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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 and 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. 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.

show all references

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. 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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