-
Title: Standard Mollifier
-
Series: Partial Differential Equations
-
Chapter: Laplace’s Equation
-
YouTube-Title: Partial Differential Equations 8 | Standard Mollifier
-
Bright video: Watch on YouTube
-
Dark video: Watch on YouTube
-
Ad-free video: Watch Vimeo video
-
Forum: Ask a question in Mattermost
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: pde08_sub_eng.srt missing
-
Download bright video: Link on Vimeo
-
Download dark video: Link on Vimeo
-
Related videos:
-
Timestamps (n/a)
-
Subtitle in English (n/a)
-
Quiz Content
Q1: What is not a property of $\eta$?
A1: $\eta(0) = 1$.
A2: $\int_{\mathbb{R}^n} \eta(x) d^n x = 1$.
A3: $\eta$ is $C^\infty$.
A4: $\eta$ has compact support.
A5: $\eta(x) \geq 0$ for all $x \in \mathbb{R}^n$.
Q2: Let’s define $\eta_{\varepsilon}(x) = \frac{1}{\varepsilon^n} \eta(x/\varepsilon)$ where $\eta(y) = c \cdot \exp\big( \frac{1}{\| y \|^2 - 1} \big)$ for $y$ inside the unit ball. What is correct for any continuous function $u : \mathbb{R}^n \rightarrow \mathbb{R}$?
A1: $\eta_{\varepsilon} \ast u \in C^{\infty}(\mathbb{R}^n)$
A2: $\eta_{\varepsilon} \ast u \in C_c(\mathbb{R}^n)$
A3: $(\eta_{\varepsilon} \ast u) (x) \geq 0$ for all $x \in \mathbb{R}^n$.
A4: $\eta_{\varepsilon} \ast u$ is not defined everywhere on $\mathbb{R}^n$.
-
Date of video: 2026-03-02
-
Last update: 2026-03
