Andrzej Szulkin: A Sobolev-type inequality for the curl operator and ground states for the curl-curl equation with critical Sobolev exponent

Abstract

Let \(\Omega\)  be a domain in \(\mathbb{R}^3\)  and let

\( S(\Omega) := \inf\{|\nabla u|_2^2/|u|_6^2: u\in C_0^\infty(\Omega)\setminus \{0\}\} \)

be the Sobolev constant with respect to the embedding \(\mathcal{D}^{1,2}_0(\Omega)\hookrightarrow L^6(\Omega)\) . As is well known, \(S(\Omega)\)  is independent of \(\Omega\) , is attained if and only if \(\Omega=\mathbb{R}^3\)  and the infimum is taken by ground state solutions for the equation \(-\Delta u = |u|^4u\)  in \(\mathcal{D}^{1,2}(\mathbb{R}^3)\)  (the Aubin-Talenti instantons).

In this talk we will be concerned with the curl operator \(\nabla\times \cdot\) . In order to define a Sobolev-type constant it seems natural to replace \(S(\Omega)\)  by

\( \overline{S}(\Omega) := \inf\{|\nabla\times u|_2^2/|u|_6^2: u\in C_0^\infty(\Omega,\mathbb{R}^3)\setminus \{0\}\}. \)

However, since the kernel of curl is nontrivial (\(\nabla\times u=0\ \forall\,u=\nabla\varphi\) ), this constant would always be \(0\).

After discussing the physical background we define another constant, \(S_{\text{curl}}(\Omega)\) , as a certain infimum. It has the following properties: \(S_{\text{curl}}(\Omega)> S(\Omega)\) ; \(S_{\text{curl}}(\Omega)\)  is independent of \(\Omega\) ; the infimum is attained when \(\Omega=\mathbb{R}^3\)  and is taken by a ground state solution to the equation \(\nabla\times(\nabla\times u) = |u|^4u\)  (which is related to Maxwell's equations).