Potential-capacity and some applications

Article type: Research Article

Authors: Rakotoson, Jean Michel^{; *}

Affiliations: Laboratoire de Mathématiques et Applications, Université de Poitiers, UMR CNRS 7348 – SP2MI, Bat H3 – Bd Marie et Pierre Curie, Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France. E-mail: rako@math.univ-poitiers.fr

Correspondence: [*] Corresponding author. E-mail: rako@math.univ-poitiers.fr.

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 LVu=−Δu+U·∇u+Vu=f has a solution in L1(V)∩L01(Ω)={g measurable,∫Ω|g|Vdx is finite and limε→0∫{x:dist(x;∂Ω)⩽ε}|g|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 Cc2(Ω∖K) in C02(Ω‾) whenever K has a zero potential-capacity.

Keywords: Capacity, Schrödinger equations, potential, measures

DOI: 10.3233/ASY-191523

Journal: Asymptotic Analysis, vol. 114, no. 3-4, pp. 225-252, 2019