The strong Legendre condition and the well-posedness of mixed Robin problems on manifolds with bounded geometry

Abstract

Let $M$ be a smooth manifold with boundary $\partial M$ and bounded geometry, ${\partial }_{D}M\subset \partial M$ be an open and closed subset, $P$ be a second order differential operator on $M$, and $b$ be a first order differential operator on $\partial M\setminus {\partial }_{D}M$. We prove the regularity and well-posedness of the mixed Robin boundary value problem $$Pu = f \mbox{ in } M,\ u = 0 \mbox{ on } \partial_D M,\ \partial^P_νu + bu = 0 \mbox{ on } \partial M \setminus \partial_D M$$ under some natural assumptions. Our operators act on sections of a vector bundle $E\to M$ with bounded geometry. Our well-posedness result is in the Sobolev spaces $H^s(M;E)$, $s\ge 0$. The main novelty of our results is that they are formulated on a non-compact manifold. We include also some extensions of our main result in different directions. First, the finite width assumption is required for the Poincaré inequality on manifolds with bounded geometry, a result for which we give a new, more general proof. Second, we consider also the case when we have a decomposition of the vector bundle $E$ (instead of a decomposition of the boundary). Third, we also consider operators with non-smooth coefficients, but, in this case, we need to limit the range of $s$. Finally, we also consider the case of uniformly strongly elliptic operators. In this case, we introduce a \emph{uniform Agmon condition} and show that it is equivalent to the Gårding inequality. This extends an important result of Agmon (1958).

arXiv commentSome of the results of this paper were first circulated in arXiv 1611.00281v1. The current version of arXiv 1611.00281 concentrates on the geometry part and the Laplacian with Dirichlet boundary condition while in this paper we work more on analytic questions and in particular include more general operators and Robin boundary conditions. v2: small changes, final version

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Cited by in corpus (1)

• Maximal Regularity of Parabolic Transmission Problems