be the Sobolev constant with respect to the embedding D^1,2(Ω)→L⁶(Ω). As is well known, S(Ω) is independent of Ω, it is attained if and only if Ω = R³ and the infimum is taken by ground state solutions for the equation −∆u = |u|⁴u in D^1,2(R³) (the Aubin-Talenti instantons).

In this talk we will be concerned with the curl operator ∇×·. In order to define a Sobolev-type constant it seems natural to replace S(Ω) by

S(Ω):=inf{|∇ × u|²₂/|u|²₆: u∈C₀^∞(Ω,R³)\{0}}.

However, since the kernel of curl is nontrivial (∇×u = 0 ∀u =∇φ), this constant would always be 0.

After discussing the physical background we will define another constant, Scurl(Ω), as a certain infimum. It has the following properties: S_curl(Ω) > S(Ω); S_curl(Ω) is independent of Ω; the infimum is attained when Ω = R³ and is taken by a ground state solution to the equation ∇×(∇×u)=|u|⁴u (which is related to Maxwell’s equations). The problem of (non)existence of the minimum for Ω ̸=R³ remains open.

If time permits, we shall briefly discuss the Brezis-Nirenberg problem for the curl-curl operator on bounded domains.

This is joint work with Jarosl􏰟aw Mederski.