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On the validity of the Euler-Lagrange system
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 |
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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