Potential-Capacity And Some Applications

We introduce a new capacity associated to a non negative function V. We apply this notion to the study of a necessary and sufficient condition to ensure the existence and uniqueness of a Schrodinger type equation with measure data and with an operator whose coefficients are discontinuous. Namely, for a potential V, f a bounded Radon measure on Omega, then the equation L(V)u = -Delta u + U . del u + Vu = f has a solution inL-1 (V) &AND L-0(1) (Omega) = {g measurable, integral(Omega)vertical bar g vertical bar V dx is finite and lim(epsilon -> 0) integral({x:dist(x;partial derivative Omega)<=epsilon)(}) vertical bar g vertical bar dx = 0}if and only if f does not charge "irregular points" of V, provided that the set of "irregular points" have a zero potential capacity. As a byproduct of our results, we have the non existence of a Green operator for some Lv.Our method is also based on a new topology and density of C-c(2) (Omega\K) in C-0(2) ((Omega) over bar) whenever K has a zero potential-capacity.