A Sobolev-Type Inequality for the Curl Operator and Ground States for the Curl–Curl Equation with Critical Sobolev Exponent

Let Ω⊂R$^{3}$ be a Lipschitz domain and let S$_{curl}$(Ω) be the largest constant such that

∫$_{R^{3}}$|∇×u|$^{2}$dx≥S$_{curl}$(Ω) infw ∈W$^{6}$$_{0}$ (curl;R$^{3}$)∇×w=0(∫$_{R^{3}}$|u+w|$^{6}$dx)$^{1/3}$
for any u in W$^{6}$$_{0}$(curl;Ω)⊂W$^{6}$$_{0}$(curl;R$^{3}$), where W$^{6}$$_{0}$(curl;Ω) is the closure of C$^{∞}_{0}$(Ω,R$^{3}$) in {u∈L$^{6}$(Ω,R$^{3}$):∇×u∈L$^{2}$(Ω,R$^{3}$)} with respect to the norm (|u|$^{2}_{6}$+|∇×u|$^{2}_{2}$)$^{1/2}$. We show that S$_{curl}$(Ω) is strictly larger than the classical Sobolev constant S in R$^{3}$. Moreover, S$_{curl}$(Ω) is independent of Ω and is attained by a ground state solution to the curl–curl problem

∇×(∇×u)=|u|$^{4}$u
if Ω=R$^{3}$. With the aid of these results we also investigate ground states of the Brezis–Nirenberg-type problem for the curl–curl operator in a bounded domain Ω

∇×(∇×u)+λu=|u|$^{4}$u in Ω,
with the so-called metallic boundary condition ν×u=0 on ∂Ω, where ν is the exterior normal to ∂Ω.