Minimal thinness in an isolated singularity of the Schrödinger equation and application to the Picard principle (Q1320313)

The problem studied in this paper is one of modern potential theory, applied to the generalized Schrödinger equation \((\Delta-\mu)u=0\), where \(\mu\) is a Radon positive measure from Kato's class, locally in \(\mathbb{R}^ 2-\{0\}\). The material of the paper is disposed in four chapters, the first of them, notations and preliminaries, very useful for a good understanding of the whole paper. The principal results: (i) For every \(x,y\in U\), the inequality \(G_ y(x)\leq G_{T(y)} [T(x)]\) is valid, where \(U= \{0<| x|<1\}\), \(G(\cdot)\) is the Green's function on \(U\), \(T(\cdot)\), the application \(\mathbb{R}^ 2\to \mathbb{R}^ 2: x\to (| x|,0)\), when \(\mu\) is a radial measure of Radon; (ii) If \(E\subset U\) and \(E\) is \(\mu\)-thin, then there exists a sequence \((r_ n)_ n\) of positive numbers converging to zero, such that \(\partial B(0,r_ n)\subset CE\); (iii) If the measure \(\nu\) satisfies \(\int_{\Omega-E} e_ \mu h_ \mu d\nu<\infty\), for a minimal thin, relatively to \(h_ \mu\), set \(E\), then the Picard principle is valid for the measure \(\mu+\nu\).