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

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January 2015, 14(1): 63-82. doi: 10.3934/cpaa.2015.14.63

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

Vincenzo Ferone ^1, , Carlo Nitsch ^2, and Cristina Trombetti ^3,

1.	Dipartimento di Matematica e Applicazioni "R. Caccioppoli, Italy
2.	Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, Monte S. Angelo, I-80126 Napoli
3.	Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Complesso Universitario Monte S. Angelo, via Cintia, 80126 Napoli

Received February 2014 Revised February 2014 Published September 2014

Abstract
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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.

Keywords: isoperimetric inequalities for eigenvalues, weighted isoperimetric inequalities., Sharp trace embeddings.

Mathematics Subject Classification: Primary: 46E35, 35P15; Secondary: 28A7.

Citation: Vincenzo Ferone, Carlo Nitsch, Cristina Trombetti. On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue. Communications on Pure and Applied Analysis, 2015, 14 (1) : 63-82. doi: 10.3934/cpaa.2015.14.63

References:

[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.
[2]	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.
[3]	F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs AMS, 4 (1976).
[4]	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.
[5]	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.
[6]	M. Bareket, On an isoperimetric inequality for the first eigenvalue of a boundary value problem, SIAM J. Math. Anal., 8 (1977), 280-287.
[7]	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.
[8]	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.
[9]	L. Brasco, G. De Philippis and B. Velichkov, Faber-Krahn inequalities in sharp quantitative form,, preprint, ().
[10]	F. Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem, Z. Angew. Math. Mech., 8 (2001), 69-71.
[11]	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.
[12]	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.
[13]	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.
[14]	A. Cianchi, V. Ferone, C. Nitsch and C. Trombetti, Balls minimize trace constants in BV,, preprint, ().
[15]	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.
[16]	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.
[17]	D. Daners, Principal eigenvalues for generalized indefinite Robin problems, Potential Anal., 38 (2013), 1047-1069. doi: 10.1007/s11118-012-9306-9.
[18]	D. Daners, Krahn's proof of the Rayleigh conjecture revisited, Arch. Math., 96 (2011), 187-199. doi: 10.1007/s00013-010-0218-x.
[19]	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.
[20]	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.
[21]	L. Esposito, C. Nitsch and C. Trombetti, Best constants in Poincaré inequalities for convex domains, J. Convex Anal., 20(2013), 253-264.
[22]	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.
[23]	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.
[24]	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.
[25]	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.
[26]	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.
[27]	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.
[28]	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.
[29]	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.
[30]	A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006.
[31]	B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., vol. 1150, Springer-Verlag, Berlin, 1985.
[32]	S. Kesavan, Symmetrization & Applications, Series in Analysis, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812773937.
[33]	E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann., 94 (1925), 97-100. doi: 10.1007/BF01208645.
[34]	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.
[35]	F. Maggi and C. Villani, Balls have the worst best Sobolev inequalities, J. Geom. Anal., 15 (2005), 83-121. doi: 10.1007/BF02921860.
[36]	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.
[37]	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.
[38]	C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, 17, Springer-Verlag, Berlin-New York, 1966.
[39]	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.
[40]	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.
[41]	G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal., 3 (1954), 343-356.
[42]	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.
[43]	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.
[44]	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.
[45]	H. F. Weinberger, An isoperimetric inequality for the $N$ dimensional free membrane problem, J. Rational Mech. Anal., 5 (1956), 633-636.
[46]	R. Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3 (1954), 745-753.

show all references

References:

[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.
[2]	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.
[3]	F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs AMS, 4 (1976).
[4]	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.
[5]	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.
[6]	M. Bareket, On an isoperimetric inequality for the first eigenvalue of a boundary value problem, SIAM J. Math. Anal., 8 (1977), 280-287.
[7]	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.
[8]	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.
[9]	L. Brasco, G. De Philippis and B. Velichkov, Faber-Krahn inequalities in sharp quantitative form,, preprint, ().
[10]	F. Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem, Z. Angew. Math. Mech., 8 (2001), 69-71.
[11]	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.
[12]	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.
[13]	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.
[14]	A. Cianchi, V. Ferone, C. Nitsch and C. Trombetti, Balls minimize trace constants in BV,, preprint, ().
[15]	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.
[16]	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.
[17]	D. Daners, Principal eigenvalues for generalized indefinite Robin problems, Potential Anal., 38 (2013), 1047-1069. doi: 10.1007/s11118-012-9306-9.
[18]	D. Daners, Krahn's proof of the Rayleigh conjecture revisited, Arch. Math., 96 (2011), 187-199. doi: 10.1007/s00013-010-0218-x.
[19]	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.
[20]	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.
[21]	L. Esposito, C. Nitsch and C. Trombetti, Best constants in Poincaré inequalities for convex domains, J. Convex Anal., 20(2013), 253-264.
[22]	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.
[23]	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.
[24]	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.
[25]	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.
[26]	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.
[27]	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.
[28]	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.
[29]	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.
[30]	A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006.
[31]	B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., vol. 1150, Springer-Verlag, Berlin, 1985.
[32]	S. Kesavan, Symmetrization & Applications, Series in Analysis, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812773937.
[33]	E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann., 94 (1925), 97-100. doi: 10.1007/BF01208645.
[34]	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.
[35]	F. Maggi and C. Villani, Balls have the worst best Sobolev inequalities, J. Geom. Anal., 15 (2005), 83-121. doi: 10.1007/BF02921860.
[36]	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.
[37]	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.
[38]	C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, 17, Springer-Verlag, Berlin-New York, 1966.
[39]	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.
[40]	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.
[41]	G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal., 3 (1954), 343-356.
[42]	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.
[43]	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.
[44]	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.
[45]	H. F. Weinberger, An isoperimetric inequality for the $N$ dimensional free membrane problem, J. Rational Mech. Anal., 5 (1956), 633-636.
[46]	R. Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3 (1954), 745-753.

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