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

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

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.
Citation: Vincenzo Ferone, Carlo Nitsch, Cristina Trombetti. On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue. Communications on Pure & Applied Analysis, 2015, 14 (1) : 63-82. doi: 10.3934/cpaa.2015.14.63
References:
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M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,, National Bureau of Standards Applied Mathematics Series, (1964). Google Scholar

[2]

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

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

[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. doi: 10.1016/j.na.2004.07.051. Google Scholar

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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, (2007), 105. doi: 10.1090/pspum/076.1/2310200. Google Scholar

[6]

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

[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. doi: 10.1007/BF00948733. Google Scholar

[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. doi: 10.1007/s00013-010-0102-8. Google Scholar

[9]

L. Brasco, G. De Philippis and B. Velichkov, Faber-Krahn inequalities in sharp quantitative form,, preprint, (). Google Scholar

[10]

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

[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, 4 (2007), 1022. doi: 10.4171/OWR/2007/18. Google Scholar

[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. doi: 10.1007/s00526-009-0252-3. Google Scholar

[13]

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

[14]

A. Cianchi, V. Ferone, C. Nitsch and C. Trombetti, Balls minimize trace constants in BV,, preprint, (). Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[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. doi: 10.1081/PDE-100107818. Google Scholar

[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. doi: 10.1007/s00205-012-0545-0. Google Scholar

[21]

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

[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. Google Scholar

[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. doi: 10.1016/j.anihpc.2003.09.005. Google Scholar

[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. doi: 10.3934/cpaa.2002.1.359. Google Scholar

[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. doi: 10.1112/S0024609304003819. Google Scholar

[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. doi: 10.4171/RLM/640. Google Scholar

[27]

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

[28]

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

[29]

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

[30]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators,, Frontiers in Mathematics, (2006). Google Scholar

[31]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE,, Lecture Notes in Math., (1150). Google Scholar

[32]

S. Kesavan, Symmetrization & Applications,, Series in Analysis, (2006). doi: 10.1142/9789812773937. Google Scholar

[33]

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

[34]

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

[35]

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

[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. doi: 10.1155/S108533750200088X. Google Scholar

[37]

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

[38]

C. Müller, Spherical Harmonics,, Lecture Notes in Mathematics, (1966). Google Scholar

[39]

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

[40]

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

[41]

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

[42]

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

[43]

I. Tamanini, Regularity results for almost minimal oriented hypersurfaces in $R^N$,, Quaderni del Dipartimento di Matematica dell'Università del Salento, (1984). Google Scholar

[44]

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

[45]

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

[46]

R. Weinstock, Inequalities for a classical eigenvalue problem,, \emph{J. Rational Mech. Anal.}, 3 (1954), 745. Google Scholar

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, (1964). Google Scholar

[2]

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

[3]

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

[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. doi: 10.1016/j.na.2004.07.051. Google Scholar

[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, (2007), 105. doi: 10.1090/pspum/076.1/2310200. Google Scholar

[6]

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

[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. doi: 10.1007/BF00948733. Google Scholar

[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. doi: 10.1007/s00013-010-0102-8. Google Scholar

[9]

L. Brasco, G. De Philippis and B. Velichkov, Faber-Krahn inequalities in sharp quantitative form,, preprint, (). Google Scholar

[10]

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

[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, 4 (2007), 1022. doi: 10.4171/OWR/2007/18. Google Scholar

[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. doi: 10.1007/s00526-009-0252-3. Google Scholar

[13]

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

[14]

A. Cianchi, V. Ferone, C. Nitsch and C. Trombetti, Balls minimize trace constants in BV,, preprint, (). Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[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. doi: 10.1081/PDE-100107818. Google Scholar

[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. doi: 10.1007/s00205-012-0545-0. Google Scholar

[21]

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

[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. Google Scholar

[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. doi: 10.1016/j.anihpc.2003.09.005. Google Scholar

[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. doi: 10.3934/cpaa.2002.1.359. Google Scholar

[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. doi: 10.1112/S0024609304003819. Google Scholar

[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. doi: 10.4171/RLM/640. Google Scholar

[27]

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

[28]

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

[29]

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

[30]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators,, Frontiers in Mathematics, (2006). Google Scholar

[31]

B. Kawohl, Rearrangements and Convexity of Level Sets in PDE,, Lecture Notes in Math., (1150). Google Scholar

[32]

S. Kesavan, Symmetrization & Applications,, Series in Analysis, (2006). doi: 10.1142/9789812773937. Google Scholar

[33]

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

[34]

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

[35]

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

[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. doi: 10.1155/S108533750200088X. Google Scholar

[37]

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

[38]

C. Müller, Spherical Harmonics,, Lecture Notes in Mathematics, (1966). Google Scholar

[39]

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

[40]

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

[41]

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

[42]

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

[43]

I. Tamanini, Regularity results for almost minimal oriented hypersurfaces in $R^N$,, Quaderni del Dipartimento di Matematica dell'Università del Salento, (1984). Google Scholar

[44]

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

[45]

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

[46]

R. Weinstock, Inequalities for a classical eigenvalue problem,, \emph{J. Rational Mech. Anal.}, 3 (1954), 745. Google Scholar

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