Speaker
Andrzej Szulkin, Stockholm University
Abstract
The classical Sobolev inequality in \(\mathbb{R}^3\) states that \[ \int_{\mathbb{R}^3}|\nabla u|^p\ge S_{p}\Big(\int_{\mathbb{R}^3}|u|^{p^*}\Big)^{\frac{p}{p^*}} \] for all \(u\in C_0^\infty(\mathbb{R}^3)\). Here \(p\in(1,3)\), \(p^* := \frac{3p}{3-p}\) and \(S_p\) is the largest constant for which the inequality above holds. It is also known that \(S_p\) is attained by a positive solution \(u\in\mathcal{D}^{1,p}(\mathbb{R}^3) := \{u\in L^{p^*}(\mathbb{R}^3): \int_{\mathbb{R}^3}|\nabla u|^p < \infty\}\) to the equation \(\Delta_pu := -\text{div}(|\nabla u|^{p-2}\nabla u)=|u|^{p^*-2}u\). In this talk we address the problem of existence of a similar constant and of ground states for the curl operator \(\nabla\times \cdot\). Such constant, \(S_{p,\text{curl}}\), has been defined by Mederski and myself for \(p=2\), and by Frank and Loss for \(p=3/2\). Both cases have a different physical background which we briefly discuss. Next we define a constant \(S_{p,\text{curl}}\) for a general \(p\in(1,3)\). Then we discuss the choice of spaces, a suitable functional-analytic framework and sketch the proof that the constant \(S_{p,\text{curl}}\) is attained by a solution \(u : \mathbb{R}^3\to\mathbb{R}^3\) to the equation \(\nabla\times (|\nabla\times u|^{p-2}\nabla\times u)= |u|^{p^*-2}u\). If time permits, we also briefly discuss existence of solutions with prescribed symmetry. The talk is based on joint work with Jaros\l aw Mederski.