OxPDE Lunchtime Seminar
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Thu, 28/10/2010 12:30 |
Lisa Harris (University of Warwick) |
OxPDE Lunchtime Seminar  |
Gibson 1st Floor SR |
| It has long been known that many materials are crystalline when in their energy-minimizing states. Two of the most common crystalline structures are the face-centred cubic (fcc) and hexagonal close-packed (hcp) crystal lattices. Here we introduce the problem of crystallization from a mathematical viewpoint and present an outline of a proof that the ground state of a large system of identical particles, interacting under a suitable potential, behaves asymptotically like fcc or hcp, as the number of particles tends to infinity. An interesting feature of this result is that it holds under no initial assumption on the particle positions. The talk is based upon a joint work in progress with Florian Theil. | |||
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Fri, 26/11/2010 12:30 |
Richard Gratwick (University of Warwick) |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
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Tonelli gave the first rigorous treatment of one-dimensional variational problems, providing conditions for existence and regularity of minimizers over the space of absolutely continuous functions. He also proved a partial regularity theorem, asserting that a minimizer is everywhere differentiable, possible with infinite derivative, and that this derivative is continuous as a map into the extended real line. Some recent work has lowered the smoothness assumptions on the Lagrangian for this result to various Lispschitz and H\"older conditions. In this talk we will discuss the partial regularity result, construct examples showing that mere continuity of the Lagrangian is an insufficient condition. |
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