# Weakly semibounded boundary problems and sesquilinear forms

Annales de l'institut Fourier (1973)

- Volume: 23, Issue: 4, page 145-194
- ISSN: 0373-0956

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topGrubb, 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 -

## References

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- [2] S. AGMON, A. DOUGLIS and L. NIRENBERG, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II, Comm. Pure Appl. Math., 17 (1964), 35-92. Zbl0123.28706MR28 #5252
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- [7] G. GRUBB, Problèmes aux limites semi-bornés pour les systèmes elliptiques, C.R. Acad. Sci. (Série A), 274 (1972), 320-323. Zbl0243.35032MR45 #7525
- [8] G. GRUBB, Properties of normal boundary problems for elliptic even-order systems, Copenh. Mat. Inst. Preprint Ser. 1973 no. 4, to appear. Zbl0309.35034
- [9] D. GUEDES DE FIGUEIREDO, The coerciveness problem for forms over vector valued functions, Comm. Pure Appl. Math., 16 (1963), 63-94. Zbl0136.09502MR26 #6578
- [10] L. HÖRMANDER, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math., 83 (1966), 129-209. Zbl0132.07402MR38 #1387
- [11] J. L. LIONS et E. MAGENES, Problèmes aux limites non homogènes et applications, vol. 1, Ed. Dunod, Paris, 1968. Zbl0165.10801
- [12] R. SEELEY, Fractional powers of boundary problems, Actes Congrès Intern., 1970, Nice, vol. 2, 795-801. Zbl0224.35029MR58 #23157

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