$H^{\infty }$ functional calculus for an elliptic operator on a half-space with general boundary conditions

Dore, Giovanni ; Venni, Alberto

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 3, pp. 487-543

Résumé

Let $A$ be the $L^p$ realization ($1<p<\infty$) of a differential operator $P(D_x,D_t)$ on ${ℝ}^n\times {ℝ}^{+}$ with general boundary conditions $B_k(D_x,D_t)u(x,0)=0$ ($1\le k\le m$). Here $P$ is a homogeneous polynomial of order $2m$ in $n+1$ complex variables that satisfies a suitable ellipticity condition, and for $1\le k\le m$ $B_k$ is a homogeneous polynomial of order $m_k<2m$; it is assumed that the usual complementing condition is satisfied. We prove that $A$ is a sectorial operator with a bounded $H^{\infty }$ functional calculus.

Détail

Classification : 47A60, 35J40, 47F05