Regularity results for the best-Sobolev-constant function
Let \(\Omega\) be a bounded, smooth domain of \(\mathbb R^N\), \(1<p<N\) and \(p^*=\frac{Np}{N-p}\). This paper deals with new regularity results for the best Sobolev constant defined by \[ q\in[1,p^*]\to \lambda_q=\inf\{\frac{||\nabla u||_p^p}{||u||^p_q}:u\in W^{1,p}_o(\Omega)\backslash{0}\}, \] where \(||\overset{.}{~}||_s\) denotes the usual norm in \(L^s(\Omega)\). Let \[ E_q=\{u\in W^{1,p}_o(\Omega):||u||_q=1\text{ and } ||\nabla u||_p=\root p\of{\lambda_q}\} \] be the set of the \(L^q\)-normalized extremal functions corresponding to \(\lambda_q\). In the abstract, the author writes: ``In this paper, we prove that the derivative \(\lambda_q^{\prime}\) exists if and only if the functional \[ I_q (u)=\int_{\Omega}|u|^q \log |u| dx \] is constant on \(E_q\). Moreover, we prove that the existence of \(\lambda^{\prime}_q\) is also equivalent to the continuity, at \(q\), of the function \(s\in [1,p^*)\to I_s(u_s)\), where \(u_s\) is any function in \(E_s\). It follows from these results that \(\lambda_q^{\prime}\) exists and is continuous if \(q\in [1,p]\) and \(\Omega\) is a general bounded domain and also if \(q \in (p,p^*)\) and \(\Omega\) is a ball. After deriving some estimates for \(I_q(u_q)\), we also prove, under an expected asymptotic behavior of \(u_q\) as \(q\to p^*\), that \(\lambda_q\) is \(\alpha\)-Hölder continuous in \([1,p^*]\) for any \(0<\alpha<1\). As a consequence, this Hölder regularity holds for a general bounded domain \(\Omega\) when \(p=2\) and also for a ball when \(p>1\).''
scientific article; zbMATH DE number 6482013
scientific article