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- CONTENTS 1987 - VOLUME 305 - SECTION I - N° 9
- .......... Page(s) .......... 357
- .......... Page(s) .......... 357
- In this Note, k is a p-adic field of characteristic zero, G the group of k-rational points of an almost absolutely simple k-group , of k-rank and a discrete cocompact subgroup of G. We state some finiteness properties of the set of such triples for which has a volume bounded by a given constant, with respect to a suitably normalized Haar measure, when or the order q of the residue field k of k vary.
- .......... Page(s) .......... 363
- .......... Page(s) .......... 363
- We prove exact controllability in a finite time of the solution to the initial boundary value problem for the wave equation by means of a Dirichlet control on a part of the boundary and a Neumann control on the complementary. The solution has a singular behaviour produced by the mixed Dirichlet-Neumann boundary condition. This makes it necessary to impose drastic restrictions of geometrical character (that we make explicit in this Note) to allow the performance of the multiplier technique in the Hilbert uniqueness method introduced by J. - L. Lions. Similar results are derived for domains with cuts.
- .......... Page(s) .......... 367
- The existence of nonlinear Neumann problems with inhomogeneous boundary conditions is established and the solution set is described. Then the asymptotic behaviour of the time-dependent parabolic equation is studied.
- .......... Page(s) .......... 371
- We obtain an optimal estimate for the elementary spherical function
- .......... Page(s) .......... 375
- We give an example of a convolution operator of "strongly" singular type for the Sobolev space on . The result extends to .
- .......... Page(s) .......... 377
- .......... Page(s) .......... 377
- One considers a matrix of partial differential operators with hyperbolic characteristic determinant and constant multiplicity; it is reduced to the cases of simple microlocal matrices. Under this form, the necessary and sufficient hyperbolicity conditions are stated, when the rank is constant or when the rank change, if the operators are analytic and the dimension is two.
- .......... Page(s) .......... 385
- The de Rham Hodge Operator has not a uniform minoration of its first eigenvalue on all conformal metrics with unit volume.
- .......... Page(s) .......... 389
- .......... Page(s) .......... 389
- We introduce the notions of
-radial, radial and weakly radial family of differentiable submanifolds and we study local arcs on closed unions of such families. We deduce a sufficient condition for a compact connected subset of to have finite geodesic diameter; Whitney stratified sets satisfy this condition.
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