American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms
  • CPAA Home
  • This Issue
  • Next Article
    Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities
September  2019, 18(5): 2819-2833. doi: 10.3934/cpaa.2019126

A symmetry result for elliptic systems in punctured domains

Stefano Biagi 1, , Enrico Valdinoci 2, and  Eugenio Vecchi 3,

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.

Keywords: Elliptic systems, moving plane method, symmetry of solutions.
Mathematics Subject Classification: 35J47, 35B06, 31B30, 35J40.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[1]

Giuseppe Riey, Berardino Sciunzi. One dimensional symmetry of solutions to some anisotropic quasilinear elliptic equations in the plane. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1157-1166. doi: 10.3934/cpaa.2012.11.1157
[2]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045
[3]

Francesco Esposito. Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 549-577. doi: 10.3934/dcds.2020022
[4]

Serena Dipierro. Geometric inequalities and symmetry results for elliptic systems. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3473-3496. doi: 10.3934/dcds.2013.33.3473
[5]

Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745
[6]

Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886
[7]

Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051
[8]

Yan Deng, Junfang Zhao, Baozeng Chu. Symmetry of positive solutions for systems of fractional Hartree equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3085-3096. doi: 10.3934/dcdss.2021079
[9]

Xavier Cabré. Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 425-457. doi: 10.3934/dcds.2008.20.425
[10]

Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685
[11]

Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41
[12]

Alberto Farina. Some symmetry results for entire solutions of an elliptic system arising in phase separation. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2505-2511. doi: 10.3934/dcds.2014.34.2505
[13]

Xianjin Chen, Jianxin Zhou. A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems. Conference Publications, 2009, 2009 (Special) : 151-160. doi: 10.3934/proc.2009.2009.151
[14]

Shuang Liu, Xinfeng Liu. Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 141-159. doi: 10.3934/dcdsb.2019176
[15]

Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083
[16]

Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071
[17]

Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869
[18]

Cristina Tarsi. Perturbation from symmetry and multiplicity of solutions for elliptic problems with subcritical exponential growth in $\mathbb{R} ^2$. Communications on Pure & Applied Analysis, 2008, 7 (2) : 445-456. doi: 10.3934/cpaa.2008.7.445
[19]

Florin Catrina, Zhi-Qiang Wang. Asymptotic uniqueness and exact symmetry of k-bump solutions for a class of degenerate elliptic problems. Conference Publications, 2001, 2001 (Special) : 80-87. doi: 10.3934/proc.2001.2001.80
[20]

Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (190)
  • HTML views (204)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences