Potential-capacity and some applications

Abstract

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 Schrödinger type equation with measure data and with an operator whose coefficients are discontinuous. Namely, for a potential V, f a bounded Radon measure on Ω, then the equation ${\mathcal{L}}_{\mathit{V}}\mathit{u}=-\mathrm{\Delta }\mathit{u}+\mathit{U}·\nabla \mathit{u}+\mathit{V}\mathit{u}=\mathit{f}$ has a solution in $$\begin{array}{c}{\mathit{L}}^1\mathrm{(}\mathit{V}\mathrm{)}\cap {\mathit{L}}_0^1\mathrm{(}\mathrm{\Omega }\mathrm{)}=\{\mathit{g}\text{measurable}\mathrm{,}{{\displaystyle \int }}_{\mathrm{\Omega }}|\mathit{g}|\mathit{V}\mathit{d}\mathit{x}\text{is finite and }\underset{\mathit{\epsilon }\to 0}{lim}{{\displaystyle \int }}_{\{\mathit{x}:dist\mathrm{(}\mathit{x};\partial \mathrm{\Omega }\mathrm{)}⩽\mathit{\epsilon }\}}|\mathit{g}|\mathit{d}\mathit{x}=0\}\end{array}$$ 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 ${\mathcal{L}}_{\mathit{V}}$ .

Our method is also based on a new topology and density of ${\mathit{C}}_{\mathit{c}}^2\mathrm{(}\mathrm{\Omega }\setminus \mathit{K}\mathrm{)}$ in ${\mathit{C}}_0^2\mathrm{(}\bar{\mathrm{\Omega }}\mathrm{)}$ whenever K has a zero potential-capacity.