Criterion for the $L^p$-dissipativity of second order differential operators with complex coefficients

Abstract

We prove that the algebraic condition $|p-2||<\mathcal{I}m\mathcal{A}\xi ,\xi >|\le 2\sqrt{p-1}<\mathcal{R}e\mathcal{A}\xi ,\xi >$ (for any $\xi \in {\mathbb{R}}^n$) is necessary and sufficient for the $L^p$-dissipativity of the Dirichlet problem for the differential operator ${\nabla }^t(\mathcal{A}\nabla )$, where $\mathcal{A}$ is a matrix whose entries are complex measures and whose imaginary part is symmetric. This result is new even for smooth coefficients, when it implies a criterion for the $L^p$-contractivity of the corresponding semigroup. We consider also the operator ${\nabla }^t(\mathcal{A}\nabla )+\mathbf{b}\nabla +a$, where the coefficients are smooth and $\mathcal{I}m\mathcal{A}$ may be not symmetric. We show that the previous algebraic condition is necessary and sufficient for the $L^p$-quasi-dissipativity of this operator. The same condition is necessary and sufficient for the $L^p$-quasi-contractivity of the corresponding semigroup. We give a necessary and sufficient condition for the $L^p$-dissipativity in ${\mathbb{R}}^n$ of the operator ${\nabla }^t(\mathcal{A}\nabla )+\mathbf{b}\nabla +a$ with constant coefficients.

37 pages, LaTeX, no figures