On Principal Eigenvalue of Stationary Diffusion Problem with Nonsymmetric Coefficients

Abstract

We consider the eigenvalue problem

$$\left\{ \begin{gathered} - div (A\nabla u) = \lambda \rho u \hfill \\ u \in H_0^1\left( \Omega \right)\hfill \\ \end{gathered} \right.$$

where Ω ∈ R ^d is open and bounded, ρ ∈ L∞(Ω) and A ∈ L∞(ΩM _d×d ) satisfying

$$A\left( x \right)\xi \cdot \xi \geqslant \alpha \xi \cdot \xi ,\rho \left( x \right) \geqslant c, \xi \in {R^d}, a.e.x \in \Omega ,$$

for some α, c > 0.

We show that, under appropriate conditions on smoothness of coefficients, the principal eigenvalue depends continuously on coefficients with respect to Htopology for A and L∞ weak * topology for ρ. An application of this result in optimal shape design problem of optimising the principal eigenvalue is presented.

Moreover, in the same topology for coefficients, we obtain the continuity of corresponding singular values.

This work is supported in part by the Croatian Ministry of Science and Technology through project 037 015.