# American Institute of Mathematical Sciences

• Previous Article
Existence and nonexistence of positive solutions to an integral system involving Wolff potential
• CPAA Home
• This Issue
• Next Article
The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited
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 & 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,, \emph{Spectral Theory and Geometry}, (1998), 95. doi: 10.1017/CBO9780511566165.007. Google Scholar [2] M. S. Ashbaugh and R. Benguria, Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature,, \emph {J. Lond. Math. Soc. (2)}, 52 (1995), 402. doi: 10.1112/jlms/52.2.402. Google Scholar [3] C. Bandle, Isoperimetric Inequalities and Applications,, Monographs and Studies in Mathematics 7, (1980). Google Scholar [4] R. Benguria and H. Linde, A second eigenvalue bound for the Dirichlet Schrödinger operator,, \emph{Comm. Math. Phys.}, 267 (2006), 741. doi: 10.1007/s00220-006-0041-1. Google Scholar [5] M. F. Betta, F. Brock, A. Mercaldo and M. R. Posteraro, Weighted isoperimetric inequalities on $\mathbbR^n$ and applications to rearrangements,, \emph{Math. Nachr.}, 281 (2008), 466. doi: 10.1002/mana.200510619. Google Scholar [6] B. Brandolini, F. Chiacchio, D. Krejčiřík and C. Trombetti, The equality case in a Poincaré-Wirtinger type inequality,, \arXiv{1410.0676}., (). Google Scholar [7] B. Brandolini, F. Chiacchio, A. Henrot A. and C. Trombetti, Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift,, \emph{J. Differential Equations}, 259 (2015), 708. doi: 10.1016/j.jde.2015.02.028. Google Scholar [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.}, (). Google Scholar [9] F. Brock, F. Chiacchio and A. Mercaldo, Weighted isoperimetric inequalities in cones and applications,, \emph{Nonlinear Anal.}, 75 (2012), 5737. doi: 10.1016/j.na.2012.05.011. Google Scholar [10] F. Brock, A. Mercaldo and M. R. Posteraro, On isoperimetric inequalities with respect to infinite measures,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 665. doi: 10.4171/RMI/734. Google Scholar [11] I. Chavel, Lowest-eigenvalue inequalities,, in \emph{Proc. Sympos. Pure Math., (1980), 79. Google Scholar [12] I. Chavel, Eigenvalues in Riemannian Geometry,, New York, (2001). Google Scholar [13] F. Chiacchio and G. di Blasio, Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space,, \emph{Ann. I. H. Poincar\'e -AN}, 29 (2012), 199. doi: 10.1016/j.anihpc.2011.10.002. Google Scholar [14] K. M. Chong and N. M. Rice, Equimeasurable rearrangements of functions,, in \emph{Queen's Papers in Pure and Applied Mathematics}, (1971). Google Scholar [15] R. Courant and D. Hilbert, Methods of Mathematical Physics vol. I and II,, Interscience Publichers New York-London, (1966). Google Scholar [16] F. Della Pietra and N. Gavitone, Faber-Krahn inequality for anisotropic eigenvalue problems with robin boundary conditions,, \emph{Potential Analysis}, 41 (2014), 1147. doi: 10.1007/s11118-014-9412-y. Google Scholar [17] F. Della Pietra and N. Gavitone, Stability results for some fully nonlinear eigenvalue estimates,, \emph{Communications in Contemporary Mathematics}, 16 (2014). doi: 10.1142/S0219199713500399. Google Scholar [18] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators,, Frontiers in Mathematics. Birkh\, (2006). Google Scholar [19] A. Henrot and M. Pierre, Variation et optimisation de formes. Une analyse géométrique,, Math\'ematiques & Applications, (2005). doi: 10.1007/3-540-37689-5. Google Scholar [20] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE,, Lecture Notes in Mathematics 1150. New York: Springer Verlag, (1150). Google Scholar [21] S. Kesavan, Symmetrization & Applications,, Series in Analysis, (2006). doi: 10.1142/9789812773937. Google Scholar [22] E. T. Kornhauser and I. Stakgold, A variational theorem for $\nabla ^{2}u+\lambda u =0$ and its applications,, \emph{J. Math. Phys.}, 31 (1952), 45. Google Scholar [23] R. S. Laugesen and B. A. Siudeja, Maximizing Neumann fundamental tones of triangles,, \emph{J. Math. Phys.}, 50 (2009). doi: 10.1063/1.3246834. Google Scholar [24] C. Müller, Spherical Harmonics,, Lecture Notes in Mathematics, (1966). Google Scholar [25] Y. Naito and T. Suzuki, Radial symmetry of self-similar solutions for semilinear heat equations,, \emph{J. Differential Equations}, 163 (2000), 407. doi: 10.1006/jdeq.1999.3742. Google Scholar [26] J. M. Rakotoson and B. Simon, Relative rearrangement on a measure space application to the regularuty of weighted monotone rearrangement, I, II,, \emph{Appl. Math. Lett.}, 6 (1993), 75. doi: 10.1016/0893-9659(93)90152-D. Google Scholar [27] G. Szegö, Inequalities for certain eigenvalues of a membrane of given area,, \emph{J. Rational Mech. Anal.}, 3 (1954), 343. Google Scholar [28] M. E. Taylor, Partial Differential Equations Vol.II,, Qualitative Studies of Linear Equations. Appl. Math. Sciences 116, (1996). doi: 10.1007/978-1-4757-4187-2. Google Scholar [29] H. F. Weinberger, An isoperimetric inequality for the N-dimensional free membrane problem,, \emph{J. Rational Mech. Anal.}, 5 (1956), 633. Google Scholar

show all references

##### References:
 [1] M. S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues,, \emph{Spectral Theory and Geometry}, (1998), 95. doi: 10.1017/CBO9780511566165.007. Google Scholar [2] M. S. Ashbaugh and R. Benguria, Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature,, \emph {J. Lond. Math. Soc. (2)}, 52 (1995), 402. doi: 10.1112/jlms/52.2.402. Google Scholar [3] C. Bandle, Isoperimetric Inequalities and Applications,, Monographs and Studies in Mathematics 7, (1980). Google Scholar [4] R. Benguria and H. Linde, A second eigenvalue bound for the Dirichlet Schrödinger operator,, \emph{Comm. Math. Phys.}, 267 (2006), 741. doi: 10.1007/s00220-006-0041-1. Google Scholar [5] M. F. Betta, F. Brock, A. Mercaldo and M. R. Posteraro, Weighted isoperimetric inequalities on $\mathbbR^n$ and applications to rearrangements,, \emph{Math. Nachr.}, 281 (2008), 466. doi: 10.1002/mana.200510619. Google Scholar [6] B. Brandolini, F. Chiacchio, D. Krejčiřík and C. Trombetti, The equality case in a Poincaré-Wirtinger type inequality,, \arXiv{1410.0676}., (). Google Scholar [7] B. Brandolini, F. Chiacchio, A. Henrot A. and C. Trombetti, Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift,, \emph{J. Differential Equations}, 259 (2015), 708. doi: 10.1016/j.jde.2015.02.028. Google Scholar [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.}, (). Google Scholar [9] F. Brock, F. Chiacchio and A. Mercaldo, Weighted isoperimetric inequalities in cones and applications,, \emph{Nonlinear Anal.}, 75 (2012), 5737. doi: 10.1016/j.na.2012.05.011. Google Scholar [10] F. Brock, A. Mercaldo and M. R. Posteraro, On isoperimetric inequalities with respect to infinite measures,, \emph{Rev. Mat. Iberoam.}, 29 (2013), 665. doi: 10.4171/RMI/734. Google Scholar [11] I. Chavel, Lowest-eigenvalue inequalities,, in \emph{Proc. Sympos. Pure Math., (1980), 79. Google Scholar [12] I. Chavel, Eigenvalues in Riemannian Geometry,, New York, (2001). Google Scholar [13] F. Chiacchio and G. di Blasio, Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space,, \emph{Ann. I. H. Poincar\'e -AN}, 29 (2012), 199. doi: 10.1016/j.anihpc.2011.10.002. Google Scholar [14] K. M. Chong and N. M. Rice, Equimeasurable rearrangements of functions,, in \emph{Queen's Papers in Pure and Applied Mathematics}, (1971). Google Scholar [15] R. Courant and D. Hilbert, Methods of Mathematical Physics vol. I and II,, Interscience Publichers New York-London, (1966). Google Scholar [16] F. Della Pietra and N. Gavitone, Faber-Krahn inequality for anisotropic eigenvalue problems with robin boundary conditions,, \emph{Potential Analysis}, 41 (2014), 1147. doi: 10.1007/s11118-014-9412-y. Google Scholar [17] F. Della Pietra and N. Gavitone, Stability results for some fully nonlinear eigenvalue estimates,, \emph{Communications in Contemporary Mathematics}, 16 (2014). doi: 10.1142/S0219199713500399. Google Scholar [18] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators,, Frontiers in Mathematics. Birkh\, (2006). Google Scholar [19] A. Henrot and M. Pierre, Variation et optimisation de formes. Une analyse géométrique,, Math\'ematiques & Applications, (2005). doi: 10.1007/3-540-37689-5. Google Scholar [20] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE,, Lecture Notes in Mathematics 1150. New York: Springer Verlag, (1150). Google Scholar [21] S. Kesavan, Symmetrization & Applications,, Series in Analysis, (2006). doi: 10.1142/9789812773937. Google Scholar [22] E. T. Kornhauser and I. Stakgold, A variational theorem for $\nabla ^{2}u+\lambda u =0$ and its applications,, \emph{J. Math. Phys.}, 31 (1952), 45. Google Scholar [23] R. S. Laugesen and B. A. Siudeja, Maximizing Neumann fundamental tones of triangles,, \emph{J. Math. Phys.}, 50 (2009). doi: 10.1063/1.3246834. Google Scholar [24] C. Müller, Spherical Harmonics,, Lecture Notes in Mathematics, (1966). Google Scholar [25] Y. Naito and T. Suzuki, Radial symmetry of self-similar solutions for semilinear heat equations,, \emph{J. Differential Equations}, 163 (2000), 407. doi: 10.1006/jdeq.1999.3742. Google Scholar [26] J. M. Rakotoson and B. Simon, Relative rearrangement on a measure space application to the regularuty of weighted monotone rearrangement, I, II,, \emph{Appl. Math. Lett.}, 6 (1993), 75. doi: 10.1016/0893-9659(93)90152-D. Google Scholar [27] G. Szegö, Inequalities for certain eigenvalues of a membrane of given area,, \emph{J. Rational Mech. Anal.}, 3 (1954), 343. Google Scholar [28] M. E. Taylor, Partial Differential Equations Vol.II,, Qualitative Studies of Linear Equations. Appl. Math. Sciences 116, (1996). doi: 10.1007/978-1-4757-4187-2. Google Scholar [29] H. F. Weinberger, An isoperimetric inequality for the N-dimensional free membrane problem,, \emph{J. Rational Mech. Anal.}, 5 (1956), 633. Google Scholar
 [1] Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264 [2] Jin-Cheng Jiang, Chengbo Wang, Xin Yu. Generalized and weighted Strichartz estimates. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1723-1752. doi: 10.3934/cpaa.2012.11.1723 [3] Selma Yildirim Yolcu, Türkay Yolcu. Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2209-2225. doi: 10.3934/dcds.2015.35.2209 [4] Francisco Odair de Paiva, Humberto Ramos Quoirin. Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1219-1227. doi: 10.3934/dcds.2009.25.1219 [5] Raffaela Capitanelli, Maria Agostina Vivaldi. Uniform weighted estimates on pre-fractal domains. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1969-1985. doi: 10.3934/dcdsb.2014.19.1969 [6] Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801 [7] David Cruz-Uribe, SFO, José María Martell, Carlos Pérez. Sharp weighted estimates for approximating dyadic operators. Electronic Research Announcements, 2010, 17: 12-19. doi: 10.3934/era.2010.17.12 [8] Feng Du, Adriano Cavalcante Bezerra. Estimates for eigenvalues of a system of elliptic equations with drift and of bi-drifting laplacian. Communications on Pure & Applied Analysis, 2017, 6 (2) : 475-491. doi: 10.3934/cpaa.2017024 [9] Antonio Vitolo. $H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1315-1329. doi: 10.3934/cpaa.2011.10.1315 [10] Barbara Brandolini, Francesco Chiacchio, Jeffrey J. Langford. Estimates for sums of eigenvalues of the free plate via the fourier transform. Communications on Pure & Applied Analysis, 2020, 19 (1) : 113-122. doi: 10.3934/cpaa.2020007 [11] Marcone C. Pereira, Ricardo P. Silva. Error estimates for a Neumann problem in highly oscillating thin domains. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 803-817. doi: 10.3934/dcds.2013.33.803 [12] Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745 [13] Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure & Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043 [14] Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210 [15] Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041 [16] Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011 [17] Annalisa Cesaroni, Matteo Novaga. The isoperimetric problem for nonlocal perimeters. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 425-440. doi: 10.3934/dcdss.2018023 [18] Jorge A. Becerril, Javier F. Rosenblueth. Necessity for isoperimetric inequality constraints. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1129-1158. doi: 10.3934/dcds.2017047 [19] Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731 [20] Yi Cao, Dong Li, Lihe Wang. The optimal weighted $W^{2, p}$ estimates of elliptic equation with non-compatible conditions. Communications on Pure & Applied Analysis, 2011, 10 (2) : 561-570. doi: 10.3934/cpaa.2011.10.561

2018 Impact Factor: 0.925