TITLE

$C^\infty$ regularity of the free boundary for a two-dimensional optimal compliance problem

AUTHOR(S)
Antonin Chambolle; Christopher J. Larsen
PUB. DATE
September 2003
SOURCE
Calculus of Variations & Partial Differential Equations;Sep2003, Vol. 18 Issue 1, p77
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
We study the regularizing effect of perimeter penalties for a problem of optimal compliance in two dimensions. In particular, we consider minimizers of $$\mathcal{E}(\Omega) = J(\Omega) + \lambda \vert\Omega\vert + \mu \mathcal{H}^1(\partial \Omega)$$ where $$J(\Omega) = -2 inf \left\{\frac{1}{2} \int_{\Omega} {\bf A} e(u) : e(u)- \int_\Gamma f\cdot u : u\in LD(\Omega),\ u\equiv 0 \textrm{on} D \right\}. $$ The sets $D\subset \Omega$, $\Gamma\subset \overline{\Omega}$, and the force f are given. We show that if we consider only scalar valued u and constant ${\bf A}$, or if we consider the elastic energy $\vert\nabla u\vert^2$, then $\partial \Omega$ is $C^\infty$ away from where $\Omega$ is pinned. In the scalar case, we also show that, for any ${\bf A}$ of class $C^{k,\theta}$, $\partial \Omega$ is $C^{ k+2,\theta}$. The proofs rely on a notion of weak outward curvature of $\partial \Omega$, which we can bound without considering properties of the minimizing fields, together with a bootstrap argument.
ACCESSION #
10649414

 

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