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A symmetry result for elliptic systems in punctured domains

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"

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  • 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.

    Mathematics Subject Classification: 35J47, 35B06, 31B30, 35J40.


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  • [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. BerchioF. 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. EspositoA. 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. FerreroF. 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. GidasB, 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. MontoroF. 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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