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On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue

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  • For a given bounded Lipschitz set $\Omega$, we consider a Steklov--type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding $H^1(\Omega)\hookrightarrow L^2(\partial \Omega)$. We prove that a conjectured reverse Faber--Krahn inequality holds true at least in the class of Lipschitz sets which are ``close'' to a ball in a Hausdorff metric sense. The result implies that among sets of prescribed measure, balls are local minimizers of the embedding constant.
    Mathematics Subject Classification: Primary: 46E35, 35P15; Secondary: 28A75.


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  • [1]

    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55, U.S. Government Printing Office, Washington, D.C., 1964.


    E. Acerbi, N. Fusco and M. Morini, Minimality via second variation for a nonlocal isoperimetric problem, Communications in Mathematical Physics, 322 (2013), 515-557.doi: 10.1007/s00220-013-1733-y.


    F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs AMS, 4 (1976).


    F. Andreu, J. M. Mazón, and J. D. Rossi, The best constant for the Sobolev trace embedding from $W^{1,1}(\Omega)$ into $L^1(\partial\Omega)$, Nonlinear Anal., 59 (2004), 1125-1145.doi: 10.1016/j.na.2004.07.051.


    M. S. Ashbaugh and R. D. Benguria, Isoperimetric inequalities for eigenvalues of the Laplacian, Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon's 60th Birthday, Proc. Sympos. Pure Math., 76, Part 1, Amer. Math. Soc., Providence, RI, 2007, 105-139.doi: 10.1090/pspum/076.1/2310200.


    M. Bareket, On an isoperimetric inequality for the first eigenvalue of a boundary value problem, SIAM J. Math. Anal., 8 (1977), 280-287.


    M. H. Bossel, Membranes élastiquement liées: inhomogènes ou sur une surface: une nouvelle extension du théorème isopérimétrique de Rayleigh-Faber-Krahn, Z. Angew. Math. Phys., 39 (1988), 733-742.doi: 10.1007/BF00948733.


    B. Brandolini, C. Nitsch and C. Trombetti, An upper bound for nonlinear eigenvalues on convex domains by means of the isoperimetric deficit, Arch. Math., 94 (2010), 391-400.doi: 10.1007/s00013-010-0102-8.


    L. Brasco, G. De Philippis and B. Velichkov, Faber-Krahn inequalities in sharp quantitative form, preprint, arXiv:1306.0392.


    F. Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem, Z. Angew. Math. Mech., 8 (2001), 69-71.


    F. Brock and D. Daners, Conjecture concerning a Faber-Krahn inequality for Robin problems, In Mini-Workshop: Shape Analysis for Eigenvalues (Organized by D. Bucur, G. Buttazzo and A. Henrot), Oberwolfach Rep., 4 (2007), 1022-1023.doi: 10.4171/OWR/2007/18.


    D. Bucur and D. Daners, An alternative approach to the Faber-Krahn inequality for Robin problems, Calc. Var. Partial Differential Equations, 37 (2010), 75-86.doi: 10.1007/s00526-009-0252-3.


    A. Cianchi, A sharp trace inequality for functions of bounded variation in the ball, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1179-1191.doi: 10.1017/S0308210511000758.


    A. Cianchi, V. Ferone, C. Nitsch and C. Trombetti, Balls minimize trace constants in BV, preprint, arXiv:1301.5770.


    M. Cicalese and G. P. Leonardi, A selection principle for the sharp quantitative isoperimetric inequality, Arch. Ration. Mech. Anal., 206 (2012), 617-643.doi: 10.1007/s00205-012-0544-1.


    D. Daners, A Faber-Krahn inequality for Robin problems in any space dimension, Math. Ann., 333 (2006), 767-785.doi: 10.1007/s00208-006-0753-8.


    D. Daners, Principal eigenvalues for generalized indefinite Robin problems, Potential Anal., 38 (2013), 1047-1069.doi: 10.1007/s11118-012-9306-9.


    D. Daners, Krahn's proof of the Rayleigh conjecture revisited, Arch. Math., 96 (2011), 187-199.doi: 10.1007/s00013-010-0218-x.


    M. del Pino and C. Flores, Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains, Comm. Partial Differential Equations, 26 (2001), 2189-2210.doi: 10.1081/PDE-100107818.


    L. Esposito, V. Ferone, B. Kawohl, C. Nitsch and C. Trombetti, The longest shortest fence and sharp Poincaré-Sobolev inequalities, Arch. Rational. Mech. Anal., 206 (2012), 821-851.doi: 10.1007/s00205-012-0545-0.


    L. Esposito, C. Nitsch and C. Trombetti, Best constants in Poincaré inequalities for convex domains, J. Convex Anal., 20(2013), 253-264.


    G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Münch. Ber., (1923), 169-172.


    J. Fernández Bonder, E. Lami Dozo, and J. D. Rossi, Symmetry properties for the extremals of the Sobolev trace embedding, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 21 (2004), 795-805.doi: 10.1016/j.anihpc.2003.09.005.


    J. Fernández Bonder and J. D. Rossi, Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains, Comm. Pure Appl. Anal., 1 (2002), 359-378.doi: 10.3934/cpaa.2002.1.359.


    J. Fernández Bonder and J. D. Rossi, On the existence of extremals for the Sobolev trace embedding theorem with critical exponent, Bull. London Math. Soc., 37 (2005), 119-125.doi: 10.1112/S0024609304003819.


    V. Ferone, C. Nitsch and C. Trombetti, A remark on optimal weighted Poincaré inequalities for convex domains, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 23 (2012), 467-475.doi: 10.4171/RLM/640.


    A. Figalli and F. Maggi, On the isoperimetric problem for radial log-convex densities, Calc. Var. Partial Differential Equations, 48 (2013), 447-489.doi: 10.1007/s00526-012-0557-5.


    B. Fuglede, Stability in the isoperimetric problem for convex or nearly spherical domains in $R^n$, Trans. Amer. Math. Soc., 314 (1989), 619-638.doi: 10.2307/2001401.


    N. Fusco, F. Maggi and A. Pratelli, Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 51-71.


    A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006.


    B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., vol. 1150, Springer-Verlag, Berlin, 1985.


    S. Kesavan, Symmetrization & Applications, Series in Analysis, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.doi: 10.1142/9789812773937.


    E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann., 94 (1925), 97-100.doi: 10.1007/BF01208645.


    F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics no. 135, Cambridge University Press, 2012.doi: 10.1017/CBO9781139108133.


    F. Maggi and C. Villani, Balls have the worst best Sobolev inequalities, J. Geom. Anal., 15 (2005), 83-121.doi: 10.1007/BF02921860.


    S. Martinez and J. D. Rossi, Isolation and simplicity for the first eigenvalue of the p-laplacian with a nonlinear boundary condition, Abst. Appl. Anal., 7 (2002), 287-293.doi: 10.1155/S108533750200088X.


    V. Maz'ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 342, Springer, Heidelberg, 2011.doi: 10.1007/978-3-642-15564-2.


    C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, 17, Springer-Verlag, Berlin-New York, 1966.


    C. Nitsch, An isoperimetric result for the fundamental frequency via domain derivative, Calc. Var. Partial Differential Equations, 49 (2014), 323-335.doi: 10.1007/s00526-012-0584-2.


    J. D. Rossi, First variations of the best Sobolev trace constant with respect to the domain, Canad. Math. Bull., 51 (2008), 140-145.doi: 10.4153/CMB-2008-016-5.


    G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal., 3 (1954), 343-356.


    I. Tamanini, Boundaries of Caccioppoli sets with Hölder-continuous normal vector, J. Reine Angew. Math., 334 (1982), 27-39.doi: 10.1515/crll.1982.334.27.


    I. Tamanini, Regularity results for almost minimal oriented hypersurfaces in $R^N$, Quaderni del Dipartimento di Matematica dell'Università del Salento, 1, 1984; available for download at http://siba-ese.unile.it/index.php/quadmat.


    D. Valtorta, Sharp estimate on the first eigenvalue of the $p$-Laplacian, Nonlinear Anal., 75 (2012), 4974-4994.doi: 10.1016/j.na.2012.04.012.


    H. F. Weinberger, An isoperimetric inequality for the $N$ dimensional free membrane problem, J. Rational Mech. Anal., 5 (1956), 633-636.


    R. Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3 (1954), 745-753.

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