2. Free Boundary Problems
-
Regularity of free boundary/obstacle problem
Problem 2.1.
Consider the following Cauchy problem, $$\begin{cases} \max\{\partial_tu + v\cdot \nabla_xu - \mathscr{L}_vu, u - \varphi\} = 0\\ u(0) = u_0. \end{cases}$$ Determine the regularity of solutions for $\mathscr{L} = \nabla_v^s$ (This is known in $d = 1$)
Determine the regularity of solutions of the free boundary: $\{u = \varphi\}$ -
Alt-Caffarelli type problem
Problem 2.2.
Find a function that satisfies the Kolmogorov equation $$\begin{cases} (\partial_t + v\cdot \nabla_x - \Delta_v)f = 0, & f > 0\\ |\nabla_vf| = 1, & \partial\{f > 0\}. \end{cases}$$
Derive and study the reasonable Alt-Caffarelli type problem. -
Hele-Shaw: convergence from a particle system
Consider the following, $$\begin{cases} -\Delta p + \Delta \Phi = 0, & \{p > 0\}\\ V = \eta\cdot (\nabla p - \nabla \Phi),& \partial\{p > 0\} \end{cases}$$ where $\eta$ is the normal, $p$ is the pressure, and $V$ is velocity. This problem can be restated as follows: Let $\rho(t) = \mathbf{1}_{\{p > 0\}}$. Then the continuity equation is given by $$\partial_t \rho + \nabla_x\cdot(\rho(\nabla \Phi - \nabla \rho)) = 0.$$ When $\rho \in \{0,1\}$, this is Hele-Shaw flow.
In the particle setting, consider people confined to a room with one exit such that everyone in the room is moving to the exit via a potential given by $\Phi(x) = d(x,exit)$. Let $x(t) = (x_1(t),\dots,x_N(t))$ be the positions of the individuals with $\dot{x}_i(t) = P_{C_{\rho_N}}(-\nabla \Phi)$ and $\rho_N(x,t) = \sum\limits_{i=1}^N \delta_{x_i(t)}$. The additional constraint, $|x_i - x_j|\geq 2r$ for $i \neq j$ corresponds to the “Hard Congestion Model".Problem 2.3.
[Inwon Kim] Find a particle system that converges to $\rho$. That is, $$\frac{1}{N} \sum\limits_{i=1}^N \delta_{X_i(t)} \to \mathbf{1}_{\{p > 0\}}.$$
Cite this as: AimPL: Integro-differential equations in many-particle interacting systems, available at http://aimpl.org/manyparticle.