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Title: Regularity of Harmonic Functions
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Series: Partial Differential Equations
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Chapter: Laplace’s Equation
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YouTube-Title: Partial Differential Equations 9 | Regularity of Harmonic Functions
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Subtitle on GitHub: pde09_sub_eng.srt missing
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Quiz Content
Q1: Let $\Omega \subseteq \mathbb{R}^n$ be open. What is not possible for a $C^2$-function $u: \Omega \rightarrow \mathbb{R}$?
A1: $u$ is harmonic and $u \notin C^3(\Omega)$.
A2: $u$ is harmonic and $u \in C^\infty(\Omega)$.
A3: $u$ is not harmonic and $u \in C^\infty(\Omega)$.
A4: $u$ is not harmonic and $u \notin C^3(\Omega)$.
Q2: Let $\Omega \subseteq \mathbb{R}^n$ be open. For a given $\varepsilon >0$, define $\Omega_{\varepsilon}$ as all the points in $\Omega$ that have distance greater than $\varepsilon$ from the boundary. We can use the standard $\eta_{\varepsilon}$ to smoothen a continuous function $u : \Omega \rightarrow \mathbb{R}$. What is correct?
A1: If $u$ satisfies the mean-value property, then $u(x) = (\eta_{\varepsilon} \ast u)(x)$ for all $x \in \Omega_{\varepsilon}$.
A2: If $u$ is harmonic, then $u(x) = \eta_{\varepsilon}(x)$ for all $x \in \Omega_{\varepsilon}$.
A3: If $u \in C^{\infty}(\Omega)$, then $u(x) = (\eta_{\varepsilon} \ast u)(x)$ for all $x \in \Omega_{\varepsilon}$.
A4: If $u$ is not harmonic, then $u(x) = (\eta_{\varepsilon} \ast u)(x)$ is not defined for any $x \in \Omega_{\varepsilon}$.
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Date of video: 2026-03-03
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Last update: 2026-03
