Weakly semibounded boundary problems and sesquilinear forms

Grubb, Gerd. "Weakly semibounded boundary problems and sesquilinear forms." Annales de l'institut Fourier 23.4 (1973): 145-194. <http://eudml.org/doc/74145>.

@article{Grubb1973,
abstract = {Let $A$ be a $2m$ order differential operator in a hermitian vector bundle $E$ over a compact riemannian manifold $\overline\{\Omega \}$ with boundary $\Gamma $ ; and denote by $A_B$ the realization defined by a normal differential boundary condition $B\rho u=0$ ($u\in H^\{2m\}(E)$, $\rho u=$ Cauchy data). We characterize, by an explicit condition on $A$ and $B$ near $\Gamma $, the realizations $A_B$ for which there exists an integro-differential sesquilinear form $a_B(u,\nu )$ on $H^m(E)$ such that $(Au,\nu )=a_B(u,\nu )$ on $D(A_B)$; moreover we show that these are exactly the realizations satisfying a weak semiboundedness estimate: $\{\rm Re\}\, e^\{i\theta \}(Au,u)\le c\Vert u\Vert ^2_m$ for all $u\in D(A_B)$. The theorems are generalized completely to systems $A=(A_\{st\})_\{s,t=1,\ldots ,q\}$, where $A_\{st\}$ is of order $m_s+m_r\ge 0$; several new concepts are introduced here. The results are fundamental in the study of semiboundedness and coerciveness inequalities (continued elsewhere); in particular they provide the complete characterization of the elliptic realizations satisfying Garding’s inequality, in conjunction with the works of Agmon and de Figueiredo on integro-differential forms.},
author = {Grubb, Gerd},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {4},
pages = {145-194},
publisher = {Association des Annales de l'Institut Fourier},
title = {Weakly semibounded boundary problems and sesquilinear forms},
url = {http://eudml.org/doc/74145},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Grubb, Gerd
TI - Weakly semibounded boundary problems and sesquilinear forms
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 4
SP - 145
EP - 194
AB - Let $A$ be a $2m$ order differential operator in a hermitian vector bundle $E$ over a compact riemannian manifold $\overline{\Omega }$ with boundary $\Gamma $ ; and denote by $A_B$ the realization defined by a normal differential boundary condition $B\rho u=0$ ($u\in H^{2m}(E)$, $\rho u=$ Cauchy data). We characterize, by an explicit condition on $A$ and $B$ near $\Gamma $, the realizations $A_B$ for which there exists an integro-differential sesquilinear form $a_B(u,\nu )$ on $H^m(E)$ such that $(Au,\nu )=a_B(u,\nu )$ on $D(A_B)$; moreover we show that these are exactly the realizations satisfying a weak semiboundedness estimate: ${\rm Re}\, e^{i\theta }(Au,u)\le c\Vert u\Vert ^2_m$ for all $u\in D(A_B)$. The theorems are generalized completely to systems $A=(A_{st})_{s,t=1,\ldots ,q}$, where $A_{st}$ is of order $m_s+m_r\ge 0$; several new concepts are introduced here. The results are fundamental in the study of semiboundedness and coerciveness inequalities (continued elsewhere); in particular they provide the complete characterization of the elliptic realizations satisfying Garding’s inequality, in conjunction with the works of Agmon and de Figueiredo on integro-differential forms.
LA - eng
UR - http://eudml.org/doc/74145
ER -