A Sobolev-type inequality for the curl operator and ground states for the curl-curl equation

Resumen:  Let \(\Omega\) be a domain in \(\mathbb{R}^3\) and let
\begin{equation}
S(\Omega) := \inf\{|\nabla u|_2^2/|u|_6^2: u\in C_0^\infty(\Omega)\setminus \{0\}\}
\end{equation}
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
\begin{equation}
\overline{S}(\Omega) := \inf\{|\nabla\times u|_2^2/|u|_6^2: u\in C_0^\infty(\Omega,\mathbb{R}^3)\setminus \{0\}\}.
\end{equation}
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). If time permits, we briefly consider a related Brezis-Nirenberg problem.

We end this talk by discussing some open problems.

This is joint work with Jaroslaw Mederski, ARMA 241 (2021), 1815--1842.

https://cuaieed-unam.zoom.us/j/89946525336
Meeting ID: 899 4652 5336