Criterion for the $L^{p}$-dissipativity of second order differential operators with complex coefficients
| dc.creator | Cialdea, Alberto | |
| dc.creator | Maz'ya, Vladimir | |
| dc.date | 2004-12-11 | |
| dc.date.accessioned | 2026-07-07T05:15:13Z | |
| dc.date.available | 2026-07-07T05:15:13Z | |
| dc.description | We prove that the algebraic condition $|p-2| |< {\mathscr Im}{\mathscr A}ξ,ξ>| \leq 2 \sqrt{p-1} < {\mathscr Re}{\mathscr A}ξ,ξ>$ (for any $ξ\in\mathbb{R}^{n}$) is necessary and sufficient for the $L^{p}$-dissipativity of the Dirichlet problem for the differential operator $\nabla^{t}({\mathscr A}\nabla)$, where ${\mathscr A}$ 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 $\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$, where the coefficients are smooth and ${\mathscr Im}{\mathscr A}$ 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 $\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$ with constant coefficients. | |
| dc.description | 37 pages, LaTeX, no figures | |
| dc.identifier | https://arxiv.org/abs/math/0412225 | |
| dc.identifier | http://arxiv.org/abs/math/0412225 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/73559 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 47D03 (Primary) 47D06, 47B44 (Secondary) | |
| dc.title | Criterion for the $L^{p}$-dissipativity of second order differential operators with complex coefficients | |
| dc.type | text |