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

Abstract

Let $\Omega \subset {\mathbb{R}}^3$ be a Lipschitz domain and let $S_{\mathrm{curl}}(\Omega )$ be the largest constant such that $${\int }_{{\mathbb{R}}^3}|\nabla \times u{|}^{2}dx\ge S_{\mathrm{curl}}(\Omega )\underset{\begin{array}{c}w\in W_0^6(\mathrm{curl};{\mathbb{R}}^3)\\ \nabla \times w=0\end{array}}{\inf }({\int }_{{\mathbb{R}}^3}|u+w{|}^{6}dx{)}^{\frac{1}{3}}$$ for any $u$ in $W_0^6(\mathrm{curl};\Omega )\subset W_0^6(\mathrm{curl};{\mathbb{R}}^3)$ where $W_0^6(\mathrm{curl};\Omega )$ is the closure of ${\mathcal{C}}_0^{\infty }(\Omega ,{\mathbb{R}}^3)$ in $\{u\in L^6(\Omega ,{\mathbb{R}}^3):\nabla \times u\in L^2(\Omega ,{\mathbb{R}}^3)\}$ with respect to the norm $(|u{|}_6^2+|\nabla \times u{|}_2^2{)}^{1/2}$. We show that $S_{\mathrm{curl}}(\Omega )$ is strictly larger than the classical Sobolev constant $S$ in ${\mathbb{R}}^3$. Moreover, $S_{\mathrm{curl}}(\Omega )$ is independent of $\Omega$ and is attained by a ground state solution to the curl-curl problem $$\nabla \times (\nabla \times u)=|u{|}^{4}u$$ if $\Omega ={\mathbb{R}}^3$. With the aid of those results, we also investigate ground states of the Brezis-Nirenberg-type problem for the curl-curl operator in a bounded domain $\Omega$ $$\nabla \times (\nabla \times u)+\lambda u=|u{|}^{4}uin\; \Omega$$ with the so-called metallic boundary condition $\nu \times u=0$ on $\partial \Omega$, where $\nu$ is the exterior normal to $\partial \Omega$.

Archive for Rational Mechanics and Analysis (2021)

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