Programa de Pós-Graduação em Matemática

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

Prof. Andrzej Szulkin

Date: 17th February 11:00 am - 11:45 am (UTC -3) Link: PPGM/UFSCar Youtube channel

Abstract: Let Ω be a domain in R3 and let

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

be the Sobolev constant with respect to the embedding D1,2(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 D1,2(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: Scurl(Ω) > S(Ω); Scurl(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.