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:
[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.  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), 515-557. 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), 1125-1145. 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, Proc. Sympos. Pure Math., 76, Part 1, Amer. Math. Soc., Providence, RI, 2007, 105-139. 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), 280-287.  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), 733-742. 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), 391-400. 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), 69-71.  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, G. Buttazzo and A. Henrot), Oberwolfach Rep., 4 (2007), 1022-1023. 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), 75-86. 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), 1179-1191. 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), 617-643. 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), 767-785. doi: 10.1007/s00208-006-0753-8.  Google Scholar

[17]

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

[18]

D. Daners, Krahn's proof of the Rayleigh conjecture revisited, Arch. Math., 96 (2011), 187-199. 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), 2189-2210. 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), 821-851. 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), 253-264.  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), 169-172. 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), 795-805. 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), 359-378. 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), 119-125. 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), 467-475. 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), 447-489. 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), 619-638. doi: 10.2307/2001401.  Google Scholar

[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.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann., 94 (1925), 97-100. 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, Cambridge University Press, 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), 83-121. 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), 287-293. doi: 10.1155/S108533750200088X.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[40]

J. D. Rossi, First variations of the best Sobolev trace constant with respect to the domain, Canad. Math. Bull., 51 (2008), 140-145. 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), 343-356.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[45]

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

[46]

R. Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal., 3 (1954), 745-753.  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, 55, U.S. Government Printing Office, Washington, D.C., 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), 515-557. 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), 1125-1145. 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, Proc. Sympos. Pure Math., 76, Part 1, Amer. Math. Soc., Providence, RI, 2007, 105-139. 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), 280-287.  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), 733-742. 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), 391-400. 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), 69-71.  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, G. Buttazzo and A. Henrot), Oberwolfach Rep., 4 (2007), 1022-1023. 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), 75-86. 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), 1179-1191. 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), 617-643. 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), 767-785. doi: 10.1007/s00208-006-0753-8.  Google Scholar

[17]

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

[18]

D. Daners, Krahn's proof of the Rayleigh conjecture revisited, Arch. Math., 96 (2011), 187-199. 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), 2189-2210. 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), 821-851. 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), 253-264.  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), 169-172. 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), 795-805. 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), 359-378. 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), 119-125. 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), 467-475. 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), 447-489. 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), 619-638. doi: 10.2307/2001401.  Google Scholar

[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.  Google Scholar

[30]

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

[31]

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

[32]

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

[33]

E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann., 94 (1925), 97-100. 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, Cambridge University Press, 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), 83-121. 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), 287-293. doi: 10.1155/S108533750200088X.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[40]

J. D. Rossi, First variations of the best Sobolev trace constant with respect to the domain, Canad. Math. Bull., 51 (2008), 140-145. 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), 343-356.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[45]

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

[46]

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

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