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The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited
Optimal Szegö-Weinberger type inequalities
1. | Universitä Rostock, Institut für Mathematik, Ulmenstraße 69, 18057 Rostock, Germany |
2. | Università degli Studi di Napoli Federico II, Dipartimento di Matematica e Applicazioni "R. Caccioppoli, Italy |
3. | Seconda Università degli Studi di Napoli, Dipartimento di Matematica e Fisica, Via Vivaldi, 81100 Caserta, Italy |
References:
[1] |
M. S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, Spectral Theory and Geometry, Edinburgh, 1998, in: London Math. Soc. Lecture Note Ser., vol. 273, Cambridge Univ. Press, Cambridge, 1999, 95-139.
doi: 10.1017/CBO9780511566165.007. |
[2] |
M. S. Ashbaugh and R. Benguria, Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature, J. Lond. Math. Soc. (2), 52 (1995), 402-416.
doi: 10.1112/jlms/52.2.402. |
[3] |
C. Bandle, Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics 7, Pitman, Boston, Mass.-London, 1980. |
[4] |
R. Benguria and H. Linde, A second eigenvalue bound for the Dirichlet Schrödinger operator, Comm. Math. Phys., 267 (2006), 741-755.
doi: 10.1007/s00220-006-0041-1. |
[5] |
M. F. Betta, F. Brock, A. Mercaldo and M. R. Posteraro, Weighted isoperimetric inequalities on $\mathbbR^n$ and applications to rearrangements, Math. Nachr., 281 (2008), 466-498.
doi: 10.1002/mana.200510619. |
[6] |
B. Brandolini, F. Chiacchio, D. Krejčiřík and C. Trombetti, The equality case in a Poincaré-Wirtinger type inequality,, \arXiv{1410.0676}., ().
|
[7] |
B. Brandolini, F. Chiacchio, A. Henrot A. and C. Trombetti, Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift, J. Differential Equations, 259 (2015), 708-727.
doi: 10.1016/j.jde.2015.02.028. |
[8] |
L. Brasco, C. Nitsch and C. Trombetti, An inequality à la Szegö-Weinberger for the $p-$Laplacian on convex sets,, \emph{Commun. Contemp. Math.}, ().
|
[9] |
F. Brock, F. Chiacchio and A. Mercaldo, Weighted isoperimetric inequalities in cones and applications, Nonlinear Anal., 75 (2012), 5737-5755.
doi: 10.1016/j.na.2012.05.011. |
[10] |
F. Brock, A. Mercaldo and M. R. Posteraro, On isoperimetric inequalities with respect to infinite measures, Rev. Mat. Iberoam., 29 (2013), 665-690.
doi: 10.4171/RMI/734. |
[11] |
I. Chavel, Lowest-eigenvalue inequalities, in Proc. Sympos. Pure Math., XXXVI, Geometry of the Laplace operator, pp. 79-89, Amer. Math. Soc., Providence, R.I., (1980). |
[12] |
I. Chavel, Eigenvalues in Riemannian Geometry, New York, Academic Press, 2001. |
[13] |
F. Chiacchio and G. di Blasio, Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space, Ann. I. H. Poincaré -AN, 29 (2012), 199-216.
doi: 10.1016/j.anihpc.2011.10.002. |
[14] |
K. M. Chong and N. M. Rice, Equimeasurable rearrangements of functions, in Queen's Papers in Pure and Applied Mathematics, No. 28, Queen's University, (1971). |
[15] |
R. Courant and D. Hilbert, Methods of Mathematical Physics vol. I and II, Interscience Publichers New York-London, 1966. |
[16] |
F. Della Pietra and N. Gavitone, Faber-Krahn inequality for anisotropic eigenvalue problems with robin boundary conditions, Potential Analysis, 41 (2014), 1147-1166.
doi: 10.1007/s11118-014-9412-y. |
[17] |
F. Della Pietra and N. Gavitone, Stability results for some fully nonlinear eigenvalue estimates, Communications in Contemporary Mathematics, 16 (2014) 1350039 (23 pages).
doi: 10.1142/S0219199713500399. |
[18] |
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006. |
[19] |
A. Henrot and M. Pierre, Variation et optimisation de formes. Une analyse géométrique, Mathématiques & Applications, vol. 48, Springer, Berlin, 2005.
doi: 10.1007/3-540-37689-5. |
[20] |
B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics 1150. New York: Springer Verlag, 1985. |
[21] |
S. Kesavan, Symmetrization & Applications, Series in Analysis, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
doi: 10.1142/9789812773937. |
[22] |
E. T. Kornhauser and I. Stakgold, A variational theorem for $\nabla ^{2}u+\lambda u =0 $ and its applications, J. Math. Phys., 31 (1952), 45-54. |
[23] |
R. S. Laugesen and B. A. Siudeja, Maximizing Neumann fundamental tones of triangles, J. Math. Phys., 50 (2009), 112903, 18 pp.
doi: 10.1063/1.3246834. |
[24] |
C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, 17, Springer-Verlag, Berlin-New York 1966. |
[25] |
Y. Naito and T. Suzuki, Radial symmetry of self-similar solutions for semilinear heat equations, J. Differential Equations, 163 (2000), 407-428.
doi: 10.1006/jdeq.1999.3742. |
[26] |
J. M. Rakotoson and B. Simon, Relative rearrangement on a measure space application to the regularuty of weighted monotone rearrangement, I, II, Appl. Math. Lett., 6 (1993), 75-78, 79-82.
doi: 10.1016/0893-9659(93)90152-D. |
[27] |
G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal., 3 (1954), 343-356. |
[28] |
M. E. Taylor, Partial Differential Equations Vol.II, Qualitative Studies of Linear Equations. Appl. Math. Sciences 116, Springer, N.Y. (1996).
doi: 10.1007/978-1-4757-4187-2. |
[29] |
H. F. Weinberger, An isoperimetric inequality for the N-dimensional free membrane problem, J. Rational Mech. Anal., 5 (1956), 633-636. |
show all references
References:
[1] |
M. S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, Spectral Theory and Geometry, Edinburgh, 1998, in: London Math. Soc. Lecture Note Ser., vol. 273, Cambridge Univ. Press, Cambridge, 1999, 95-139.
doi: 10.1017/CBO9780511566165.007. |
[2] |
M. S. Ashbaugh and R. Benguria, Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature, J. Lond. Math. Soc. (2), 52 (1995), 402-416.
doi: 10.1112/jlms/52.2.402. |
[3] |
C. Bandle, Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics 7, Pitman, Boston, Mass.-London, 1980. |
[4] |
R. Benguria and H. Linde, A second eigenvalue bound for the Dirichlet Schrödinger operator, Comm. Math. Phys., 267 (2006), 741-755.
doi: 10.1007/s00220-006-0041-1. |
[5] |
M. F. Betta, F. Brock, A. Mercaldo and M. R. Posteraro, Weighted isoperimetric inequalities on $\mathbbR^n$ and applications to rearrangements, Math. Nachr., 281 (2008), 466-498.
doi: 10.1002/mana.200510619. |
[6] |
B. Brandolini, F. Chiacchio, D. Krejčiřík and C. Trombetti, The equality case in a Poincaré-Wirtinger type inequality,, \arXiv{1410.0676}., ().
|
[7] |
B. Brandolini, F. Chiacchio, A. Henrot A. and C. Trombetti, Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift, J. Differential Equations, 259 (2015), 708-727.
doi: 10.1016/j.jde.2015.02.028. |
[8] |
L. Brasco, C. Nitsch and C. Trombetti, An inequality à la Szegö-Weinberger for the $p-$Laplacian on convex sets,, \emph{Commun. Contemp. Math.}, ().
|
[9] |
F. Brock, F. Chiacchio and A. Mercaldo, Weighted isoperimetric inequalities in cones and applications, Nonlinear Anal., 75 (2012), 5737-5755.
doi: 10.1016/j.na.2012.05.011. |
[10] |
F. Brock, A. Mercaldo and M. R. Posteraro, On isoperimetric inequalities with respect to infinite measures, Rev. Mat. Iberoam., 29 (2013), 665-690.
doi: 10.4171/RMI/734. |
[11] |
I. Chavel, Lowest-eigenvalue inequalities, in Proc. Sympos. Pure Math., XXXVI, Geometry of the Laplace operator, pp. 79-89, Amer. Math. Soc., Providence, R.I., (1980). |
[12] |
I. Chavel, Eigenvalues in Riemannian Geometry, New York, Academic Press, 2001. |
[13] |
F. Chiacchio and G. di Blasio, Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space, Ann. I. H. Poincaré -AN, 29 (2012), 199-216.
doi: 10.1016/j.anihpc.2011.10.002. |
[14] |
K. M. Chong and N. M. Rice, Equimeasurable rearrangements of functions, in Queen's Papers in Pure and Applied Mathematics, No. 28, Queen's University, (1971). |
[15] |
R. Courant and D. Hilbert, Methods of Mathematical Physics vol. I and II, Interscience Publichers New York-London, 1966. |
[16] |
F. Della Pietra and N. Gavitone, Faber-Krahn inequality for anisotropic eigenvalue problems with robin boundary conditions, Potential Analysis, 41 (2014), 1147-1166.
doi: 10.1007/s11118-014-9412-y. |
[17] |
F. Della Pietra and N. Gavitone, Stability results for some fully nonlinear eigenvalue estimates, Communications in Contemporary Mathematics, 16 (2014) 1350039 (23 pages).
doi: 10.1142/S0219199713500399. |
[18] |
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006. |
[19] |
A. Henrot and M. Pierre, Variation et optimisation de formes. Une analyse géométrique, Mathématiques & Applications, vol. 48, Springer, Berlin, 2005.
doi: 10.1007/3-540-37689-5. |
[20] |
B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics 1150. New York: Springer Verlag, 1985. |
[21] |
S. Kesavan, Symmetrization & Applications, Series in Analysis, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
doi: 10.1142/9789812773937. |
[22] |
E. T. Kornhauser and I. Stakgold, A variational theorem for $\nabla ^{2}u+\lambda u =0 $ and its applications, J. Math. Phys., 31 (1952), 45-54. |
[23] |
R. S. Laugesen and B. A. Siudeja, Maximizing Neumann fundamental tones of triangles, J. Math. Phys., 50 (2009), 112903, 18 pp.
doi: 10.1063/1.3246834. |
[24] |
C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, 17, Springer-Verlag, Berlin-New York 1966. |
[25] |
Y. Naito and T. Suzuki, Radial symmetry of self-similar solutions for semilinear heat equations, J. Differential Equations, 163 (2000), 407-428.
doi: 10.1006/jdeq.1999.3742. |
[26] |
J. M. Rakotoson and B. Simon, Relative rearrangement on a measure space application to the regularuty of weighted monotone rearrangement, I, II, Appl. Math. Lett., 6 (1993), 75-78, 79-82.
doi: 10.1016/0893-9659(93)90152-D. |
[27] |
G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal., 3 (1954), 343-356. |
[28] |
M. E. Taylor, Partial Differential Equations Vol.II, Qualitative Studies of Linear Equations. Appl. Math. Sciences 116, Springer, N.Y. (1996).
doi: 10.1007/978-1-4757-4187-2. |
[29] |
H. F. Weinberger, An isoperimetric inequality for the N-dimensional free membrane problem, J. Rational Mech. Anal., 5 (1956), 633-636. |
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