Divergence L²-Coercivity Inequalities

Abstract

This paper derives sharp L ²-coercivity inequalities for the divergence operator on bounded Lipschitz regions in ℝ ⁿ . They hold for fields in H(div,Ω) that are orthogonal to N(div). The optimal constants in the inequality are defined by a variational principle and are identified as the least eigenvalue of a nonstandard boundary value problem for a linear biharmonic type operator. The dependence of the optimal constant under dilations of the region is described and a generalization that involves weighted surface integrals is also proved. When n = 2, this also yields a similar coercivity result for the curl operator.

Keywords:

• Biharmonic eigenproblems
• Divergence inequalities
• L ²-coercivity

AMS Subject Classification:

• 47A50
• 35Q60
• 49R50

ACKNOWLEDGMENT

This paper was partially based on work supported by the National Science Foundation while working at the foundation. All opinions or conclusions expressed in this paper are those of the author and do not necessarily reflect the views of the NSF.