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

Abstract

AbstractLet $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 be a Lipschitz domain and let $$S_\mathrm {curl}(\Omega )$$ S curl ( Ω ) be the largest constant such that $$\begin{aligned} \int _{\mathbb {R}^3}|\nabla \times u|^2\, \mathrm{d}x\ge S_{\mathrm {curl}}(\Omega ) \inf _{\begin{array}{c} w\in W_0^6(\mathrm {curl};\mathbb {R}^3)\\ \nabla \times w=0 \end{array}}\Big (\int _{\mathbb {R}^3}|u+w|^6\,\mathrm{d}x\Big )^{\frac{1}{3}} \end{aligned}$$ ∫ R 3 | ∇ × u | 2 d x ≥ S curl ( Ω ) inf w ∈ W 0 6 ( curl ; R 3 ) ∇ × w = 0 ( ∫ R 3 | u + w | 6 d x ) 1 3 for any u in $$W_0^6(\mathrm {curl};\Omega )\subset W_0^6(\mathrm {curl};\mathbb {R}^3)$$ W 0 6 ( curl ; Ω ) ⊂ W 0 6 ( curl ; R 3 ) , where $$W_0^6(\mathrm {curl};\Omega )$$ W 0 6 ( curl ; Ω ) is the closure of $$\mathcal {C}_0^{\infty }(\Omega ,\mathbb {R}^3)$$ C 0 ∞ ( Ω , R 3 ) in $$\{u\in L^6(\Omega ,\mathbb {R}^3): \nabla \times u\in L^2(\Omega ,\mathbb {R}^3)\}$$ { u ∈ L 6 ( Ω , R 3 ) : ∇ × u ∈ L 2 ( Ω , R 3 ) } with respect to the norm $$(|u|_6^2+|\nabla \times u|_2^2)^{1/2}$$ ( | u | 6 2 + | ∇ × u | 2 2 ) 1 / 2 . We show that $$S_{\mathrm {curl}}(\Omega )$$ S curl ( Ω ) is strictly larger than the classical Sobolev constant S in $$\mathbb {R}^3$$ R 3 . Moreover, $$S_{\mathrm {curl}}(\Omega )$$ S curl ( Ω ) is independent of $$\Omega $$ Ω and is attained by a ground state solution to the curl–curl problem $$\begin{aligned} \nabla \times (\nabla \times u) = |u|^4u \end{aligned}$$ ∇ × ( ∇ × u ) = | u | 4 u if $$\Omega =\mathbb {R}^3$$ 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 $$\Omega $$ Ω $$\begin{aligned} \nabla \times (\nabla \times u) +\lambda u = |u|^4u\quad \hbox {in }\Omega , \end{aligned}$$ ∇ × ( ∇ × u ) + λ u = | u | 4 u in Ω , with the so-called metallic boundary condition $$\nu \times u=0$$ ν × u = 0 on $$\partial \Omega $$ ∂ Ω , where $$\nu $$ ν is the exterior normal to $$\partial \Omega $$ ∂ Ω .