# PDE and applications

- Date:
- Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64119
- Lecturer: Andrzej Szulkin
- Contact person: Kaj Nyström
- Seminarium

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

The seminar will take place at Ångström Laboratory, but it will also be possible to attend on Zoom.

Abstract: Let $\Omega$ be a domain in $\mathbb{R}^3$ and let

\[

S(\Omega) := \inf\{\|\nabla u\|_{L^2(\Omega)}^2/\|u\|_{L^6(\Omega)}^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 $\Omega$, it 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 $\nabla\times \cdot$. After discussing the physical background we

define a Sobolev-type constant, $S_{\text{curl}}(\Omega)$, as a certain infimum. As we shall see, it is not possible to

simply replace $\|\nabla u\|_{L^2(\Omega)}$ by $\|\nabla\times u\|_{L^2(\Omega)}$ in the definition above.

$S_{\text{curl}}$ 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).

This is joint work with Jaroslaw Mederski (Institute of Mathematics of the Polish Academy of Sciences).