On principal eigenvalues for periodic parabolic Steklov problems

Abstract

Let $\Omega$ be a $C^{2+\gamma }$ domain in ${ℝ}^N$, $N\ge 2$, . Let $T>0$ and let $L$ be a uniformly parabolic operator $Lu=\partial u/\partial t-{\sum }_{i,j}\left(\partial /\partial x_i\right)\left(a_{ij}\left(\partial u/\partial x_j\right)\right)+{\sum }_j{b}_j\left(\partial u/\partial x_i\right)+a_{0}u$, $a_0\ge 0$, whose coefficients, depending on $\left(x,t\right)\in \Omega \times ℝ$, are $T$ periodic in $t$ and satisfy some regularity assumptions. Let $A$ be the $N\times N$ matrix whose $i,j$ entry is $a_{ij}$ and let $\nu$ be the unit exterior normal to $\partial \Omega$. Let $m$ be a $T$-periodic function (that may change sign) defined on $\partial \Omega$ whose restriction to $\partial \Omega \times ℝ$ belongs to $W_q^{2-1/q,1-1/2q}\left(\partial \Omega \times \left(0,T\right)\right)$ for some large enough $q$. In this paper, we give necessary and sufficient conditions on $m$ for the existence of principal eigenvalues for the periodic parabolic Steklov problem $Lu=0$ on $\Omega \times ℝ$, $\langle A\nabla u,\nu \rangle =\lambda mu$ on $\partial \Omega \times ℝ$, $u\left(x,t\right)=u\left(x,t+T\right)$, $u>0$ on $\Omega \times ℝ$. Uniqueness and simplicity of the positive principal eigenvalue is proved and a related maximum principle is given.