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Semilinear elliptic equations involving mixed local and nonlocal operators

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

Published online by Cambridge University Press:  14 October 2020

Stefano Biagi ,
Eugenio Vecchi ,
Serena Dipierro  and
Enrico Valdinoci
Stefano Biagi
Affiliation:
Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy (stefano.biagi@polimi.it; eugenio.vecchi@polimi.it)
Eugenio Vecchi
Affiliation:
Politecnico di Milano - Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy (stefano.biagi@polimi.it; eugenio.vecchi@polimi.it)
Serena Dipierro
Affiliation:
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA 6009, Crawley, Australia (serena.dipierro@uwa.edu.au; enrico.valdinoci@uwa.edu.au)
Enrico Valdinoci
Affiliation:
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA 6009, Crawley, Australia (serena.dipierro@uwa.edu.au; enrico.valdinoci@uwa.edu.au)
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Abstract

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In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form − Δ + ( − Δ)s, with s ∈ (0, 1). We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.

Keywords

Operators of mixed orderexistencesymmetrymoving planequalitative properties of solutions

MSC classification

Primary: 35A01: Existence problems
Secondary: 35B65: Smoothness and regularity of solutions 35R11: Fractional partial differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 5 , October 2021 , pp. 1611 - 1641
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

References

Abatangelo, N., Cozzi, M.. An elliptic boundary value problem with fractional nonlinearity, preprint.Google Scholar
Barles, G. and Imbert, C.. Second-order elliptic integro-differential equations viscosity solutions’ theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 567–585.CrossRefGoogle Scholar
Barles, G., Chasseigne, E., Ciomaga, A. and Imbert, C.. Lipschitz regularity of solutions for mixed integro-differential equations. J. Differ. Equ. 252 (2012), 6012–6060.CrossRefGoogle Scholar
Barles, G., Chasseigne, E., Ciomaga, A. and Imbert, C.. Large time behavior of periodic viscosity solutions for uniformly parabolic integro-differential equations. Calc. Var. Partial Differential Equations 50 (2014), 283–304.CrossRefGoogle Scholar
Barlow, M. T., Bass, R. F. and Gui, C.. The Liouville property and a conjecture of De Giorgi. Commun. Pure Appl. Math. 53 (2000), 1007–1038.3.0.CO;2-U>CrossRefGoogle Scholar
Barrios, B., Montoro, L. and Sciunzi, B.. On the moving plane method for nonlocal problems in bounded domains. J. Anal. Math. 135 (2018), 37–57.CrossRefGoogle Scholar
Berestycki, H. and Nirenberg, L.. On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), 1–37.CrossRefGoogle Scholar
Berestycki, H., Hamel, F. and Monneau, R.. One-dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math. J. 103 (2000), 375–396.CrossRefGoogle Scholar
Biagi, S., Dipierro, S., Valdinoci, E., Vecchi, E., Mixed local, nonlocal elliptic operators. regularity and maximum principles, preprint.Google Scholar
Biagi, S., Valdinoci, E. and Vecchi, E.. A symmetry result for elliptic systems in punctured domains. Commun. Pure Appl. Anal. 18 (2019), 2819–2833.CrossRefGoogle Scholar
Biagi, S., Valdinoci, E. and Vecchi, E.. A symmetry result for cooperative elliptic systems with singularities. Publ. Mat. 64 (2020), 621–652.CrossRefGoogle Scholar
Biswas, I. H., Jakobsen, E. R. and Karlsen, K. H.. Viscosity solutions for a system of integro-PDEs and connections to optimal switching and control of jump-diffusion processes. Appl. Math. Optim. 62 (2010), 47–80.CrossRefGoogle Scholar
Biswas, I. H., Jakobsen, E. R. and Karlsen, K. H.. Difference-quadrature schemes for nonlinear degenerate parabolic integro-PDE. SIAM J. Numer. Anal. 48 (2010), 1110–1135.CrossRefGoogle Scholar
Blazevski, D. and del-Castillo-Negrete, D.. Local, nonlocal anisotropic transport in reversed shear magnetic fields Shearless Cantori and nondiffusive transport. Phys, Rev. E 87 (2013), 063106.CrossRefGoogle ScholarPubMed
Brezis, H.. Functional analysis, sobolev spaces and partial differential equations (New York: Universitext. Springer, 2011).CrossRefGoogle Scholar
Cabré, X., Dipierro, S. and Valdinoci, E.. The Bernstein technique for integro-differential equations, preprint.Google Scholar
Cabré, X. and Serra, J.. An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions. Nonlinear Anal. 137 (2016), 246–265.CrossRefGoogle Scholar
Cabré, X. and Sire, Y.. Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions. Trans. Amer. Math. Soc. 367 (2015), 911–941.CrossRefGoogle Scholar
Caffarelli, L. and Valdinoci, E.. A priori bounds for solutions of a nonlocal evolution PDE, Analysis and numerics of partial differential equations 141–163, Springer INdAM Ser., 4 (Milan: Springer, 2013).Google Scholar
Caffarelli, L., Li, Y. Y. and Nirenberg, L.. Some remarks on singular solutions of nonlinear elliptic equations. II. Symmetry and monotonicity via moving planes, Advances in geometric analysis (Somerville, MA: Int. Press, 2012), vol. 21, 97–105.Google Scholar
Chen, Z.-Q., Kim, P., Panki, R. and Vondraček, Z.. Sharp Green function estimates for Δ + Δα/2 in C 1, 1 open sets and their applications. Illinois J. Math. 54 (2010), 981–1024.(2012).CrossRefGoogle Scholar
Chen, Z.-Q., Kim, P., Panki, R. and Vondraček, Z.. Boundary Harnack principle for Δ + Δα/2. Trans. Amer. Math. Soc. 364 (2012), 4169–4205.CrossRefGoogle Scholar
Ciomaga, A.. On the strong maximum principle for second-order nonlinear parabolic integro-differential equations. Adv. Differ. Equ. 17 (2012), 635–671.Google Scholar
Colasuonno, F. and Vecchi, E.. Symmetry and rigidity in the hinged composite plate problem. J. Differential Equations 266 (2019), 4901–4924.CrossRefGoogle Scholar
de la Llave, R. and Valdinoci, E.. A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1309–1344.CrossRefGoogle Scholar
del Teso, F., Endal, J. and Jakobsen, E. R.. On distributional solutions of local and non-local problems of porous medium type. C. R. Math. Acad. Sci. Paris 355 (2017), 1154–1160.CrossRefGoogle Scholar
Dell'Oro, F., Pata, V.. Second order linear evolution equations with general dissipation, pre-print, available at https://arxiv.org/pdf/1811.07667.pdf.Google Scholar
Dipierro, S., Montoro, L., Peral, I. and Sciunzi, B.. Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential. Calc. Var. Part. Differ. Equ. 55 (2016), 29, Art. 99.CrossRefGoogle Scholar
Dipierro, S.. Proietti Lippi, E. and Valdinoci, E.. (Non)local logistic equations with Neumann conditions, preprint.Google Scholar
Dipierro, S., Soave, N. and Valdinoci, E.. On fractional elliptic equations in Lipschitz sets, epigraphs regularity, monotonicity and rigidity results. Math. Ann. 369 (2017), 1283–1326.CrossRefGoogle Scholar
Dipierro, S., Valdinoci, E. and Vespri, V.. Decay estimates for evolutionary equations with fractional time-diffusion. J. Evol. Equ. 19 (2019), 435–462.CrossRefGoogle Scholar
Esposito, F.. Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities. Discrete Contin. Dyn. Syst. 40 (2020), 549–577.CrossRefGoogle Scholar
Fall, M. M. and Jarohs, S.. Overdetermined problems with fractional Laplacian. ESAIM Control Optim. Calc. Var. 21 (2015), 924–938.CrossRefGoogle Scholar
Farina, A.. Symmetry for solutions of semilinear elliptic equations in ℝN and related conjectures. Papers in memory of Ennio De Giorgi. Ricerche Mat. 48 suppl. (1999), 129–154.Google Scholar
Farina, A.. Propriétés de monotonie et de symétrie unidimensionnelle pour les solutions de Δ u + f(u) = 0 avec des fonctions f éventuellement discontinues. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), 973–978.CrossRefGoogle Scholar
Farina, A.. Monotonicity and one-dimensional symmetry for the solutions of Δu + f(u) = 0 in ℝN with possibly discontinuous nonlinearity. Adv. Math. Sci. Appl. 11 (2001), 811–834.Google Scholar
Farina, A. and Valdinoci, E.. 1D symmetry for solutions of semilinear and quasilinear elliptic equations. Trans. Amer. Math. Soc. 363 (2011), 579–609.CrossRefGoogle Scholar
Farina, A. and Valdinoci, E.. Rigidity results for elliptic PDEs with uniform limits an abstract framework with applications. Indiana Univ. Math. J. 60 (2011), 121–141.CrossRefGoogle Scholar
Gibbons, G. W. and Townsend, P. K.. Bogomol'nyi equation for intersecting domain walls. Phys. Rev. Lett. 83 (1999), 1727–1730.CrossRefGoogle Scholar
Gidas, B., Ni, B. W. M. and Nirenberg, L.. Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68 (1979), 209–243.CrossRefGoogle Scholar
Jakobsen, E. R. and Karlsen, K. H.. Continuous dependence estimates for viscosity solutions of integro-PDEs. J. Differ. Equ. 212 (2005), 278–318.CrossRefGoogle Scholar
Jakobsen, E. R. and Karlsen, K. H.. A ‘maximum principle for semicontinuous functions’ applicable to integro-partial differential equations. NoDEA Nonlinear Differ. Equ. Appl. 13 (2006), 137–165.CrossRefGoogle Scholar
Jarohs, S. and Weth, T.. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete Contin. Dyn. Syst. 34 (2014), 2581–2615.CrossRefGoogle Scholar
Jarohs, S. and Weth, T.. Symmetry via antisymmetric maximum principles in nonlocal problems of variable order. Ann. Mat. Pura Appl. 195 (2016), 273–291.CrossRefGoogle Scholar
Leoni, G.. A first course in Sobolev spaces. Graduate studies in mathematics, 105. (Providence, RI: American Mathematical Society, 2009), xvi+607.Google Scholar
Montoro, L., Punzo, F. and Sciunzi, B.. Qualitative properties of singular solutions to nonlocal problems. Ann. Mat. Pura Appl. 197 (2018), 941–964.CrossRefGoogle Scholar
Poláčik, P. and Terracini, S.. Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains. Proc. Amer. Math. Soc. 142 (2014), 1249–1259.CrossRefGoogle Scholar
Ros-Oton, X. and Serra, J.. Nonexistence results for nonlocal equations with critical and supercritical nonlinearities. Commun. Partial Differ. Equ. 40 (2015), 115–133.CrossRefGoogle Scholar
Sciunzi, B.. On the moving plane method for singular solutions to semilinear elliptic equations. J. Math. Pures Appl. 108 (2017), 111–123.CrossRefGoogle Scholar
Soave, N. and Valdinoci, E.. Overdetermined problems for the fractional Laplacian in exterior and annular sets. J. Anal. Math. 137 (2019), 101–134.CrossRefGoogle Scholar
Troy, W. C.. Symmetry properties in systems of semilinear elliptic equations. J. Differ. Equ. 42 (1981), 400–413.CrossRefGoogle Scholar