Asymptotics for the best Sobolev constants and their extremal functions (Q2825992)

This work studies the behavior of the best constant \(C\Lambda_p(\Omega)\) in the Sobolev inequality NEWLINE\[NEWLINE\| u\| ^p_{\infty} \leq \Lambda_p(\Omega)\| \nabla u\| ^pNEWLINE\]NEWLINE in a bounded domain \(\Omega\) as \(p\) goes to infinity, as well as the extremal functions. The main result is that NEWLINE\[NEWLINE\Lambda_p(\Omega)^{1/p} \rightarrow \| \rho\| _{\infty}^{-1}NEWLINE\]NEWLINE where \(\rho\) is the distance function to the boundary of \(\Omega\). The authors also show that there is a subsequence \(p_n\) such that the extremizers of the Sobolev inequality for \(p_n\) converge to a solution of an \(\infty\)-Laplacian PDE with a boundary condition involving \(\rho\). The proof uses a characterization of extremizers via the Euler-Lagrange formulation of the minimization problem associated to the Sobolev inequality for fixed \(p\), and the fact that the distance function satisfies \(\| \rho\| _{\infty}^{-1} = \min \left\{\frac{\| \nabla \phi\| _{\infty}}{\| \phi\| _{\infty}}; \phi \in W^{1, \infty}, \phi \neq 0\right\}\).