Absolute continuity of the best Sobolev constant. (English) Zbl 1304.46030
Summary: Let \(\lambda_q:=\inf\biggl\{||\nabla u||_{L^p(\Omega)}^p/||u||_{L^q(\Omega)}^p:u\in W_0^{1,p}(\Omega)\setminus \{0\}\biggr\}\) be the best Sobolev constant of the immersion \(W_0^{1,p}(\Omega)\hookrightarrow L^q(\Omega)\), where \(\Omega\) is a bounded and smooth domain of \(\mathbb R^N\), \(1<p<N\) and \(1\leq q\leq p^\star:=\frac{Np}{N-p}\). We prove that the function \(q\in [1,p^\star]\mapsto\lambda_q\) is absolutely continuous.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
absolute continuity; Lipschitz continuity; \(p\)-Laplacian; Rayleigh quotient; Sobolev best constantsReferences:
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