Application of the Trace Inequality to the Poisson Equation

Abstract

The purpose of this paper is to show that solutions of the Poisson equation $$-\Delta u=f$$ where f be a complex-valued distribution on \({\mathbb{R}}^{d}\), d ≥ 3 and \(f{\in} {\mathcal{M}}\left( \overset{.}{H}^{1}{\rightarrow} \overset{.}{H} ^{-1}\right) \) satisfy the coercivity property : \(D^{k} u\in \mathcal{M}\left( \overset{.}{H}^{1}\rightarrow \overset{.}{H}^{-1}\right)\) for all \(k,\left\vert k\right\vert =2\) . The coercivity of this equation is well studied by Maz’ ya and Verbitsky [14] in the case where f belongs to the class of positive Borel measures.