Symmetry and rigidity results for composite membranes and plates

Eugenio Vecchi

Abstract


The composite membrane problem is an eigenvalue optimization problem deeply studied from the beginning of the '00's. In this note we survey most of the results proved by several authors over the last twenty years, up to the recent paper [14] written in collaboration with Giovanni Cupini.

We finally introduce an eigenvalue optimization problem for a fourth order operator, called composite plate problem and we present the symmetry and rigidity results obtained in this framework. These last mentioned results are part of the papers [12,13], written in collaboration with Francesca Colasuonno.


Keywords


Faber-Krahn inequality; Navier boundary conditions; moving plane method

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References


E. Berchio and A. Falocchi, Maximizing the ratio of eigenvalues of non-homogeneous partially hinged plates, arXiv:1907.11097 (2019, preprint).

E. Berchio, A. Falocchi, A. Ferrero and D. Ganguly, On the first frequency of reinforced partially hinged plates, Commun. Contemp. Math. (2019), 1950074 (in press).

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.

S. Biagi, E. Valdinoci and E. Vecchi, A symmetry result for elliptic systems in punctured domains, Commun. Pure Appl. Anal. 18(5), (2019), 2819–2833.

S. Biagi, E. Valdinoci and E. Vecchi, A symmetry result for cooperative elliptic systems with singularities, to appear in Publ. Mat.

J.E. Brothers and W.P. Ziemer, Minimal rearrangements of Sobolev functions, J. reine angew. Math. 384, (1988), 153–179.

S. Chanillo, Conformal geometry and the composite membrane problem, Anal. Geom. Metr. Spaces 1, (2013), 31–35.

S. Chanillo, D. Grieser, M. Imai, K. Kurata and I. Ohnishi, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Comm. Math. Phys. 214, (2000), 315–337.

S. Chanillo, D. Grieser and K. Kurata, The free boundary problem in the optimization of composite membranes, Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999), Amer. Math. Soc., Providence, RI, 2000, 268, 61–81.

S. Chanillo and C. E. Kenig, Weak uniqueness and partial regularity for the composite membrane problem, J. Eur. Math. Soc. (JEMS) 10, (2008), 705–737.

S. Chanillo, C. E. Kenig and T. To, Regularity of the minimizers in the composite membrane problem in R2, J. Funct. Anal. 255, (2008), 2299–2320.

[[12]] F. Colasuonno and E. Vecchi, Symmetry in the composite plate problem, Commun. Contemp. Math. 21(2), (2019), 1850019.

[[13]] F. Colasuonno and E. Vecchi, Symmetry and rigidity for the hinged composite plate problem, J. Differential Equations 266(8), (2019), 4901–4924.

[[14]] G. Cupini and E. Vecchi, Faber-Krahn and Lieb-type inequalities for the composite membrane problem, Commun. Pure Appl. Anal. 18(5), (2019), 2679–2691.

L. Damascelli and F. Pacella, Symmetry results for cooperative elliptic systems via linearization, SIAM J. Math. Anal. 45, (2013), 1003–1026.

D. G. De Figueiredo, Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differential Equations Appl. 1, (1994), 119–123.

A. Ferrero, F. Gazzola and T. Weth, Positivity, symmetry and uniqueness for minimizers of second-order Sobolev inequalities, Ann. Mat. Pura Appl. 186(4), (2007), 565–578.

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic boundary value problems, Springer- Verlag, Berlin, 2010, 1991, pages xviii+423.

B. Gidas, B, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68, (1979), 209–243.

S. Kesavan, Some remarks on a result of Talenti, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15(4), (1988), 453–465.

S. Kesavan, Symmetrization and applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2006), xii+148 pp.

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43, (1971), 304–318.

H. Shahgholian, The singular set for the composite membrane problem, Comm. Math. Phys. 271, (2007), 93–101.

G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (4), (1976), 697–718.

W.C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations 42, (1981), 400–413.




DOI: 10.6092/issn.2240-2829/10587

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