English | 2023 | ISBN: 3985470529 | 164 Pages | True PDF | 1.05 MB
This book treats the classical problem, posed by Kazdan and Warner, of prescribing a given function on a closed manifold as the scalar curvature of a metric within a conformal class. Since both critical equations and obstructions to the existence of solutions appear, the problem is particularly challenging. The author's focus is to present a general approach for understanding the matter, particularly the issue of loss of compactness. The task of establishing the existence of solutions is attacked, combining several tools: the variational structure of the problem, Liouville-type theorems, blow-up analysis, elliptic regularity theory, and topological arguments. Treating different aspects of the subject and containing several references to up-to-date research directions and perspectives, the book will be useful to graduate students and researchers interested in geometric analysis and partial differential equations.
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