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

Tid: On 2020-10-28 kl 13.15 - 14.15

Medverkande: Andrzej Szulkin, Stockholms universitet

### Abstract

Let $$\Omega$$  be a domain in $$\mathbb{R}^3$$  and let

$$S(\Omega) := \inf\{|\nabla u|_2^2/|u|_6^2: u\in C_0^\infty(\Omega)\setminus \{0\}\}$$

be the Sobolev constant with respect to the embedding $$\mathcal{D}^{1,2}_0(\Omega)\hookrightarrow L^6(\Omega)$$ . As is well known, $$S(\Omega)$$  is independent of $$\Omega$$ , is attained if and only if $$\Omega=\mathbb{R}^3$$  and the infimum is taken by ground state solutions for the equation $$-\Delta u = |u|^4u$$  in $$\mathcal{D}^{1,2}(\mathbb{R}^3)$$  (the Aubin-Talenti instantons).

In this talk we will be concerned with the curl operator $$\nabla\times \cdot$$ . In order to define a Sobolev-type constant it seems natural to replace $$S(\Omega)$$  by

$$\overline{S}(\Omega) := \inf\{|\nabla\times u|_2^2/|u|_6^2: u\in C_0^\infty(\Omega,\mathbb{R}^3)\setminus \{0\}\}.$$

However, since the kernel of curl is nontrivial ($$\nabla\times u=0\ \forall\,u=\nabla\varphi$$ ), this constant would always be $$0$$.

After discussing the physical background we define another constant, $$S_{\text{curl}}(\Omega)$$ , as a certain infimum. It has the following properties: $$S_{\text{curl}}(\Omega)> S(\Omega)$$ ; $$S_{\text{curl}}(\Omega)$$  is independent of $$\Omega$$ ; the infimum is attained when $$\Omega=\mathbb{R}^3$$  and is taken by a ground state solution to the equation $$\nabla\times(\nabla\times u) = |u|^4u$$  (which is related to Maxwell's equations).