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By François Alouges (auth.), Jean-Michel Coron, Jean-Michel Ghidaglia, Frédéric Hélein (eds.)

This quantity (>Ie) NEMATICS Mathematical and actual features constitutes the lawsuits of a workshop which was once held at l'Universite de Paris Sud (Orsay) in might 1990. This assembly used to be a complicated learn Workshop backed by way of NATO. We gratefully recognize the assistance and help of the NATO technological know-how Committee. extra help has been supplied through the Ministere des affaires etrangeres (Paris) and by way of the path des Recherches et Etudes thoughts (Paris). additionally logistic aid has been supplied by way of the organization des Numericiens d'Orsay. (*) those court cases are released within the framework of the "Contrat DRET W 90/316/ AOOO". v Contents (*) FOREWORD v advent 1. M. CORON, 1. M. GHIDAGLIA, F. HELEIN xi AN ENERGY-DECREASING set of rules FOR HARMONIC MAPS F. ALOUGES 1 A COHOMOLOGICAL CRITERION FOR DENSITY OF delicate MAPS IN SOBOLEV areas among MANIFOLDS F. BETHUEL, 1. M. CORON, F. DEMENGEL, F. HELEIN 15 at the MATHEMATICAL MODELING OF TEXTURES IN POLYMERIC LIQUID CRYSTALS M. C. CAmERER 25 A outcome at the international lifestyles for warmth FLOWS OF HARMONIC MAPS FROM D2 INTO S2 okay. C. CHANG, W. Y. DING 37 BLOW-UP research for warmth circulate OF HARMONIC MAPS Y. CHEN forty nine T AYLOR-COUETTE INSTABILITY IN NEMATIC LIQUID CRYSTALS P. E. ClADIS sixty five ON a category OF ideas within the concept OF NEMATIC levels B. D. COLEMAN, 1. T. JENKINS ninety three RHEOLOGY OF THERMOTROPIC NEMATIC LIQUID CRYSTALLINE POLYMERS M. M. DENN, 1. A.

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2) exists for all time, it may encounter singularties at T = +00, so that it fails to converge asymptotically. We can see this point from the example produced by Eells and Wood [8] in the case that M has no boundary. If M has a smooth boundary, this is obvious. when the initial map is in a nontrivial homotopic class with finite energy and constant boundary value, the flow must develop a singularity at T = +00. This is due to the fact that any smooth harmonic map with constant boundary value has to be a constant.

Let p be a smooth map from IR to IR with compact support and with weight fJRP(s)ds equal to 1. Set for f > 0 PE(S) Since = ~p CD and (JE(X) = PE[dist (x,P)]d[dist (x,P)]. 11) where a is the pull-back n" a of a by the orthogonal projection II onto P. 11). 10). D. 2 ~ Now let p be in C1(IR,IR) and such that p(s) = 0 for s ~ 1 and p(s) = 1 for s. 12) is zero. The second term tends trivially to zero when e goes to zero. 12) also goes to zero when e goes to zero. 2. Proof of Theorems 1 and 2 in the limiting case V = m - I, with M and M' = cm.

1. Ik' since Ik The proof of Theorem 2 for V = m - I and u' = cm is the same as in Section 3. M M'=C m • c m and It follows the outlines of the proof in [Bel]. Recall that in [Bel], the hypothesis TI[pJ(N) = 0 is used to approximate I restricted on [vI-dimensional sets (with prescribed boundary conditions). In our situation we replace this hypothesis using Theorem I and 2 in dimension [V] + 1. In order to do so, we use an induction argument and Lemma 3 above. 4. M' are any manifolds. 2) in the special case 8M' = P(a,ej).

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