Calculus of Variations and Geometric Measure Theory

M. Petrache - J. Van Schaftingen

Controlled singular extension of critical trace Sobolev maps from spheres to compact manifolds

created by petrache on 04 Sep 2015



Inserted: 4 sep 2015
Last Updated: 4 sep 2015

Pages: 26
Year: 2015


Given $n \in \mathbb N_*$, a compact Riemannian manifold $M$ and a Sobolev map $u \in W^{n/(n + 1), n + 1} (\mathbb S^n; M)$, we construct a map $U$ in the Sobolev-Marcinkiewicz (or Lorentz-Sobolev) space $W^{1, (n + 1, \infty)} (\mathbb{B}^{n + 1}; M)$ such that $u = U$ in the sense of traces on $\mathbb{S}^{n} = \partial \mathbb{B}^{n + 1}$ and whose derivative is controlled: for every $\lambda > 0$, \[ \lambda^{n + 1} \left\vert\left\{ x \in \mathbb B^{n + 1} :\ \vert D U (x)\vert > \lambda\right\}\right\vert \le \gamma \left(\int_{\mathbb S^n}\int_{\mathbb S^n} \frac{\vert u (y) - u (z)\vert^{n + 1}}{\vert y - z\vert^{2 n}} \ d y \ d z \right), \] where the function $\gamma : [0, \infty) \to [0, \infty)$ only depends on the dimension $n$ and on the manifold $M$. The construction of the map $U$ relies on a smoothing process by hyperharmonic extension and radial extensions on a suitable covering by balls.