On symmetry of nonnegative solutions of elliptic equations

Poláčik, P.

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 1, pp. 1-19.

Nous considérons le problème de Dirichlet pour une classe dʼéquations elliptiques complètement non-linéaires sur un domaine borné Ω. Nous supposons que Ω est symétrique par rapport à un hyperplan H et convexe dans la direction perpendiculaire à H. Par un résultat bien connu de Gidas, Ni et Nirenberg ainsi que par ses généralisations, toutes les solutions positives sont symétriques par rapport à une réflextion de H et décroissent à partir de leur distance de lʼhyperplan dans la direction orthogonale à H. Pour les solutions non-négatives, ce résultat nʼest pas toujours vrai. Nous montrons que, néanmoins, le résultat sur la symétrie reste valable pour les solutions positives : toute solution non-négative u est symétrique par rapport à H. En outre, nous montrons que si $u\neg \equiv 0$, alors lʼensemble nodal de u divise le domaine Ω en un nombre fini de sous-domaines symétriques sous réflextion dans lesquels u possède la symétrie habituelle de Gidas–Ni–Nirenberg et des propriétés de monotonie. Nous montrons aussi plusieurs exemples de solutions positives avec un ensemble nodal intérieur non vide.

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction orthogonal to H. For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution u is symmetric about H. Moreover, we prove that if $u\neg \equiv 0$, then the nodal set of u divides the domain Ω into a finite number of reflectionally symmetric subdomains in which u has the usual Gidas–Ni–Nirenberg symmetry and monotonicity properties. We also show several examples of nonnegative solutions with a nonempty interior nodal set.

[1] A.D. Alexandrov, A characteristic property of spheres, Ann. Math. Pura Appl. 58 (1962), 303-354 | MR

[2] J.M. Arrieta, Domain dependence of elliptic operators in divergence form, Workshop on Differential Equations and Nonlinear Analysis Águas de Lindóia, 1996 Resenhas 3 (1997), 107-122 | MR | Zbl

[3] I. Babuška, R. Výborný, Continuous dependence of eigenvalues on the domain, Czechoslovak Math. J. 15 no. 90 (1965), 169-178 | EuDML | MR | Zbl

[4] H. Berestycki, Qualitative properties of positive solutions of elliptic equations, Partial Differential Equations, Praha, 1998, Chapman & Hall/CRC Res. Notes Math. vol. 406, Chapman & Hall/CRC, Boca Raton, FL (2000), 34-44 | MR | Zbl

[5] H. Berestycki, L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), 1-37 | MR | Zbl

[6] F. Brock, Continuous rearrangement and symmetry of solutions of elliptic problems, Proc. Indian Acad. Sci. Math. Sci. 110 (2000), 157-204 | MR | Zbl

[7] X. Cabré, On the Alexandroff–Bakelman–Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 48 (1995), 539-570 | MR | Zbl

[8] A. Castro, R. Shivaji, Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric, Comm. Partial Differential Equations 14 (1989), 1091-1100 | MR | Zbl

[9] F. Da Lio, B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. European Math. Soc. 9 (2007), 317-330 | EuDML | MR | Zbl

[10] L. Damascelli, F. Pacella, M. Ramaswamy, A strong maximum principle for a class of non-positone singular elliptic problems, Nonlinear Differential Equations Appl. 10 no. 2 (2003), 187-196 | MR | Zbl

[11] E.N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations 74 (1988), 120-156 | MR | Zbl

[12] E.N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc. 46 (1992), 425-434 | MR | Zbl

[13] J. Dolbeault, P. Felmer, Monotonicity up to radially symmetric cores of positive solutions to nonlinear elliptic equations: local moving planes and unique continuation in a non-Lipschitz case, Nonlinear Anal. 58 (2004), 299-317 | MR | Zbl

[14] Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, vol. 1, Series in Partial Differential Equations and Applications vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006) | MR

[15] J. Földes, On symmetry properties of parabolic equations in bounded domains, preprint. | MR

[16] L.E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Tracts in Mathematics vol. 128, Cambridge University Press, Cambridge (2000) | MR | Zbl

[17] I. Fragalà, F. Gazzola, J. Lamboley, M. Pierre, Counterexamples to symmetry for partially overdetermined elliptic problems, Analysis (Munich) 29 no. 1 (2009), 85-93 | MR | Zbl

[18] B. Gidas, W.-M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243 | MR | Zbl

[19] P. Hartman, A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math. 75 (1953), 449-476 | MR | Zbl

[20] B. Helffer, T. Hoffmann-Ostenhof, S. Terracini, Nodal domains and spectral minimal partitions, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 101-138 | EuDML | Numdam | MR | Zbl

[21] P. Hess, P. Poláčik, Symmetry and convergence properties for non-negative solutions of nonautonomous reaction–diffusion problems, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 573-587 | MR | Zbl

[22] L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Classics in Mathematics, Springer, Berlin (2007) | MR

[23] B. Kawohl, Symmetrization – or how to prove symmetry of solutions to a PDE, Partial Differential Equations, Praha, 1998, Chapman & Hall/CRC Res. Notes Math. vol. 406, Chapman & Hall/CRC, Boca Raton, FL (2000), 214-229 | MR | Zbl

[24] O. Lopes, Radial symmetry of minimizers for some translation and rotation invariant functionals, J. Differential Equations 124 (1996), 378-388 | MR | Zbl

[25] C. Miranda, Partial Differential Equations of Elliptic Type, Ergebnisse der Mathematik und ihrer Grenzgebiete Band 2, Springer-Verlag, New York (1970) | MR | Zbl

[26] W.-M. Ni, Qualitative properties of solutions to elliptic problems, M. Chipot, P. Quittner (ed.), Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 1, Elsevier (2004), 157-233 | MR

[27] P. Poláčik, Symmetry properties of positive solutions of parabolic equations: a survey, W.-Y. Lin, Y. Du, H. Ishii (ed.), Recent Progress on Reaction–Diffusion Systems and Viscosity Solutions, World Scientific (2009), 170-208

[28] P. Poláčik, Symmetry of nonnegative solutions of elliptic equations via a result of Serrin, Comm. Partial Differential Equations 36 (2011), 657-669 | MR | Zbl

[29] P. Pucci, J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications vol. 73, Birkhäuser Verlag, Basel (2007) | MR | Zbl

[30] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304-318 | MR | Zbl