Green's function-based framework for pointwise enclosures: Extending sub- and super-solutions of Poisson's equation

Kazuaki Tanaka (Waseda University)

Ryoga Iwanami (Waseda University) Kaname Matsue (Kyushu University) Hiroyuki Ochiai (Kyushu University)

About me

TANAKA, Kazuaki

Ph.D., Associate Professor Global Center for Science and Engineering, Waseda University, Tokyo, Japan

Born in Nagasaki Prefecture, raised in Yamaguchi Prefecture

• Interests: Delicious food and drinks

  🇫🇷 Great respect for French culinary culture!!

  

  

  

  

Enclosing solutions of differential equations

$$ \Huge\left\|u-\hat{u}\right\|\leq r $$

             ***mid-rad type*** error estimate

            **Recent our interest (Sub- and Super-solutions are useful)**

Application

• ODE: Initial Value Problems (1st order) · Boundary Value Problems (2nd order, 4th order) ← We haven't done 4th order yet, but theoretically it's OK
• Partial Differential Equations: Parabolic Type · Elliptic Type
• Others

Sub- and super- Solution of Elliptic Differential Equation

$$ \begin{align} \left\{\begin{array}{l l} -\Delta u-g(u)=f &\rm{in~}\Omega\\ u=0 &\mathrm{on} ~\partial\Omega\\ \end{array}\right. \end{align} $$

<aside> 💾 Definition (existing definition)

We call $\overline{u}\in H^{1}\left(\Omega\right)$ a super-solution of (1) if

1. $(\nabla\overline{u},\nabla v)+\langle -g(\overline{u}),v \rangle\geq \langle f,v \rangle~~~~~\forall~v\in H_{0}^{1}\left(\Omega\right)\cap L_{+}^{2}\left(\Omega\right)$
2. $\overline{u}\geq 0$ on $\partial\Omega$ </aside>

Sub-solution: Reversed inequalities

Note:

If $\overline{u}$ is $H^2$ regular, then checking the following two conditions is sufficient:

1. $-\Delta u - g(u) \geq f$
2. $\overline{u}\geq 0$ on $\partial\Omega$

<aside> 🖊️ Comparison Theorem

Let $\underline{u}, \overline{u}\in H^{1}\left(\Omega\right)$ be sub- and super-solutions of (1) if $\underline{u}\leq \overline{u}$ then, there exists a solution $u$ of (1) satisfing

$$ \underline{u}\leq u \leq \overline{u} \text{\rm on}\Omega $$

</aside>