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March  2016, 15(2): 367-383. doi: 10.3934/cpaa.2016.15.367

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

Received  February 2015 Revised  September 2015 Published  January 2016

Denote with $\mu _{1}(\Omega ;e^{h( |x|) })$ the first nontrivial eigenvalue of the Neumann problem \begin{eqnarray} &-div( e^{h( |x|) }\nabla u) =\mu e^{h(|x|) }u \quad in \ \Omega \\ &\frac{\partial u}{\partial \nu }=0 \quad on \ \partial \Omega, \end{eqnarray} where $\Omega$ is a bounded and Lipschitz domain in $\mathbb{R}^{N}$. Under suitable assumption on $h$ we prove that the ball centered at the origin is the unique set maximizing $\mu _{1}(\Omega ;e^{h( |x|)})$ among all Lipschitz bounded domains $\Omega$ of $\mathbb{R}^{N}$ of prescribed $e^{h( |x|) }dx$-measure and symmetric about the origin. Moreover, an example in the model case $h( |x|) =|x|^{2},$ shows that, in general, the assumption on the symmetry of the domain cannot be dropped. In the one-dimensional case, i.e. when $\Omega$ reduces to an interval $(a,b),$ we consider a wide class of weights (including both Gaussian and anti-Gaussian). We then describe the behavior of the eigenvalue as the interval $(a,b)$ slides along the $x$-axis keeping fixed its weighted length.
Citation: Friedemann Brock, Francesco Chiacchio, Giuseppina di Blasio. Optimal Szegö-Weinberger type inequalities. Communications on Pure and Applied Analysis, 2016, 15 (2) : 367-383. doi: 10.3934/cpaa.2016.15.367
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 $\mathbb{R}^{N}$ 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, Commun. Contemp. Math., to appear. DOI: 10.1142/S0219199715500868 [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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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 $\mathbb{R}^{N}$ 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, Commun. Contemp. Math., to appear. DOI: 10.1142/S0219199715500868 [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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