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Title: Maximum Principle for 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 5 | Maximum Principle for Harmonic Functions
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Subtitle on GitHub: pde05_sub_eng.srt missing
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $\Omega \subseteq \mathbb{R}^n$ be connected. What is correct for a non-empty subset $A \subseteq \Omega$?
A1: If $A$ is closed, then $A = \Omega$.
A2: If $A$ is closed and open in $\Omega$, then $A = \Omega$.
A3: If $A$ is open in $\Omega$, then $A = \Omega$.
A4: If $A$ is compact and closed in $\Omega$, then $A = \Omega$.
Q2: Let $\Omega \subseteq \mathbb{R}^n$ be open and connected. What is correct?
A1: $\Omega$ is also closed.
A2: $\Omega$ is also path-connected.
A3: $\Omega$ is also compact.
A4: $\Omega$ is also empty.
Q3: Let $u: \mathbb{R}^n \rightarrow \mathbb{R}$ be harmonic and non-constant. What is not correct?
A1: There is an open set $\Omega$ and a point $x_0 \in \Omega$ such that $u(x_0) = \max_{x \in \overline{\Omega} } u(x)$.
A2: $\Delta u = 0$
A3: $u$ is continuous.
A4: $\max_{x \in K} u(x) = \max_{x \in \partial K} u(x)$ for every compact set $K \subseteq \mathbb{R}^n$.
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Date of video: 2026-01-14
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Last update: 2026-01
