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By Veblen O., Whitehead J. H.

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This can be done, for example, for the case of some mixed boundary-value problems when the stored-energy function is polyconvex but not quasiconvex at the boundary (see Ball and Marsden [1984]). Another approach would be to try to use Morse theory or mountain-pass methods, but it is not clear how to do this so that, for example, appropriate conditions of Palais-Smale type can be verified; for results in an interesting model problem see Zhang [2001]. More generally, one can ask for a description of how the set of equilibrium solutions varies as a function of relevant parameters such as boundary displacements or loads.

They are restricted to initial data having small total variation, and thus, via total variation estimates on the solution, to solutions of small total variation. Glimm’s original work assumed that the system was ‘genuinely nonlinear’, but this restriction was removed by Liu [1981]. Thanks to work of Bressan [1988, 1995], Bressan and Colombo [1995], Bressan [Crasta and Piccoli], Bressan and Goatin [1999], Bressan and Le Floch [1997], Bressan and Lewicka [2000], Bressan, Liu and Yang [1999] and Liu and Yang [1999b,a,c], the solutions obtained in these ways are now known to be unique in appropriate function classes.

Develop a qualitative dynamics for dynamic theories of Of course a prerequisite for such a qualitative dynamics is a global existence theory for solutions. Given such a theory, the points at issue are the usual ones for dissipative dynamical systems, namely whether solutions converge to equilibrium states as t → ∞, the structure of regions of attraction, the existence of stable and unstable manifolds of equilibria, the existence of a global attractor, and so on. In particular one can ask whether dynamic orbits generically realize suitably defined local minimizing sequences for the ballistic free energy.

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A Set of Axioms for Differential Geometry by Veblen O., Whitehead J. H.


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