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

This quantity (>Ie) NEMATICS Mathematical and actual facets 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 sophisticated study Workshop subsidized by means of NATO. We gratefully recognize the assistance and aid of the NATO technological know-how Committee. extra aid has been supplied via the Ministere des affaires etrangeres (Paris) and via the course des Recherches et Etudes strategies (Paris). additionally logistic help has been supplied via the organization des Numericiens d'Orsay. (*) those lawsuits are released within the framework of the "Contrat DRET W 90/316/ AOOO". v Contents (*) FOREWORD v creation 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 gentle 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 consequence at the international life for warmth FLOWS OF HARMONIC MAPS FROM D2 INTO S2 okay. C. CHANG, W. Y. DING 37 BLOW-UP research for warmth circulation OF HARMONIC MAPS Y. CHEN forty nine T AYLOR-COUETTE INSTABILITY IN NEMATIC LIQUID CRYSTALS P. E. ClADIS sixty five ON a category OF strategies within the idea OF NEMATIC stages B. D. COLEMAN, 1. T. JENKINS ninety three RHEOLOGY OF THERMOTROPIC NEMATIC LIQUID CRYSTALLINE POLYMERS M. M. DENN, 1. A.

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Remarks. In the case Vo > 0, the following observations apply: (1) No transitions through the isotropic state are possible. Moreover, such state cannot exist as equilibrium either. e. they approach asymptotically either 8 = 1 or s = -~. Finally, we consider the case c > 0 and Vo > o. The governing equation for s becomes nonautonomous. 8) can be interpreted as two kinds of dissipation being present in the problem. 4. Let Vo > 0 and c > o. Suppose that 81 > o. Then: (1) Orbits in the phase plane enter the corresponding ones of the case c = o.

3). 5) we see that the lemma will be proved if we can show '7 == O. Now define vCr, t) = ,,(r, t) sin her, t). Then v is continuous on [0,11 x [0, T) and it is smooth on n = {(r,t) E [0,11 x (0, T) : sin her, t) #- o}. 6) that v satisfies Vt on n. 7) = O. Therefore, if s E (O,T), n. = {(r,t) En: t ~ s}, and a·n. = an. - n. n (0,1) x {s}, then vla·n. = O. Notice that the coefficient of v is bounded above on each n.. so we can replace v by ve- ct with a large positive constant c to make the coefficient negative.

Suppose that 81 > o. Then: (1) Orbits in the phase plane enter the corresponding ones of the case c = o. 2). (3) Regions s > 0 and s < 0 are invariant. Remark Although the isotropic state cannot be reached for Vo i= 0, it can be asymptotically approached for small values of Vo. Therefore as Vo increases solutions become further appart from the isotropic state. From now on, we let x E (0,1). 3) by setting c = o. Throughout the rest of this section we will assume that (H2) is satisfied with a > 2 and (3 > 2, so as to prevent non global existence of minimizers by blow-up of 8 x • 31 First of all, we observe that if the boundary conditions satisfied by () correspond to taking Vo = 0, then the problem reduces to: Minimize 11 + ms;)dx O'(x) = o.