ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 49J40; 49J10, 35J20, 35R05 Denzler, Jochen, Technische Universität München Existence of windows with minimal heat diffusion Consider the first eigenvalue λ_{1} > 0 of the Laplacian −Δ on a fixed bounded Lipschitz domain Ω ∈ R^{n} under varying mixed boundary conditions, namely Dirichlet conditions on some measurable subset D ⊂ ∂Ω and Neumann conditions elsewhere. D varies subject to the constraint that its measure A is prescribed. We show the existence of some D minimizing λ_{1}. For Ω a ball, the optimal D is a spherical cap. In contrast, the supremum of λ_{1} is not attained. Its value is independent of A, and an approximating sequence is obtained by homogenizing D.