📂Partial Differential Equations

Mollification

Definition ¹

$f \in {L^1_{\mathrm{Loc}}( \Omega)}$ and $\epsilon>0$ with respect to $f$, the $\epsilon$-mollification is defined as follows.

$$ f^{\epsilon}(x) := \eta_{\epsilon} * f (x) =\int_{\mathbb{R}^{n}} \eta_{\epsilon}(x-y)f(y)dy, \quad x\in \Omega_{>\epsilon} $$

• Here, $f$ is a function defined as $0$ outside of $\Omega$.
• $\eta_\epsilon$ is a mollifier.
• $\ast$ is a convolution.
• $\Omega_{>\epsilon} := \left\{ x \in \Omega : \mathrm{dist}(x, \partial \Omega) > \epsilon \right\}$

Properties

• (i) $f^{\epsilon} \in C^\infty( \Omega_{>\epsilon})$
• (ii) Almost everywhere, $f^{\epsilon} \to f \text{ as } \epsilon \to 0$

Proof

Let a fixed point $x \in \Omega_{>\epsilon}$ be given. And since $\Omega_{>\epsilon}$ is an open set, there exists a very small $h>0$ that satisfies $x+he_{i} \in \Omega_{>\epsilon}$. Then, for some open set $V \Subset \Omega$, the following holds.

$$ \begin{align*} \dfrac{ f^{\epsilon} (x+he_{i})- f^{\epsilon}(x) }{ h} &= \int_\Omega \dfrac{\eta_\epsilon (x+he_{i}-y) - \eta_\epsilon (x-y) }{h} f(y) dy \\ &= \int_\Omega \dfrac{1}{\epsilon^n}\dfrac{1}{h}\left[ \eta \left( \frac{x+he_{i}-y}{\epsilon} \right) - \eta \left( \frac{x-y}{h} \right) \right] f(y) dy \\ &= \dfrac{1}{\epsilon^n}\int_{V} \dfrac{1}{h}\left[ \eta \left( \frac{x+he_{i}-y}{\epsilon} \right) - \eta \left( \frac{x-y}{h} \right) \right] f(y) dy \end{align*} $$

And for $y \in V$, the following holds.

$$ \dfrac{1}{h}\left[ \eta \left( \dfrac{x+he_{i}-y}{\epsilon} \right) - \eta \left( \dfrac{x-y}{h} \right) \right] \to \dfrac{1}{\epsilon}\eta _{x_{i}}\left( \frac{x-y}{\epsilon} \right) \text{ uniformly as } h \to 0 $$

Therefore, $f^{\epsilon}_{x_{i}}(x)$ exists and its value is as follows.

$$ \begin{align*} f^{\epsilon}_{x_{i}}(x) &= \dfrac{1}{\epsilon^n}\int _{V} \eta_{x_{i}}\left( \frac{x-y}{\epsilon} \right) f(y)dy \\ &= \int_{\Omega} (\eta_\epsilon)_{x_{i}}(x-y)f(y) dy \end{align*} $$

Similarly, for each multi-index $\alpha$, $D^\alpha f^{\epsilon}(x)$ exists and its value is as follows.

$$ D^\alpha f^{\epsilon}(x) = \int_\Omega D^\alpha \eta_\epsilon (x-y)f(y)dy, \quad x \in \Omega_{>\epsilon} $$

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