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September 2019, 18(5): 2819-2833. doi: 10.3934/cpaa.2019126

## A symmetry result for elliptic systems in punctured domains

 1 Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica della Marche, Via Brecce Bianche, 60131, Ancona, Italy 2 Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA 6009 Crawley, Australia 3 Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38123, Povo (Trento), Italy

Received  October 2018 Revised  January 2019 Published  April 2019

Fund Project: The authors are members of INdAM/GNAMPA. The first and the third author are partially supported by the INdAM-GNAMPA Project 2018 "Problemi di curvatura relativi ad operatori ellittico-degeneri". The second author is supported by the Australian Research Council Discovery Project 170104880 NEW "Nonlocal Equations at Work"

We consider an elliptic system of equations in a punctured bounded domain. We prove that if the domain is convex in one direction and symmetric with respect to the reflections induced by the normal hyperplane to such a direction, then the solution is necessarily symmetric under this reflection and monotone in the corresponding direction. As a consequence, we prove symmetry results also for a related polyharmonic problem of any order with Navier boundary conditions.

Citation: Stefano Biagi, Enrico Valdinoci, Eugenio Vecchi. A symmetry result for elliptic systems in punctured domains. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2819-2833. doi: 10.3934/cpaa.2019126
##### References:
 [1] D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer-Verlag London, Ltd., London, 2001. doi: 10.1007/978-1-4471-0233-5. [2] E. Berchio, F. Gazzola and T. Weth, Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math., 620 (2008), 165-183. doi: 10.1515/CRELLE.2008.052. [3] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N. S.), 22 (1991), 1-37. doi: 10.1007/BF01244896. [4] L. Caffarelli, Y. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. Ⅱ. Symmetry and monotonicity via moving planes, Advances in Geometric Analysis, Int. Press, Somerville, MA, 21 (2012), 97–105. [5] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. [6] F. Colasuonno and E. Vecchi, Symmetry in the composite plate problem, Commun. Contemp. Math., 21 (2019) no.2, 1850019, 34 pp. doi: 10.1142/S0219199718500190. [7] F. Colasuonno and E. Vecchi, Symmetry and rigidity in the hinged composite plate problem, J. Differential Equations, 266 (2019), 4901-4924. doi: 10.1016/j.jde.2018.10.011. [8] L. Damascelli and F. Pacella, Symmetry results for cooperative elliptic systems via linearization, SIAM J. Math. Anal., 45 (2013), 1003-1026. doi: 10.1137/110853534. [9] D. G. De Figueiredo, Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 119-123. doi: 10.1007/BF01193947. [10] F. Esposito, A. Farina and B. Sciunzi, Qualitative properties of singular solutions to semilinear elliptic problems, J. Differential Equations, 265 (2018), 1962-1983. doi: 10.1016/j.jde.2018.04.030. [11] A. Ferrero, F. Gazzola and T. Weth, Positivity, symmetry and uniqueness for minimizers of second-order Sobolev inequalities, Ann. Mat. Pura Appl., 186 (2007), 565-578. doi: 10.1007/s10231-006-0019-9. [12] F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin, 2010, 1991. doi: 10.1007/978-3-642-12245-3. [13] B. Gidas, B, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [15] L. Montoro, F. Punzo and B. Sciunzi, Qualitative properties of singular solutions to nonlocal problems, Ann. Mat. Pura Appl., 197 (2018), 941-964. doi: 10.1007/s10231-017-0710-z. [16] P. Pizzetti, Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera, Rendiconti Lincei, 18 (1909), 182-185. [17] B. Sciunzi, On the moving plane method for singular solutions to semilinear elliptic equations, J. Math. Pures Appl., 108 (2017), 111-123. doi: 10.1016/j.matpur.2016.10.012. [18] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. doi: 10.1007/BF00250468. [19] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264. [20] W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. doi: 10.1016/0022-0396(81)90113-3.

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##### References:
 [1] D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer-Verlag London, Ltd., London, 2001. doi: 10.1007/978-1-4471-0233-5. [2] E. Berchio, F. Gazzola and T. Weth, Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math., 620 (2008), 165-183. doi: 10.1515/CRELLE.2008.052. [3] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N. S.), 22 (1991), 1-37. doi: 10.1007/BF01244896. [4] L. Caffarelli, Y. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations. Ⅱ. Symmetry and monotonicity via moving planes, Advances in Geometric Analysis, Int. Press, Somerville, MA, 21 (2012), 97–105. [5] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. [6] F. Colasuonno and E. Vecchi, Symmetry in the composite plate problem, Commun. Contemp. Math., 21 (2019) no.2, 1850019, 34 pp. doi: 10.1142/S0219199718500190. [7] F. Colasuonno and E. Vecchi, Symmetry and rigidity in the hinged composite plate problem, J. Differential Equations, 266 (2019), 4901-4924. doi: 10.1016/j.jde.2018.10.011. [8] L. Damascelli and F. Pacella, Symmetry results for cooperative elliptic systems via linearization, SIAM J. Math. Anal., 45 (2013), 1003-1026. doi: 10.1137/110853534. [9] D. G. De Figueiredo, Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 119-123. doi: 10.1007/BF01193947. [10] F. Esposito, A. Farina and B. Sciunzi, Qualitative properties of singular solutions to semilinear elliptic problems, J. Differential Equations, 265 (2018), 1962-1983. doi: 10.1016/j.jde.2018.04.030. [11] A. Ferrero, F. Gazzola and T. Weth, Positivity, symmetry and uniqueness for minimizers of second-order Sobolev inequalities, Ann. Mat. Pura Appl., 186 (2007), 565-578. doi: 10.1007/s10231-006-0019-9. [12] F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin, 2010, 1991. doi: 10.1007/978-3-642-12245-3. [13] B. Gidas, B, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [15] L. Montoro, F. Punzo and B. Sciunzi, Qualitative properties of singular solutions to nonlocal problems, Ann. Mat. Pura Appl., 197 (2018), 941-964. doi: 10.1007/s10231-017-0710-z. [16] P. Pizzetti, Sulla media dei valori che una funzione dei punti dello spazio assume alla superficie di una sfera, Rendiconti Lincei, 18 (1909), 182-185. [17] B. Sciunzi, On the moving plane method for singular solutions to semilinear elliptic equations, J. Math. Pures Appl., 108 (2017), 111-123. doi: 10.1016/j.matpur.2016.10.012. [18] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. doi: 10.1007/BF00250468. [19] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264. [20] W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. doi: 10.1016/0022-0396(81)90113-3.
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