  We study the regularity of the solutions of second order boundary value
problems on manifolds with boundary and bounded geometry. We first show that
the regularity property of a given boundary value problem $(P, C)$ is
equivalent to the uniform regularity of the natural family $(P_x, C_x)$ of
associated boundary value problems in local coordinates. We verify that this
property is satisfied for the Dirichlet boundary conditions and strongly
elliptic operators via a compactness argument. We then introduce a uniform
Shapiro-Lopatinski regularity condition, which is a modification of the
classical one, and we prove that it characterizes the boundary value problems
that satisfy the usual regularity property. We also show that the natural Robin
boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity
condition, provided that our operator satisfies the strong Legendre condition.
This is achieved by proving that "well-posedness implies regularity" via a
modification of the classical "Nirenberg trick". When combining our regularity
results with the Poincare inequality of (Ammann-Grosse-Nistor, preprint 2015),
one obtains the usual well-posedness results for the classical boundary value
problems in the usual scale of Sobolev spaces, thus extending these important,
well-known theorems from smooth, bounded domains, to manifolds with boundary
and bounded geometry. As we show in several examples, these results do not hold
true anymore if one drops the bounded geometry assumption. We also introduce a
uniform Agmon condition and show that it is equivalent to the coerciveness.
Consequently, we prove a well-posedness result for parabolic equations whose
elliptic generator satisfies the uniform Agmon condition.
