A Littlewood–Paley type inequality with applications to the elliptic Dirichlet problem

Caroline Sweezy¹

¹ Department of Mathematical Sciences New Mexico State University Box 30001 3MB Las Cruces, NM 88003-8001, U.S.A.

Annales Polonici Mathematici, Tome 90 (2007) no. 2, pp. 105-130

Résumé

Let $L$ be a strictly elliptic second order operator on a bounded domain ${\mit \Omega } \subset {{\mathbb R}}^{n}$. Let $u$ be a solution to $Lu=\mathop {\rm div}\vec {f}$ in ${\mit \Omega } $, $u=0$ on $\partial {\mit \Omega } $. Sufficient conditions on two measures, $\mu $ and $\nu $ defined on ${\mit \Omega } $, are established which imply that the $L^{q}({\mit \Omega } ,d\mu )$ norm of $| \nabla u| $ is dominated by the $L^{p}({\mit \Omega } ,dv)$ norms of $\mathop {\rm div}\vec {f}$ and $| \vec {f}| $. If we replace $| \nabla u| $ by a local Hölder norm of $u$, the conditions on $\mu $ and $\nu $ can be significantly weaker.

Détail

DOI : 10.4064/ap90-2-2

Keywords: strictly elliptic second order operator bounded domain mit omega subset mathbb solution mathop div vec mit omega partial mit omega sufficient conditions measures defined mit omega established which imply mit omega norm nabla dominated mit omega norms mathop div vec vec replace nabla local lder norm conditions significantly weaker