In this article, we are concerned with the following eigenvalue problem of a linear second order elliptic operator:complemented by a general boundary condition including Dirichlet boundary condition and Robin boundary condition:where β ∈ C(∂Ω) allows to be positive, sign-changing or negative, and n(x) is the unit exterior normal to ∂Ω at x. The domain Ω ⊂ R N is bounded and smooth, the constants D > 0 and α > 0 are, respectively, the diffusive and advection coefficients, and m ∈ C 2 (Ω), V ∈ C(Ω) are given functions.We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient D → 0 or D → ∞. Our results, together with those of [4,5,10] where the Nuemann boundary case (i.e., β = 0 on ∂Ω) and Dirichlet boundary case were studied, reveal the important effect of advection and boundary conditions on the asymptotic behavior of the principal eigenvalue. We also apply our results to a reaction-diffusion-advection equation which is used to describe the evolution of a single species living in a heterogeneous stream environment and show some interesting behaviors of the species persistence and extinction caused by the buffer zone and small/large diffusion rate.

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We compute the second variation of the stream function energy of two-dimensional steady free surface gravity water waves with vorticity in the stream function formulation. We prove that for nonpositive vorticity the second variation of the stream function energy at extreme waves with Stokes corner asymptotics cannot be nonnegative in any small neighbourhood of a given isolated stagnation point. The particular form of our second variation suggests however the possibility that certain singularities in the case of nonzero vorticity might be constructible as minimizers of the stream function energy.

We consider the singular perturbation problem, β is a Lipschitz continuous function such that β > 0 in (0, 1), β ≡ 0 outside (0, 1) andKnown results suggest that this singularity must be unstable, which makes it hard to capture analytically and numerically. Our result answers a question raised by Jean-Michel Roquejoffre at the FBP'08 in Stockholm. 1 0 β(s) ds = 1 2 .