Let $ M $ be a compact Riemannian manifold with or without boundary) and let $ \Delta $ be the metric laplacian. I want to study the differential operator $ -\Delta +q $ where $ q $ is a smooth function on $ M $. Now the first thing to do is a development of the theory of weak solutions for this differential operator. I think that all existence results in Chapter 6 , paragraph 2 of Evans' book 'Partial differentail equation' can be reformulate without changement for our case if we are able to prove the Poincarè inequality in the setting of a compact Riemannian manifold. Therefore the first question is
is the poincare inequality true for a compact riannian manifold with boundary? (i think so)
Now the second problem is the proof of the well- known regularity theorem in our setting (chapter 6.3 of Evans book). Can anyone suggest me a good reference for it? I know the Aubin book but it seems not so clear at this point.\
Thanks
If you are willing to assume that $q \geq 0$, then this problem has been extensively studied by Mitrea and Taylor (cf. also the references therein for their other papers on this topic that go back to 1999). If you want $q$ to be an arbitrary smooth function, then I doubt that there is any comprehensive study to date.