On principal eigenvalues for periodic parabolic Steklov problems (Q700889)

Summary: Let \(\Omega\) be a \(C^{2+\gamma}\) domain in \({\mathbb R}^{N}\), \(N\geq 2\), \(0<\gamma <1\). Let \(T>0\) and let \(L\) be a uniformly parabolic operator \[ Lu=\partial u/\partial t-\sum_{i,j} (\partial /\partial x_{i}) (a_{ ij} (\partial u/\partial x_{j}))+\sum_{j}b_{j} (\partial u/\partial x_{i})+a_{0}u, \qquad a_{0}\geq 0, \] whose coefficients, depending on \((x,t)\in \Omega \times {\mathbb R}\), 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 \(\partial \Omega \times {\mathbb R}\) belongs to \(W_q^{2-1/q,1-1/2q}(\partial \Omega \times (0,T))\) 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 \(\Omega \times {\mathbb R}\), \(\langle A\nabla u,\nu \rangle =\lambda mu\) on \(\partial \Omega \times {\mathbb R}\), \(u (x,t)=u (x,t+T)\), \(u>0\) on \(\Omega \times {\mathbb R}\). Uniqueness and simplicity of the positive principal eigenvalue is proved and a related maximum principle is given.