Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-25T14:42:59.827Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal measures for the fundamental gap of Schrödinger operators

Published online by Cambridge University Press:  19 December 2008

Nicolas Varchon
Nicolas Varchon*
Affiliation:
Collège Condorcet de Bresles, Rue du Petit Chantilly, 60510 Bresles, France. nicolas.varchon@ac-amiens.fr
Get access

Abstract

We study the potential which minimizes the fundamental gap of the Schrödinger operator under the total mass constraint. We consider the relaxed potential and prove a regularity result for the optimal one, we also give a description of it. A consequence of this result is the existence of an optimal potential under L1 constraints.

Keywords

Schrödinger operatoreigenvalue problemsmeasure theoryshape optimization
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 1 , January 2010 , pp. 194 - 205
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ashbaugh, M.S., Harrell, E.M. and Svirsky, R., On minimal and maximal eigenvalue gaps and their causes. Pacific J. Math. 147 (1991) 1–24. CrossRef
D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and Their Applications 65. Birkhäuser, Basel, Boston (2005).
Bucur, D. and Chatelain, T., Strict monotonicity of the second eigenvalue of the Laplace operator on relaxed domain. Bull. Appl. Comp. Math. 1510–1566 (1998) 115–122.
Bucur, D. and Henrot, A., Minimization of the third eigenvalue of the Dirichlet Laplacian. Proc. Roy. Soc. London 456 (2000) 985–996.
Buttazzo, G. and Dal Maso, G., Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991) 17–49. CrossRef
Buttazzo, G., Varchon, N. and Zoubairi, H., Optimal measures for elliptic problems. Annali Mat. Pur. Appl. 185 (2006) 207–221. CrossRef
R. Courant and D. Hilbert, Methods of Mathematical Physics. Interscience Publishers (1953).
Dal Maso, G., Γ-convergence and µ-capacities. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987) 423–464.
G. Dal Maso, An introduction to Γ-convergence. Birkhäuser, Boston (1993).
Dal Maso, G. and Mosco, U., Wiener's criterion and Γ-convergence. Appl. Math. Optim. 15 (1987) 15–63. CrossRef
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1992).
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Birkhäuser Verlag, Basel, Boston, Berlin (2006).
T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag (1980).
Varchon, N., Optimal measures for nonlinear cost functionals. Appl. Mat. Opt. 54 (2006) 205–221. CrossRef
W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, Berlin (1989).