Extension and trace for nonlocal operators. (English. French summary) Zbl 1452.46023
Summary: We prove an optimal extension and trace theorem for Sobolev spaces of nonlocal operators. The extension is given by a suitable Poisson integral and solves the corresponding nonlocal Dirichlet problem. We give a Douglas-type formula for the quadratic form of the Poisson extension.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 35A15 | Variational methods applied to PDEs |
| 31C05 | Harmonic, subharmonic, superharmonic functions on other spaces |
References:
| [1] | Adams, R.; Fournier, J., Sobolev Spaces, Pure and Applied Mathematics (2003), Elsevier Science · Zbl 1098.46001 |
| [2] | Baeumer, B.; Luks, T.; Meerschaert, M. M., Space-time fractional Dirichlet problems, Math. Nachr., 291, 17-18, 2516-2535 (2018) · Zbl 1476.35294 |
| [3] | Barles, G.; Chasseigne, E.; Georgelin, C.; Jakobsen, E. R., On Neumann type problems for nonlocal equations set in a half space, Trans. Am. Math. Soc., 366, 9, 4873-4917 (2014) · Zbl 1323.35192 |
| [4] | Blumenthal, R. M.; Getoor, R. K., Markov Processes and Potential Theory, Pure and Applied Mathematics, vol. 29 (1968), Academic Press: Academic Press New York-London · Zbl 0169.49204 |
| [5] | Bogdan, K., The boundary Harnack principle for the fractional Laplacian, Stud. Math., 123, 1, 43-80 (1997) · Zbl 0870.31009 |
| [6] | Bogdan, K., Representation of α-harmonic functions in Lipschitz domains, Hiroshima Math. J., 29, 2, 227-243 (1999) · Zbl 0936.31008 |
| [7] | Bogdan, K.; Byczkowski, T., Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains, Stud. Math., 133, 1, 53-92 (1999) · Zbl 0923.31003 |
| [8] | Bogdan, K.; Byczkowski, T., Potential theory of Schrödinger operator based on fractional Laplacian, Probab. Math. Stat., 20, 2256, 293-335 (2000), (2, Acta Univ. Wratislav. No. 2256) · Zbl 0996.31003 |
| [9] | Bogdan, K.; Dyda, B.; Luks, T., On Hardy spaces of local and nonlocal operators, Hiroshima Math. J., 44, 2, 193-215 (2014) · Zbl 1312.42026 |
| [10] | Bogdan, K.; Grzywny, T.; Ryznar, M., Density and tails of unimodal convolution semigroups, J. Funct. Anal., 266, 6, 3543-3571 (2014) · Zbl 1290.60002 |
| [11] | Bogdan, K.; Grzywny, T.; Ryznar, M., Dirichlet heat kernel for unimodal Lévy processes, Stoch. Process. Appl., 124, 11, 3612-3650 (2014) · Zbl 1320.60112 |
| [12] | Bogdan, K.; Grzywny, T.; Ryznar, M., Barriers, exit time and survival probability for unimodal Lévy processes, Probab. Theory Relat. Fields, 162, 1-2, 155-198 (2015) · Zbl 1317.31016 |
| [13] | Bogdan, K.; Jakubowski, T., Estimates of the Green function for the fractional Laplacian perturbed by gradient, Potential Anal., 36, 3, 455-481 (2012) · Zbl 1238.35018 |
| [14] | Bogdan, K.; Rosiński, J.; Serafin, G.; Wojciechowski, Ł., Lévy systems and moment formulas for mixed Poisson integrals, (Stochastic Analysis and Related Topics. Stochastic Analysis and Related Topics, Progr. Probab., vol. 72 (2017), Birkhäuser/Springer: Birkhäuser/Springer Cham), 139-164 · Zbl 1409.60074 |
| [15] | Bogdan, K.; Sztonyk, P., Estimates of the potential kernel and Harnack’s inequality for the anisotropic fractional Laplacian, Stud. Math., 181, 2, 101-123 (2007) · Zbl 1223.47038 |
| [16] | Caffarelli, L.; Roquejoffre, J.-M.; Savin, O., Nonlocal minimal surfaces, Commun. Pure Appl. Math., 63, 9, 1111-1144 (2010) · Zbl 1248.53009 |
| [17] | Chen, Z.-Q., On notions of harmonicity, Proc. Am. Math. Soc., 137, 10, 3497-3510 (2009) · Zbl 1181.60118 |
| [18] | Chen, Z.-Q.; Fukushima, M., Symmetric Markov Processes, Time Change, and Boundary Theory, London Mathematical Society Monographs Series, vol. 35 (2012), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1253.60002 |
| [19] | Chen, Z.-Q.; Kim, P.; Song, R., Dirichlet heat kernel estimates for rotationally symmetric Lévy processes, Proc. Lond. Math. Soc. (3), 109, 1, 90-120 (2014) · Zbl 1304.60080 |
| [20] | Chung, K. L.; Zhao, Z. X., From Brownian Motion to Schrödinger’s Equation, Grundlehren der Mathematischen Wissenschaften, vol. 312 (1995), Springer-Verlag: Springer-Verlag Berlin · Zbl 0819.60068 |
| [21] | Dhara, R. N.; Kałamajska, A., On one extension theorem dealing with weighted Orlicz-Slobodetskii space. Analysis on Lipschitz subgraph and Lipschitz domain, Math. Inequal. Appl., 19, 2, 451-488 (2016) · Zbl 1356.46029 |
| [22] | Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 5, 521-573 (2012) · Zbl 1252.46023 |
| [23] | Ding, Z., A proof of the trace theorem of Sobolev spaces on Lipschitz domains, Proc. Am. Math. Soc., 124, 2, 591-600 (1996) · Zbl 0841.46021 |
| [24] | Dipierro, S.; Ros-Oton, X.; Valdinoci, E., Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33, 2, 377-416 (2017) · Zbl 1371.35322 |
| [25] | Dłotko, T.; Kania, M. B.; Sun, C., Quasi-geostrophic equation in \(\mathbb{R}^2\), J. Differ. Equ., 259, 2, 531-561 (2015) · Zbl 1325.35161 |
| [26] | Douglas, J., Solution of the problem of plateau, Trans. Am. Math. Soc., 33, 1, 263-321 (1931) · JFM 57.0601.01 |
| [27] | Dyda, B.; Kassmann, M., Function spaces and extension results for nonlocal Dirichlet problems, J. Funct. Anal. (2018) |
| [28] | Dynkin, E. B., Markov Processes. Vols. I, II, Die Grundlehren der Mathematischen Wissenschaften, Bände 121, 122 (1965), Academic Press Inc., Publishers/Springer-Verlag: Academic Press Inc., Publishers/Springer-Verlag New York/Berlin-Göttingen-Heidelberg, translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone · Zbl 0132.37901 |
| [29] | Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, vol. 19 (2010), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1194.35001 |
| [30] | Felsinger, M.; Kassmann, M.; Voigt, P., The Dirichlet problem for nonlocal operators, Math. Z., 279, 3-4, 779-809 (2015) · Zbl 1317.47046 |
| [31] | Fiscella, A.; Servadei, R.; Valdinoci, E., Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn., Math., 40, 1, 235-253 (2015) · Zbl 1346.46025 |
| [32] | Fukushima, M.; Oshima, Y.; Takeda, M., Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics, vol. 19 (2011), Walter de Gruyter & Co.: Walter de Gruyter & Co. Berlin · Zbl 1227.31001 |
| [33] | Grzywny, T.; Kassmann, M.; Leżaj, Ł., Remarks on the nonlocal Dirichlet problem (2018), arXiv e-prints |
| [34] | 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, 1-38 (2018) · Zbl 1386.60265 |
| [35] | Grzywny, T.; Ryznar, M.; Trojan, B., Asymptotic behaviour and estimates of slowly varying convolution semigroups, Int. Math. Res. Not. (2018) |
| [36] | Kang, J.; Kim, P., On estimates of Poisson kernels for symmetric Lévy processes, J. Korean Math. Soc., 50, 5, 1009-1031 (2013) · Zbl 1277.31013 |
| [37] | Khoshnevisan, D.; Schilling, R., From Lévy-Type Processes to Parabolic SPDEs, Advanced Courses in Mathematics - CRM Barcelona (2016), Birkhäuser/Springer: Birkhäuser/Springer Cham, edited by Lluís Quer-Sardanyons and Frederic Utzet · Zbl 1382.60005 |
| [38] | Klimsiak, T.; Rozkosz, A., Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form, NoDEA Nonlinear Differ. Equ. Appl., 22, 6, 1911-1934 (2015) · Zbl 1328.35263 |
| [39] | Koskela, P.; Soto, T.; Wang, Z., Traces of weighted function spaces: dyadic norms and Whitney extensions, Sci. China Math., 60, 11, 1981-2010 (2017) · Zbl 1406.46024 |
| [40] | Kulczycki, T.; Ryznar, M., Gradient estimates of harmonic functions and transition densities for Lévy processes, Trans. Am. Math. Soc., 368, 1, 281-318 (2016) · Zbl 1336.31012 |
| [41] | Millot, V.; Sire, Y.; Wang, K., Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces, Arch. Ration. Mech. Anal. (2018) |
| [42] | Pruitt, W. E., The growth of random walks and Lévy processes, Ann. Probab., 9, 6, 948-956 (1981) · Zbl 0477.60033 |
| [43] | Ros-Oton, X., Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat., 60, 1, 3-26 (2016) · Zbl 1337.47112 |
| [44] | Rutkowski, A., The Dirichlet problem for nonlocal Lévy-type operators, Publ. Mat., 62, 1, 213-251 (2018) · Zbl 1391.35444 |
| [45] | Sato, K.-i., Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, vol. 68 (1999), Cambridge University Press: Cambridge University Press Cambridge, translated from the 1990 Japanese original, revised by the author · Zbl 0973.60001 |
| [46] | Schilling, R. L.; Song, R.; Vondraček, Z., Bernstein Functions. Theory and Applications, De Gruyter Studies in Mathematics, vol. 37 (2012), Walter de Gruyter & Co.: Walter de Gruyter & Co. Berlin · Zbl 1257.33001 |
| [47] | Servadei, R.; Valdinoci, E., Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58, 1, 133-154 (2014) · Zbl 1292.35315 |
| [48] | Silvestre, L., Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60, 1, 67-112 (2007) · Zbl 1141.49035 |
| [49] | Sztonyk, P., On harmonic measure for Lévy processes, Probab. Math. Stat., 20, 383 (2000), (2, Acta Univ. Wratislav. No. 2256) · Zbl 0991.60067 |
| [50] | Wu, J.-M., Harmonic measures for symmetric stable processes, Stud. Math., 149, 3, 281-293 (2002) · Zbl 0990.60074 |
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