Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T07:03:03.462Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions

Part of: Miscellaneous topics - Partial differential equations Partial differential equations Integral, integro-differential, and pseudodifferential operators

Published online by Cambridge University Press:  26 January 2019

Begoña Barrios  and
Maria Medina
Begoña Barrios
Affiliation:
Departamento de Análisis Matemático Universidad de La Laguna, C/Astrofísico Francisco Sánchez s/n, La Laguna 38271, Spain ([email protected])
Maria Medina
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860 Santiago, Chile ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We present some comparison results for solutions to certain non-local elliptic and parabolic problems that involve the fractional Laplacian operator and mixed boundary conditions, given by a zero Dirichlet datum on part of the complementary of the domain and zero Neumann data on the rest. These results represent a non-local generalization of a Hopf's lemma for elliptic and parabolic problems with mixed conditions. In particular we prove the non-local version of the results obtained by Dávila and Dávila and Dupaigne for the classical case s = 1 in [23, 24] respectively.

Keywords

Non-local operatorsfractional Laplacianmixed boundary conditionsmaximum principle

MSC classification

Primary: 35B50: Maximum principles
Secondary: 35S15: Boundary value problems for pseudodifferential operators 47G20: Integro-differential operators
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 475 - 495
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Abdellaoui, B., Medina, M., Primo, A. and Peral, I.. The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian. J. Differ. Equ. 260 (2016), 8160–8206.CrossRefGoogle Scholar
2Alberti, G. and Bellettini, G.. A nonlocal anisotropic model for phase transitions. I. The optimal profile problem. Math. Ann. 310 (1998), 527–560.CrossRefGoogle Scholar
3Applebaum, D.. Lévy processes and Stochastic Calculus. Cambridge studies in Advanced Mathematics, vol. 116, 2nd edn (Cambridge: Cambridge University Press, 2009).CrossRefGoogle Scholar
4Barles, G., Chasseigne, E., Georgelin, C. and Jakobsen, E.. On Neumann type problems for nonlocal equations in a half space. Trans. Am. Math. Soc. 366 (2014), 4873–4917.CrossRefGoogle Scholar
5Barrios, B., Medina, M. and Peral, I.. Some remarks on the solvability of non local elliptic problems with the Hardy potential. Com. Contemp. Math. 16 (2014), 1350046.Google Scholar
6Barrios, B., Colorado, E., Servadei, R. and Soria, F.. A critical fractional equation with concave-convex nonlinearities. Ann. Inst. H. Poincar Anal. Non Linaire 32 (2015a), 875900.Google Scholar
7Barrios, B., Peral, I. and Vita, S., Some remarks about the summability of nonlocal nonlinear problems. Adv. Nonlinear Anal. 4 (2015b), 91–107.Google Scholar
8Barrios, B., Figalli, A. and Ros-Oton, X.. Global regularity for the free boundary in the obstacle problem for the fractional Laplacian. To appear in Am. J. Math.Google Scholar
9Barrios, B., Figalli, A. and Ros-Oton, X.. Free boundary in the parabolic fractional obstacle problem. To appear in Comm. Pure Appl. Math.Google Scholar
10Bertoin, J.. Lévy Processes. Cambridge tracts in Mathematics, vol. 121 (Cambridge: Cambridge University Press, 1996).Google Scholar
11Bogdan, K., Burdzy, K. and Chen, Z.-Q.. Censored stable processes. Probab. Theory Relat. Fields 127 (2003), 89–152.CrossRefGoogle Scholar
12Bonforte, M., Figalli, A. and Ros-Oton, X.. Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains. To appear in Comm. Pure Appl. Math.Google Scholar
13Bouchaud, J. P. and Georges, A.. Anomalous diffusion in disordered media, statistical mechanics, models and physical applications. Phys. Rep. 195 (1990), 127–293.CrossRefGoogle Scholar
14Brezis, H. and Cabré, X.. Some simple nonlinear PDE's without solutions. Boll. Unione Mat. Ital. 1-B (1998), 223–262.Google Scholar
15Caffarelli, L. and Figalli, A.. Regularity of solutions to the parabolic fractional obstacle problem. J. Reine Angew. Math., 680 (2013), 191–233.Google Scholar
16Caffarelli, L. and Vasseur, L.. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (2) 171 (2010), 1903–1930.CrossRefGoogle Scholar
17Caffarelli, L., Roquejoffre, J. M. and Sire, Y.. Variational problems in free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12 (2010), 1151–1179.CrossRefGoogle Scholar
18Chen, Z., Kim, P. and Song, R.. Heat kernel estimates for the Dirichlet fractional Laplacian. J. Eur. Math. Soc. 12 (2010), 1307–1329.CrossRefGoogle Scholar
19Colorado, E. and Peral, I.. Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions. J. Funct. Anal. 199 (2003), 468–507.CrossRefGoogle Scholar
20Constantin, P.. Euler equations, Navier-Stokes equations and turbulence. In Mathematical foundation of Turbulent Viscous Flows, Vol. 1871 of Lecture Notes in Math. (Berlin: Springer, 2006).Google Scholar
21Cont, R. and Tankov, P.. Financial modelling with Jump processes (Boca Raton, Fl: Chapman & Hall/CRC Financial Mathematics Series, 2004).Google Scholar
22Cortazar, C., Elgueta, M., Rossi, J. and Wolanski, N.. How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Rat. Mech. Anal. 187 (2008), 137–156.CrossRefGoogle Scholar
23Dávila, J.. A strong maximum principle for the Laplace equation with mixed boundary condition. J. Funct. Anal. 183 (2001), 23–244.CrossRefGoogle Scholar
24Dávila, J. and Dupaigne, L.. Comparison results for PDEs with a singular potential. Proc. R. Soc. Edinb. 133A (2003), 61–83.CrossRefGoogle Scholar
25Di Nezza, E., Palatucci, G. and Valdinoci, E.. Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521–573.CrossRefGoogle Scholar
26Dipierro, S., Figalli, A. and Valdinoci, E.. Strongly nonlocal dislocation dynamics in crystals. Comm. Partial Differ. Equ. 39 (2014), 2351–2387.CrossRefGoogle Scholar
27Dipierro, S., Ros-Oton, X. and Valdinoci, E.. Nonlocal problems with Neumann boundary conditions.Google Scholar
28Evans, L. C.. Partial differential equations. Graduate studies in Mathematics, vol. 19 (Providence, RI: American Mathematical Society, 1998).Google Scholar
29Ihnatsyeva, L., Lehrback, J., Tuominen, H. and Vahakangas, A.. Fractional Hardy inequalities and visibility of the boundary. To be published in Studia Math.Google Scholar
30Leonori, T., Peral, I., Primo, A. and Soria, F.. Basic estimates for solutions of elliptic and parabolic equations for a class of nonlocal operators. Discrete. Contin. Dyn. Syst. 35 (2015), 6031–6068.CrossRefGoogle Scholar
31Martel, Y.. Complete blow up and global behaviour of solutions of $u_t-\Delta u=g(u)$. Ann. Henr. Poincaré 15 (1998), 687–723.CrossRefGoogle Scholar
32Rodriguez-Bernal, A.. Introduction to semigroup theory for partial differential equations. Copyright (c) 1998–2005 by A. Rodriguez–Bernal.Google Scholar
33Ros-Oton, X. and Serra, J.. The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101 (2014), 275–302.CrossRefGoogle Scholar
34Signorini, A.. Questioni di elasticitá non linearizzata e semilinearizzata. Rendiconti di Matematica e delle sue applicazioni 18 (1959), 95–139.Google Scholar
35Silvestre, L., Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60 (2007), 67–112.CrossRefGoogle Scholar
36Sire, Y. and Valdinoci, E.. Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result. J. Funct. Anal. 256 (2009), 1842–1864.CrossRefGoogle Scholar
37Stein, E. M.. Singular integrals and differentiability properties of functions. Princeton Mathematical Series vol. 30 (Princeton, N.J.: Princeton University Press, 1970).Google Scholar
38Toland, J.. The Peierls-Nabarro and Benjamin-Ono equations. J. Funct. Anal. 145 (1997), 136–150.CrossRefGoogle Scholar