an alternative proof, Arch. Ration. Mech. Anal., 190, 267-280 (2008) · Zbl 1161.35025 |
| [16] | Caffarelli, L. A.; Cabre, X., (Fully Nonlinear Elliptic Equations. Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, vol. 43 (1995), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0834.35002 |
| [17] | Cianchi, Andrea; Salani, Paolo, Overdetermined anisotropic elliptic problems, Math. Ann., 345, 4, 859-881 (2009) · Zbl 1179.35107 |
| [18] | Cozzi, Matteo; Farina, Alberto; Valdinoci, Enrico, Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations, Comm. Math. Phys., 331, 1, 189-214 (2014) · Zbl 1303.49001 |
| [19] | Crandall, M. G., A visit with the \(\infty \)-Laplace equation, (Calculus of Variations and Nonlinear Partial Differential Equations. Calculus of Variations and Nonlinear Partial Differential Equations, Lecture Notes in Math., vol. 1927 (2008), Springer: Springer Berlin), 75-122 · Zbl 1357.49112 |
| [20] | Crandall, M. G.; Evans, L. C.; Gariepy, Optimal Lipschitz extensions and the infinity-Laplacian, Calc. Var. Partial Differential Equations, 13, 2, 123-139 (2001) · Zbl 0996.49019 |
| [21] | Crandall, M. G.; Gunarson, G.; Wang, P. Y., Uniqueness of \(\infty \)-harmonic functions and the eikonal equation, Comm. Partial Differential Equations, 32, 10-12, 1587-1615 (2007) · Zbl 1135.35048 |
| [22] | Crandall, M. G.; Ishii, H.; Lions, P. L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27, 1-67 (1992) · Zbl 0755.35015 |
| [23] | Crasta, C.; Malusa, A., The distance function from the boundary in a Minkowski space, Trans. Amer. Math. Soc., 359, 12, 5725-5759 (2007) · Zbl 1132.35005 |
| [24] | Dolcetta, I. Capuzzo; Vitolo, Antonio, Glaeser’s type gradient estimates for non-negative solutions of fully nonlinear elliptic equations, Discrete Contin. Dyn. Syst., 28, 2, 539-557 (2010) · Zbl 1193.35043 |
| [25] | Evans, L. C.; Savin, O., \(C^{1, \alpha}\) regularity for infinity harmonic functions in two dimensions, Calc. Var. Partial Differential Equations, 32, 325-347 (2008) · Zbl 1151.35096 |
| [26] | Evans, L. C.; Smart, C. K., Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations, 42, 1-2, 289-299 (2011) · Zbl 1251.49034 |
| [27] | Farina, A.; Kawohl, B., Remarks on an overdetermined boundary value problem, Calc. Var., 31, 351-357 (2008) · Zbl 1166.35353 |
| [28] | Farina, Alberto; Valdinoci, Enrico, Gradient bounds for anisotropic partial differential equations, Calc. Var. Partial Differential Equations, 49, 3-4, 923-936 (2014) · Zbl 1288.35131 |
| [29] | Ferone, V.; Kawohl, B., Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc., 137, 247-253 (2009) · Zbl 1161.35017 |
| [30] | Fragal, I.; Gazzola, F.; Kawohl, B., Overdetermined boundary value problems with possibly degenerate ellipticity: a geometric approach, Math. Zeitschr., 254, 117-132 (2006) · Zbl 1220.35077 |
| [31] | Igbida, Noureddine; Mazón, José M.; Rossi, Julio D.; Toledo, Julián, Optimal mass transportation for costs given by finsler distances via p-Laplacian approximations, Adv. Calc. Var., 11, 1, 1-28 (2018) · Zbl 1385.49001 |
| [32] | Imbert, Cyril, Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations, J. Differential Equations, 250, 3, 1553-1574 (2011) · Zbl 1205.35124 |
| [33] | Jensen, Robert, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal., 123, 1, 51-74 (1993) · Zbl 0789.35008 |
| [34] | Kawohl, Bernd; Manfredi, Juan; Parviainen, Mikko, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl. (9), 97, 2, 17-188 (2012) · Zbl 1236.35083 |
| [35] | Lindqvist, P.; Manfredi, J. J., The Harnack inequality for infinity-harmonic functions, Electron. J. Differential Equations, 04 (1995) · Zbl 0818.35033 |
| [36] | Lu, G.; Wang, P., Inhomogeneous infinity Laplace equation, Adv. Math., 217, 4, 1838-1868 (2008) · Zbl 1152.35042 |
| [37] | Lu, G.; Wang, P., A PDE perspective of the normalized infinity Laplacian, Comm. Partial Differential Equations, 33, 10-12, 1788-1817 (2008) · Zbl 1157.35388 |
| [38] | Lu, G.; Wang, P., Infinity Laplace Equation with non-trivial right-hand side, Electron. J. Differential Equations, 2010, 77, 1-12 (2010), (electronic) · Zbl 1194.35194 |
| [39] | Manfredi, J. J.; Parviainen, M.; Rossi, J. D., An asymptotic mean value characterization for \(p\)-harmonic functions, Proc. Amer. Math. Soc., 138, 3, 881-889 (2010) · Zbl 1187.35115 |
| [40] | Manfredi, J. J.; Parviainen, M.; Rossi, J. D., On the definition and properties of \(p\)-harmonious functions, Ann. Scuola Nor. Sup. Pisa., 11, 215-241 (2012) · Zbl 1252.91014 |
| [41] | Benyam Mebrate, Ahmed Mohammed, Harnack inequality and the mean-value property for Finsler infinity-Laplacian. https://doi.org/10.1515/acv-2018-0083. · Zbl 1469.35009 |
| [42] | Mebrate, Benyam; Mohammed, Ahmed, Infinity-Laplacian type equations and their associated Dirichlet problems, Complex Var. Elliptic Equ. (2019), to appear · Zbl 1442.35124 |
| [43] | Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D.; Repovš, Dušan D., (Nonlinear Analysis—Theory and Methods. Nonlinear Analysis—Theory and Methods, Springer Monographs in Mathematics (2019), Springer: Springer Cham) · Zbl 1414.46003 |
| [44] | Peres, Y.; Schramm, O.; Sheffield, S.; Wilson, D., Tug-of-war and the infinity-Laplacian, J. Amer. Math. Soc., 22, 1, 167-210 (2009) · Zbl 1206.91002 |
| [45] | Rossi, Julio D., Tug-of-war games and PDEs, Proc. Roy. Soc. Edinburgh Sect. A, 141, 2, 319-369 (2011) · Zbl 1242.35091 |
| [46] | Savin, O., \(C^1\) regularity for infinity harmonic functions in two dimensions, Arch. Ration. Mech. Anal., 176, 3, 351-361 (2005) · Zbl 1112.35070 |
| [47] | Schneider, R., Convex Bodies: The Brunn-Minkowski Theory (1993), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0798.52001 |
| [48] | Taylor, J., Crystalline variational problems, Bull. Amer. Math. Soc., 84, 568-588 (1978) · Zbl 0392.49022 |
| [49] | Taylor, J. E., (Lawson, B.; Tanenblat, K., Constructions and Conjectures in Crystalline Nondifferential Geometry. Constructions and Conjectures in Crystalline Nondifferential Geometry, Monographs in Pure and Applied Mathematics, vol. 55 (1991), Pitman: Pitman London), 321-336 · Zbl 0725.53011 |
| [50] | Wang, H.; He, Y., Existence of solutions of a normalized F-infinity Laplacian equation, Electron. J. Differential Equations, 2014, 109, 1-17 (2014), (electronic) · Zbl 1297.35080 |
| [51] | Wang, Guofang; Xia, Chao, A characterization of the Wulff shape by an overdetermined anisotropic PDE, Arch. Ration. Mech. Anal., 199, 1, 99-115 (2011) · Zbl 1232.35103 |
| [52] | Wulff, G., Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallfläschen, Z. Krist., 34, 449-530 (1901) |
| [53] | Zhang, Qihu; Rădulescu, Vicenţiu D., Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl., 118, 9, 159-203 (2018) · Zbl 1404.35191 |
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