  The classical local Neumann problem is well studied and solutions of this
problem lie, in general, in a Sobolev space. In this work, we focus on nonlocal
Neumann problems with measurable, nonnegative kernels, whose solutions require
less regularity assumptions. For kernels of this kind we formulate and study
the weak formulation of the nonlocal Neumann problem and we investigate a
nonlocal counterpart of the Sobolev space $H^{1}$ as well as a resulting
nonlocal trace space. We further establish, mainly for symmetric kernels,
various existence results for the weak solution of the Neumann problem and we
discuss related necessary conditions. Both, homogeneous and nonhomogeneous
Neumann boundary conditions are considered. In addition to that, we present a
new weak formulation of a Robin problem, where we reformulate the Robin problem
into a regional problem.
