OpenYear of origin: 1979
Posted online: 2018-07-08 12:17:00Z by Mohammad Safdari
Cite as: P-180708.2
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Let $U$ be a bounded open set in $\mathbb{R}^2 $. Let $u$ be the minimizer of $$\int_U |\nabla u|^2 -u\,dx \tag{$\ast$ }$$ over the set $\{u\in H^1_0(U) \mid |\nabla u|\le 1 \}$. This is the classical elastic-plastic torsion problem. Let $E:=\{|\nabla u|< 1\}$ be the elastic region, and let $P:=\{ |\nabla u|=1\}$ be the plastic region. It can be shown that there is a function $f:\partial U \to [0,\infty )$ such that the plastic region is consisted of connected components of the form $$\{x+t\nu(x) \mid x\in\partial U, f(x)\ge t>0\},$$ where $\nu$ is the inward unit normal to $\partial U$. (Note that $f$ actually parametrizes the free boundary.) In [2] it is shown that when a part of $\partial U$ is a line segment, or a segment of a circle, then the number of plastic components attached to it are finite. This is also repeated in the book [1], where also the question of finiteness of the number of all plastic components is raised.
In [3,4], it is shown that the number of plastic components attached to a line segment of $\partial U$ are finite, when the gradient constraint is replaced by $|\nabla u|_p \le 1$, where $|\;|_p $ is the p-norm. These methods can be used to prove a similar result (at least when $F,g$ are analytic) when $u$ is the minimizer of the functional $$\int_U F(\nabla u) +g(u)\,dx \tag{$\ast \ast $ }$$ over the set $\{u\in H^1_0(U) \mid \gamma (\nabla u)\le 1 \}$. Here $F,g$ are convex functions, and $\gamma $ is the gauge function of a compact convex set whose interior contains the origin. (In [5] it is shown that in this case the minimizer $u$ is $C^{1,1}$, in any dimension.)
The above proofs use the fact that part of $\partial U$ is a line segment. They also heavily use the analyticity of the solution, and the free boundary. Now an important question is what can we say about the number of plastic components of the minimizer of $ (\ast \ast )$. Are they finite in general, in any dimension? Or at least when $F,g,\gamma ,\partial U$ are analytic?
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