Minimization of the third eigenvalue of the Dirichlet Laplacian

In this paper, we prove the existence of a domain which minimizes the third Dirichlet eigenvalue λ3 of the Laplacian (–Δ) in the family of domains of RNof given volume. Some remarks are made on the existence of a minimizer for λk, k ⪖ 4 in the same family, and for a class of functionals depending on (λ1,λ2).

[1]  Umberto Mosco,et al.  Stopping times and Γ-convergence , 1987 .

[2]  A. Henrot,et al.  Some results about Schiffer's conjectures , 1999 .

[3]  Jan Sokolowski,et al.  Introduction to shape optimization , 1992 .

[4]  P. Lions.  The concentration-compactness principle in the Calculus of Variations , 1984 .

[5]  W. Ziemer.  Weakly differentiable functions , 1989 .

[6]  J. Keller,et al.  Range of the first two eigenvalues of the laplacian , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[7]  Mark S. Ashbaugh,et al.  Open Problems on Eigenvalues of the Laplacian , 1999 .

[8]  E. Davies,et al.  Heat kernels and spectral theory , 1989 .

[9]  J. Heinonen,et al.  Nonlinear Potential Theory of Degenerate Elliptic Equations , 1993 .

[10]  Giuseppe Buttazzo,et al.  An existence result for a class of shape optimization problems , 1993 .

[11]  L. Hedberg,et al.  Function Spaces and Potential Theory , 1995 .

[12]  彰 五十嵐.  N. Dunford and J. T. Schwartz (with the assistance of W. G. Bade and R. G. Bartle): Linear Operators. : Part II. Spectral Theoty. Self Adjoint Operators in Hilbert Space. Interscience. 1963. X+1065+7頁, 16×23.5cm, 14,000円。 , 1964 .

[13]  Isabel N. Figueiredo,et al.  On the attainable eigenvalues of the Laplace operator , 1999 .

[14]  Gianni Dal Maso,et al.  Wiener's criterion and Γ-convergence , 1987 .

[15]  S. C. Malik,et al.  Introduction to convergence , 1984 .

[16]  E. Krahn,et al.  Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises , 1925 .

[17]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[18]  Kilpeläinen Tero,et al.  Supersolutions to Degenerate Elliptic Equations on Quasi open Sets , 1992 .

[19]  L. Caffarelli,et al.  Existence and regularity for a minimum problem with free boundary. , 1981 .

[20]  Concentration-compacité et γ-convergence , 1998 .

[21]  Dorin Bucur,et al.  Uniform Concentration-Compactness for Sobolev Spaces on Variable Domains , 2000 .