# Tagged Questions

For questions about the use of maximum principle, either in the context of partial differential equations or complex analysis.

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### Maximum principle question

Does maximum principle exist for an elliptic equation(laplace or poisson) if the domain doesn't have a boundary, for example if you take $S^1$ (the 1-torus) in 1d?
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### “U satisfies a maximum principle” - what might be meant with this statement?

Suppose on a 2d lattice we have discretized some PDE and get something like $$U_{ij}^{n+1}=\mu(u_{i-1j}^n +u_{i j-1}^n+u_{i+1j}^n+u_{ij+1}^n)+(1-4\mu)u_{ij}^n$$ What then might be meant with a ...
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### Showing Complex Function is Constant

I am preparing for qualifying exams, and this is a question from the Penn State Qualifying Exam for Fall 2015. It is stated as follows Let $\epsilon > 0$ and let $f$ be holomorphic (analytic) on ...
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### Maximum modulus theorem proof

I do not understand the proof for the maximum modulus theorem done with the open mapping theorem. Unfortunately my notes are a little bit cryptic. What I understand: Let $z_0$ be the maximum that is ...
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### Maximum Principle variation.

Let $f$ be a non-constant holomorphic function in a bounded open connected set $\Omega$ in $\mathbb{C}$. Let $M:=\lim \sup_{n \to \infty} |f(z_n)|$ for every sequence ${z_n}$ in $\Omega$ which ...
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### Uniform convergence of n-fold composition using Schwarz lemma

Let $f$ be an analytic function mapping the unit disk $\mathbb D$ to itself with $f(0) = 0$ and $|f'(0)| < 1$. Let $f^{n} = f \circ f \circ \dots \circ f$ be the function obtained by composing $f$ ...
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### Bound on Rayleigh quotient on $H^{1}_{\text{per}}[0,1]^{n}$

Let $Q=[0,1]^{n}$. Define a $Q$-periodic function as follows: A function $f:\mathbb{R}^{n}\to\mathbb{C}$ is said to be $Q$-periodic if for any $x\in\mathbb{R}^{n}$ and any $z\in\mathbb{Z}^{n}$, it ...
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### Prove an inequality using complex analysis

If $f:\mathbb{D}\rightarrow\mathbb{D}$ is holomorphic then prove that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq|f(z)| \leq\frac{|f(0)| + |z|}{1 - |f(0)||z|}$$ I have been wracking my brain for ...
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### $L^\infty$ bound on solution of $u_t - \Delta u + cu = f$ and dependence on $c > 0$

Let $u$ be the weak solution to $$u_t - \Delta u + cu = f$$ $$u(0) = 0$$ with zero Neumann data, on a bounded domain and time interval $[0,T]$. Here $c > 0$. If $f$ is in $L^\infty$ in time and ...
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### Showing $f(z)$ is constant when $Im(f^2(z)-f(z))=0$

I don't have the full question but I'm assuming $f(z)$ must be entire for this to occur. Also note that $u>1$ where $u$ is the real part of $f(z)$. If we solve the given eqn then we are left with ...
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### An entire function such that $if(z)= \bar f(z)$ for all $z$ is constant

Let $f$ be an entire function such that $if(z)= \bar f(z)$ for all $z\in\Bbb C$. Show that $f$ is the constant function. I didn't understand it but I took some notes which seem like gibberish to me ...
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### Prove $v$ is harmonic and $\lim_{r \uparrow 1} v(re^{i\theta}) = 0$

Prove that if $v(z) = \mathrm{Im}[(\frac{1+z}{1-z})^2]$, then $v$ is harmonic on the unit disc and $\lim_{r \uparrow 1} v(re^{i\theta}) = 0$ for all $\theta \in [0,2\pi)$. Explain why this does not ...
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### Uniqueness for a mixed Dirichlet/Neumann problem

Let $L$ be a uniformly elliptic operator with $c \equiv 0$ on a bounded domain $U \subset \mathbb{R}^n$ with $C^2$-boundary, and let $\partial U = S_1 \cup S_2$. Suppose $u \in C^2 (\bar{U})$ ...
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### Hyperplane problem by Lagrange multiplier method

Solve the problem using Lagrange multiplier method. Find the point that belongs to both hyperplanes xT c = β and xT d = γ which is closest to the origin.
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### Holomorphic function vanishes on the boundary

Let $f \in \mathcal{O}(D) \cap \mathcal{C}(\overline{D})$, $D$ - a bounded region in $\mathbb{C}$. Suppose that $f(z)=0$ for $z \in \partial D$. Prove that $f(z)=0$ for all $z \in D$. At first I ...
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### Maximum minimum of $x^2$ on the open interval $-1<x<1$

The question goes like this: Does the function $f(x)=x^2$ have a maximum on the open interval $(-1,1)$ ? And a minimum? Explain Not exactly sure how to give a proper answer really. For me it makes ...
I'm struggling a bit to arrive at the conclusion that $f(z)$ is a constant. Suppose $f(z)$ is holomorphic on and inside the unit disk, and that it has no zeroes on the interior. Also, assume that \$|...