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Optimal measures for the fundamental gapof 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. [email protected]
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Abstract

We study the potential which minimizes the fundamental gap of theSchrödinger operator under the total mass constraint. We considerthe relaxed potential and prove a regularity result for the optimalone, we also give a description of it. A consequence of this resultis the existence of an optimal potential under L 1 constraints.

Keywords

Schrödinger operatoreigenvalue problemsmeasuretheoryshape 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

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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).