Coupling the Ball Construction to the Pohozaev Identity and Applications

Abstract

The key ingredient here is the Pohozaev identity for solutions of Ginzburg- Landau. This identity was already used crucially in Bethuel-Brezis- Hélein [43], Brezis-Merle-Rivière [61], and its first use on small balls goes back to Bethuel-Rivière [52] and Struwe [189]. Its consequences were also explored further in the book of Pacard-Rivière [148]. Here, the idea is to combine it with the ball-construction method in order to obtain lower bounds for the energy in terms of the potential term ∫(1 − |u|²)² instead of the degree, or equivalently, upper bounds of the potential by the energy divided by | log ε|. This method works for solutions of the Ginzburg-Landau equation, without magnetic field as well as with. We will present the two situations in parallel, in Sections 5.1 and 5.2. In the third section of the chapter, we present applications to the microscopic analysis of vortices of solutions of (GL) or (1.3). Among all these results, only Theorem 5.4 will be used later, for the study of solutions with bounded numbers of vortices: for Proposition 10.2 and in the course of the proof of Theorem 11.1.