Nonlocal operators of small order. (English) Zbl 1516.35458
Summary: In this work we study nonlocal operators and corresponding spaces with a focus on operators of order near zero. We investigate the interior regularity of eigenfunctions and of weak solutions to the associated Poisson problem depending on the regularity of the right-hand side. Our method exploits the variational structure of the problem and we prove that eigenfunctions are of class \(C^{\infty}\) if the kernel satisfies this property away from its singularity. Similarly in this case, if in the Poisson problem the right-hand is of class \(C^{\infty}\), then also any weak solution is of class \(C^{\infty}\).
MSC:
| 35R11 | Fractional partial differential equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
References:
| [1] | Antil, H.; Bartels, S., Spectral approximation of fractional PDEs in image processing and phase field modeling, Comput. Methods Appl. Math., 17, 4, 661-678 (2017) · Zbl 1431.65222 · doi:10.1515/cmam-2017-0039 |
| [2] | Biočić, I., Vondraček, Z., Wagner, V.: Semilinear equations for non-local operators: beyond the fractional Laplacian. Nonlinear Anal. 207, Paper No. 112303 (2021). doi:10.1016/j.na.2021.112303 · Zbl 1479.35914 |
| [3] | Biswas, A.; Jarohs, S., On overdetermined problems for a general class of nonlocal operators, J. Differ. Equ., 268, 5, 2368-2393 (2020) · Zbl 1435.35264 · doi:10.1016/j.jde.2019.09.010 |
| [4] | Chen, Z-Q; Song, R., Hardy inequality for censored stable processes, Tohoku Math. J., 55, 439-450 (2003) · Zbl 1070.46021 · doi:10.2748/tmj/1113247482 |
| [5] | Chen, H.; Weth, T., The Dirichlet problem for the Logarithmic Laplacian, Commun. Partial Differ. Equ., 44, 11, 1100-1139 (2019) · Zbl 1423.35390 · doi:10.1080/03605302.2019.1611851 |
| [6] | Correa, E.; de Pablo, A., Nonlocal operators of order near zero, J. Math. Anal. Appl., 461, 837-867 (2018) · Zbl 1393.45011 · doi:10.1016/j.jmaa.2017.12.011 |
| [7] | Cozzi, M., Interior regularity of solutions of non-local equations in Sobolev and Nikol’skii spaces, Ann. Mat. Pura Appl. (4), 196, 2, 555-578 (2017) · Zbl 1371.35312 · doi:10.1007/s10231-016-0586-3 |
| [8] | Dyda, B., A fractional order Hardy inequality, Ill. J. Math., 48, 2, 575-588 (2004) · Zbl 1068.26014 |
| [9] | Dyda, B., Kassmann, M.: Comparability and regularity estimates for symmetric nonlocal Dirichlet forms. Preprint (2011). arXiv:1109.6812 · Zbl 1437.35175 |
| [10] | Evans, LC; Gariepy, RF, Measure Theory and Fine Properties of Functions (1992), Taylor and Francis Group: CRC Press, Taylor and Francis Group · Zbl 0804.28001 |
| [11] | Felsinger, M.; Kassmann, M.; Voigt, P., The Dirichlet problem for nonlocal operators, Math. Z., 279, 779-809 (2015) · Zbl 1317.47046 · doi:10.1007/s00209-014-1394-3 |
| [12] | Feulefack, PA, The logarithmic Schrödinger operator and associated Dirichlet problems, J. Math. Anal. Appl., 517, 2 (2023) · Zbl 1498.60183 · doi:10.1016/j.jmaa.2022.126656 |
| [13] | Feulefack, PA; Jarohs, S.; Weth, T., Small order asymptotics of the Dirichlet eigenvalue problem for the fractional Laplacian, J. Fourier Anal. Appl., 28, 2, 1-44 (2022) · Zbl 1485.35304 · doi:10.1007/s00041-022-09908-8 |
| [14] | Grzywny, T., On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes, Potential Anal., 41, 1-29 (2014) · Zbl 1302.60109 · doi:10.1007/s11118-013-9360-y |
| [15] | Grzywny, T.; Kassmann, M.; Leżaj, Ł., Remarks on the nonlocal Dirichlet problem, Potential Anal., 54, 119-151 (2021) · Zbl 1456.35059 · doi:10.1007/s11118-019-09820-9 |
| [16] | Grzywny, T.; Kwaśnicki, M., Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes, Stoch. Process. Appl., 128, 1-38 (2018) · Zbl 1386.60265 · doi:10.1016/j.spa.2017.04.004 |
| [17] | Hernández Santamaría, V.; Saldaña, A., Small order asymptotics for nonlinear fractional problems, Calc. Var., 61, 3, 1-26 (2022) · Zbl 1486.35053 · doi:10.1007/s00526-022-02192-w |
| [18] | Jarohs, S.; Saldaña, A.; Weth, T., A new look at the fractional Poisson problem via the Logarithmic Laplacian, J. Funct. Anal., 279, 11, 108732 (2020) · Zbl 1450.35269 · doi:10.1016/j.jfa.2020.108732 |
| [19] | Jarohs, S.; Weth, T., On the strong maximum principle for nonlocal operators, Math. Z., 293, 1-2, 81-111 (2019) · Zbl 1455.35032 · doi:10.1007/s00209-018-2193-z |
| [20] | Jarohs, S.; Weth, T., Local compactness and nonvanishing for weakly singular nonlocal quadratic forms, Nonlinear Anal., 193 (2020) · Zbl 1444.47064 · doi:10.1016/j.na.2019.01.021 |
| [21] | Kassmann, M.; Mimica, A., Intrinsic scaling properties for nonlocal operators, J. Eur. Math. Soc., 19, 4, 983-1011 (2017) · Zbl 1371.35316 · doi:10.4171/JEMS/686 |
| [22] | Kim, P.; Mimica, A., Green function estimates for subordinate Brownian motions: stable and beyond, Trans. Am. Math. Soc., 366, 8, 4383-4422 (2014) · Zbl 1327.60165 · doi:10.1090/S0002-9947-2014-06017-0 |
| [23] | Mimica, A., On harmonic functions of symmetric Lévy processes, Ann. Inst. Henri Poincar Probab. Stat., 50, 214-235 (2014) · Zbl 1298.60054 · doi:10.1214/12-AIHP508 |
| [24] | Pellacci, B.; Verzini, G., Best dispersal strategies in spatially heterogeneous environments: optimization of the principal eigenvalue for indefinite fractional Neumann problems, J. Math. Biol., 76, 1357-1386 (2018) · Zbl 1390.35404 · doi:10.1007/s00285-017-1180-z |
| [25] | Temgoua, R.Y., Weth, T.: The eigenvalue problem for the regional fractional Laplacian in the small order limit. Preprint (2021). arXiv:2112.08856 |
| [26] | Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1995), Heidelberg: Johann Ambrosius Barth, Heidelberg · Zbl 0830.46028 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
