| Summary: | In part I of this thesis, we prove some regularity and uniqueness results of the minimizer for the problem$$\inf\{\int\sb\Omega \phi(Dv) + \int\sb{\partial\Omega} \vert{v - g}\vert dH\sp{n-1} : v \in BV(\Omega), g \in L\sp1(\partial\Omega)\},$$where $\Omega$ is a domain in $R\sp{n}, \phi(p)$ = 1/2$\vert p\vert\sp2$ for $\vert p\vert \le$ 1, $\phi(p)$ = $\vert p\vert \ -$ 1/2 for $\vert p\vert \ge$ 1. The integrand arises in the study of anti-planar shear in elastic/plastic deformation, (K), (KT), (HK1). We also study the behaviors of the static minimizer at the singular point.
In part II, we use the weak formulation analogous to that of (GC1) and (LT1) to study the evolution problems associated with $\phi$, formally written as $\partial u/\partial t$ = div$\sb{x}\nabla\sb{p}\phi(\nabla u),$ with Dirichlet and Neumann boundary data. We prove the existence, uniqueness and various regularities, plus the asymptotic behavior of the weak solution.
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