The sliding method for the nonlocal Monge-Ampère operator. (English) Zbl 1437.35419
Summary: We develop a sliding method for the nonlocal Monge-Ampère operator. We first establish a narrow region principle in bounded domains, which plays an important role in the sliding method. Then we consider the equation with the nonlocal Monge-Ampère operator and derive the monotonicity of solutions in both bounded domains and the whole space. We use a new idea-estimating the singular integrals defining the nonlocal Monge-Ampère operator along a sequence of approximate maximum points.
MSC:
| 35J96 | Monge-Ampère equations |
Keywords:
nonlocal Monge-Ampère operator; sliding method; narrow region principle; monotonicity of solutionsReferences:
| [1] | Berestycki, H.; Caffarelli, L.; Nirenberg, L., Inequalitites for second-order elliptic equations with applications to unbounded domains, I, Duke Math. J., 81, 467-494 (1996) · Zbl 0860.35004 |
| [2] | Berestycki, H.; Caffarelli, L.; Nirenberg, L., Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50, 1089-1111 (1997) · Zbl 0906.35035 |
| [3] | Berestycki, H.; Nirenberg, L., Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5, 237-275 (1988) · Zbl 0698.35031 |
| [4] | Berestycki, H.; Nirenberg, L., Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, (Analysis, Et Cetera (1990), Academic Press), 115-164 · Zbl 0705.35004 |
| [5] | Berestycki, H.; Nirenberg, L., On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat., 22, 1-37 (1991) · Zbl 0784.35025 |
| [6] | Cabré, X.; Cinti, E., Sharp energy estimates for nonlinear fractional diffusion equations, Calc. Var. Partial Differ. Equ., 49, 1-2, 233-269 (2014) · Zbl 1282.35399 |
| [7] | Cabré, X.; Sire, Y., Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. Henri Poincare C Non Linear Anal., 31, 1, 23-53 (2014), Elsevier Masson · Zbl 1286.35248 |
| [8] | Cabré, X.; Sire, Y., Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367, 2, 911-941 (2015) · Zbl 1317.35280 |
| [9] | Caffarelli, L.; Charro, F., On a fractional Monge-Ampère operator, Ann. PDE., 1, 4 (2015) · Zbl 1393.35268 |
| [10] | Caffarelli, L.; Salsa, S.; Silvestre, L., Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171, 2, 425-461 (2008) · Zbl 1148.35097 |
| [11] | Caffarelli, L.; Silvestre, L., An extension problem related to the fractional Laplacian, Comm. Partial Differ. Equ., 32, 8, 1245-1260 (2007) · Zbl 1143.26002 |
| [12] | Cai, M.; Ma, L., Moving planes for nonlinear fractional Laplacian equation with negative powers, Discrete Contin. Dyn. Syst., 38, 9, 4603-4615 (2018) · Zbl 1397.35330 |
| [13] | Chen, W.; Fang, Y.; Yang, R., Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274, 167-198 (2015) · Zbl 1372.35332 |
| [14] | Chen, W.; Li, C., Maximum principles for the fractional \(p\)-Laplacian and symmetry of solutions, Adv. Math., 335, 735-758 (2018) · Zbl 1395.35055 |
| [15] | Chen, W.; Li, C.; Li, Y., A direct method of moving planes for the fractional Laplacian, Adv. Math., 308, 404-437 (2017) · Zbl 1362.35320 |
| [16] | Chen, W.; Li, C.; Li, G., Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differ. Equ., 56, 2, 29 (2017) · Zbl 1368.35110 |
| [17] | Chen, W.; Li, Y.; Ma, P., The Fractional Laplacian (2019), World Scientific Publishing Co. |
| [18] | Dipierro, S.; Soave, N.; Valdinoci, E., On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results, Math. Ann., 369, 1283-1326 (2017) · Zbl 1388.35208 |
| [19] | Felmer, P.; Quaas, A.; Tan, J., Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Edinb. Math. Soc. A, 142, 1237-1262 (2012) · Zbl 1290.35308 |
| [20] | Felmer, P.; Wang, Y., Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commun. Contemp. Math., 16, 01, Article 1350023 pp. (2014) · Zbl 1286.35016 |
| [21] | Liu, Z.; Chen, W., Maximum principles and monotonicity of solutions for fractional \(p\)-equations in unbounded domains (2019), arXiv preprint arXiv:1905.06493 |
| [22] | Musina, R.; Nazarov, A. I., On fractional Laplacians, Comm. Partial Differential Equations, 39, 9, 1780-1790 (2014) · Zbl 1304.47061 |
| [23] | Ros-Oton, X.; Serra, J., The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101, 3, 275-302 (2014) · Zbl 1285.35020 |
| [24] | Wu, L.; Chen, W., Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities (2019), arXiv preprint arXiv:1905.09999v2 |
| [25] | Wu, L.; Chen, W., The sliding methods for the fractional \(p\)-Laplacian, Adv. Math., 361, Article 106933 pp. (2020) · Zbl 1507.35334 |
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