Let $d\in\mathbb{N}$ and let $\Omega$ be a non-empty bounded arcwise connected open subset of $\mathbb{R}^d$ with regular boundary. Denote by $C^1(\bar\Omega)$ the set of differentiable continuous functions on $\Omega$ with uniformly continuous gradient so that the normal exterior derivative $\frac{\partial u}{\partial n}$ of $u\in C^1(\bar\Omega)$ is well defined. Denote by $\mathcal{M}(\Omega)$ the set of real signed measures on $\Omega$ and by $\mathcal{M}_0(\Omega)=\{\mu\in \mathcal{M}(\Omega) : \mu(\Omega)=0\}$. If $\mu\in \mathcal{M}(\Omega)$ denote by $\|\mu\|$ the variation norm of $\mu$. If $u\in C^1(\bar\Omega)$, denote by $\Delta u$ the distributional laplacian of $u$. Suppose that $z_1,\ldots,z_m\in\Omega$ are distinct points of $\Omega$ and $a_1,\ldots,a_m\in\mathbb{R}$. Define $$V:=\left\{u\in C^1(\bar\Omega) : \left(\Delta u\in \mathcal{M}_0(\Omega)\right)\land \left(\frac{\partial u}{\partial n}\Big|_{\partial\Omega}\!\!\!=0\right) \land (u(z_1)=a_1)\land\ldots\land(u(z_m)=a_m) \right\}.$$
I'm wondering if under these conditions there exists $\bar u\in V$ such that: $$\|\Delta \bar u\| = \min _{u\in V} \|\Delta u\|.$$
Now, I'm going to describe the way by which I came to this problem.
- Suppose that we are in a room (the open bounded arcwise connected set $\Omega$), with prefectly isolating walls (the Neumann condition $\frac{\partial u}{\partial n}\big|_{\partial\Omega}=0$).
- Suppose we want to study the temperature inside the room (the function $u$) and there are some heat sources and heat wells in the room modeled by a function $v:\Omega\to\mathbb{R}$.
Then, up to structural constants that can multiply each terms, the dynamic of the temperature is described (with a somewhat abuse of notation due to the fact that I'm dealing with measures) by the equation: $$\partial_t u (z,t) =-\Delta_zu (z,t)+ v(z).$$ Now, if there is a positive (or negative) amount of energy transferred in the room by $v$, i.e. if $\int_\Omega v>0\ (<0)$, then the room is going to burn (freeze) to a $+\infty \ (-\infty)$ temperature. But suppose that $\int_\Omega v=0$. Then, as $t\to +\infty$, the intuition suggests that we are going to find a dynamic equilibrium, say $u_\infty$, satisfying: $$\Delta u_\infty (z) = v (z).$$
Now, suppose that we actually don't know $v$, but we know that we have reached the dynamic equilibrium $u_\infty$ and we have measured the temperature in the points $z_1,...,z_m\in\Omega$ recording the values $a_1,...,a_m \in \mathbb{R}$. So, we want to guess the temperature in the other points in the room and clearly this is an ill-posed question. However, suppose that we want to find the "simplest" $v$ that could generate this situation, the one with less heating sources and wells, i.e. the one with less variation. So, we finally come to the minimizing problem I was asking about.
Final questions:
- If the previous question has a positive answer, denoting by $u_m$ a solution corrisponding to the points $z_1,...,z_m$ and values $a_1,...,a_m$, is it true that $u_m\to u_\infty, m\to \infty$ in some way? Maybe should we add some condition on the sequence $(z_m)_{m\in\mathbb{N}}$?
- Is it know anything about this kind of problems? Maybe choosing specific domains, or changing boundary conditions (i.e. Dirichlet, Robin), or maybe working in the whole $\mathbb{R}^d$ or in others unbounded domain...
- Can anyone give me references about problem like this? Books or articles are welcome.
