Optimal measures for the fundamental gap of Schrödinger operators
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 1, p. 194-205

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.

DOI : https://doi.org/10.1051/cocv:2008069
Classification:  35J10,  49K20,  35J20,  35B20
Keywords: Schrödinger operator, eigenvalue problems, measure theory, shape optimization
@article{COCV_2010__16_1_194_0,
     author = {Varchon, Nicolas},
     title = {Optimal measures for the fundamental gap of Schr\"odinger operators},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {1},
     year = {2010},
     pages = {194-205},
     doi = {10.1051/cocv:2008069},
     zbl = {1183.35092},
     mrnumber = {2598095},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2010__16_1_194_0}
}
Varchon, Nicolas. Optimal measures for the fundamental gap of Schrödinger operators. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 1, pp. 194-205. doi : 10.1051/cocv:2008069. http://www.numdam.org/item/COCV_2010__16_1_194_0/

[1] M.S. Ashbaugh, E.M. Harrell and R. Svirsky, On minimal and maximal eigenvalue gaps and their causes. Pacific J. Math. 147 (1991) 1-24. | Zbl 0734.35061

[2] D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and Their Applications 65. Birkhäuser, Basel, Boston (2005). | Zbl 1117.49001

[3] D. Bucur and T. Chatelain, Strict monotonicity of the second eigenvalue of the Laplace operator on relaxed domain. Bull. Appl. Comp. Math. 1510-1566 (1998) 115-122.

[4] D. Bucur and A. Henrot, Minimization of the third eigenvalue of the Dirichlet Laplacian. Proc. Roy. Soc. London 456 (2000) 985-996. | Zbl 0974.35082

[5] G. Buttazzo and G. Dal Maso, Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991) 17-49. | Zbl 0762.49017

[6] G. Buttazzo, N. Varchon and H. Zoubairi, Optimal measures for elliptic problems. Annali Mat. Pur. Appl. 185 (2006) 207-221.

[7] R. Courant and D. Hilbert, Methods of Mathematical Physics. Interscience Publishers (1953). | Zbl 0051.28802

[8] G. Dal Maso, Γ-convergence and µ-capacities. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987) 423-464. | Numdam | Zbl 0657.49005

[9] G. Dal Maso, An introduction to Γ-convergence. Birkhäuser, Boston (1993). | Zbl 0816.49001

[10] G. Dal Maso and U. Mosco, Wiener's criterion and Γ-convergence. Appl. Math. Optim. 15 (1987) 15-63. | Zbl 0644.35033

[11] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1992). | Zbl 0804.28001

[12] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators. Birkhäuser Verlag, Basel, Boston, Berlin (2006). | Zbl 1109.35081

[13] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag (1980). | Zbl 0435.47001

[14] N. Varchon, Optimal measures for nonlinear cost functionals. Appl. Mat. Opt. 54 (2006) 205-221. | Zbl 1118.49034

[15] W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, Berlin (1989). | Zbl 0692.46022