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Strong maximum principles for fractional Laplacians

Part of: Partial differential equations Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  16 January 2019

Roberta Musina  and
Alexander I. Nazarov
Roberta Musina
Affiliation:
Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, via delle Scienze, 206 – 33100, Udine, Italy ([email protected])
Alexander I. Nazarov
Affiliation:
St. Petersburg Department of Steklov Institute, Fontanka 27, St. Petersburg 191023, Russia and St. Petersburg State University, Universitetskii pr. 28, St. Petersburg 198504, Russia ([email protected])
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Abstract

We give a unified approach to strong maximum principles for a large class of nonlocal operators of order s ∈ (0, 1) that includes the Dirichlet, the Neumann Restricted (or Regional) and the Neumann Semirestricted Laplacians.

Keywords

Fractional Laplace operatorsmaximum principle

MSC classification

Primary: 35R11: Fractional partial differential equations
Secondary: 35B50: Maximum principles
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 5 , October 2019 , pp. 1223 - 1240
Copyright
Copyright © Royal Society of Edinburgh 2019 

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References

1Arendt, W., ter Elst, A. F. M. and Warma, M.. Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator. Comm. Partial Differ. Equ. 43 (2018), 1–24.Google Scholar
2Barrios, B. and Medina, M.. Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, preprint arXiv:1607.01505v2 (2016).Google Scholar
3Bogdan, K., Burdzy, K. and Chen, Z.-Q.. Censored stable processes. Probab. Theory Related Fields 127 (2003), 89–152.Google Scholar
4Caffarelli, L. A. and Stinga, P. R.. Fractional elliptic equations, Caccioppoli estimates and regularity. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), 767–807.Google Scholar
5Capella, A., Dávila, J., Dupaigne, L. and Sire, Y.. Regularity of radial extremal solutions for some non-local semilinear equations. Comm. Partial Differ. Equ. 36 (2011), 1353–1384.Google Scholar
6Del Pezzo, L. M., Quaas, A., Del Pezzo, L. M. and Quaas, A.. A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian. J. Differ. Equ. 263 (2017), 765–778.Google Scholar
7Di Castro, A., Kuusi, T. and Palatucci, G.. Local behavior of fractional p-minimizers. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), 1279–1299.Google Scholar
8Di Nezza, E., Palatucci, G. and Valdinoci, E.. Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521–573.Google Scholar
9Dipierro, S., Ros-Oton, X. and Valdinoci, E.. Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam. 33 (2017), 377–416.Google Scholar
10Gal, C. G. and Warma, M.. Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evol. Equ. Control Theory 5 (2016), 61–103.Google Scholar
11Grubb, G.. Regularity of spectral fractional Dirichlet and Neumann problems. Math. Nachr. 289 (2016), 831–844.Google Scholar
12Guan, Q. Y.. Integration by parts formula for regional fractional Laplacian. Comm. Math. Phys. 266 (2006), 289–329.Google Scholar
13Guan, Q. Y. and Ma, Z. M.. Boundary problems for fractional Laplacians. Stoch. Dyn. 5 (2005), 385–424.Google Scholar
14Iannizzotto, A., Mosconi, S. and Squassina, M.. H s versus C 0-weighted minimizers. NoDEA Nonlinear Differ. Equ. Appl. 22 (2015), 477–497.Google Scholar
15Kwaśnicki, M.. Ten equivalent definitions of the fractional laplace operator. Fract. Calc. Appl. Anal. 20 (2017), 7–51.Google Scholar
16Musina, R. and Nazarov, A. I.. On fractional Laplacians. Comm. Partial Differ. Equ. 39 (2014), 1780–1790.Google Scholar
17Ros-Oton, X.. Nonlocal elliptic equations in bounded domains: a survey. Publ. Mat. 60 (2016), 3–26.Google Scholar
18Ros-Oton, X. and Serra, J.. The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. (9) 101 (2014), 275–302.Google Scholar
19Silvestre, L.. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60 (2007), 67–112.Google Scholar
20Stinga, P. R. and Torrea, J. L.. Extension problem and Harnack's inequality for some fractional operators. Comm. Partial Differ. Equ. 35 (2010), 2092–2122.Google Scholar
21Triebel, H.. Interpolation theory, function spaces, differential operators (Berlin: Deutscher Verlag Wissensch, 1978).Google Scholar
22Warma, M.. The p-Laplace operator with the nonlocal Robin boundary conditions on arbitrary open sets. Ann. Mat. Pura Appl. (4) 193 (2014), 203–235.Google Scholar
23Warma, M.. The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets. Potential Anal. 42 (2015a), 499–547.Google Scholar
24Warma, M.. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Commun. Pure Appl. Anal. 14 (2015b), 2043–2067.Google Scholar