Ian Schindler
Cyril Tintarev

The limiting Dirac-Sobolev inequality

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 68, 3 - 12 (2013)

MSC: 35Q40   Equations from quantum mechanics
  35J50   Variational methods for elliptic systems
  46E35   Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Abstract:   We prove the critical Dirac-Sobolev inequality for p ∈ (1, 3). It follows that the Dirac Sobolev spaces are equivalent to classical Sobolev spaces if and only if p ∈ (1, 3). We prove the cocompactness of Lp* (R3 ) in H1,p (R3 ). As an application, we prove the existence of minimizers to a class of isoperimetric problems.

Keywords:   cocompact imbeddings, concentration compactness, Dirac operator, minimizers, Sobolev imbeddings, critical exponent

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