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).