Plastic components attached to the fixed boundary

Problem's Description

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?

1. BookIs an originVariational principles and free-boundary problems

   • Avner Friedman

2. ArticleIs an originThe Free Boundary for Elastic-Plastic Torsion Problems

   • L.A. Caffarelli,
   • Avner Friedman

   Transactions of the American Mathematical Society 252, 65-97, 1979

3. Article On the shape of the free boundary of variational inequalities with gradient constraints

   • Mohammad Safdari

   Interfaces and Free Boundaries (IFB) 19 (2), 183-200, 2017arXiv

4. Article The free boundary of variational inequalities with gradient constraints

   • Mohammad Safdari

   Nonlinear Analysis: Theory, Methods & Applications 123-124, 1-22, 2015arXiv

5. Article Global optimal regularity for variational problems with nonsmooth non-strictly convex gradient constraints

   • Mohammad Safdari

   arXiv