Carnegie Mellon
Department of Mathematical Sciences

Christopher Larsen, Department of Mathematics, Worcester Polytechnic Institute.

"Avoiding the Neumann sieve in a problem of fracture growth"

In proving existence for mathematical models of brittle fracture growth, the following problem arrises:

Suppose $u_n\in SBV(\Omega)$ minimizes

\begin{displaymath}E(v):=\int_\Omega \vert\nabla v\vert^2 dx + {\cal{H}}^{N-1}(S_v\backslash S_{u_n})\end{displaymath}

subject to $v=u_n$ on $\partial \Omega$, where $\Omega\subset I\!\!R^N$. If $u_n\rightarrow u$ (in the sense of SBV compactness), does the corresponding minimality hold for $u$?

The problem of the "Neumann sieve" suggests that this minimality may be false, though Dal Maso and Toader found a way around it, assuming that the number of components of $S_{u_n}$ is bounded and that $N=2$. It turns out that there is a way to completely avoid the issue of the Neumann sieve, which I will describe in this talk. This is part of joint work with G. Francfort.

Location: PPB 300
Time: 1:30 P.M.

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