# Questions tagged [maximum-principle]

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

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### What is the domain in the unit disk such that a holomorphic function f remains biholomorphic? [duplicate]

Let $\mathbb{D}$ denote the unit disk in the complex plane $\mathbb{C}$. Suppose that $f: \mathbb{D}\rightarrow\mathbb{D}$ is holomorphic and $f(0)=0$, $f'(0)=a>0$. How can I prove that $f$ is ...
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### Minimum value on the boundary of U help

Question: Let $U \subset\mathbb{R}^2$ be an open and bounded set and $u\in C^2(U)\cap C(U)$ satisfy $\Delta u(x)=\frac{-1}{1+|x|}$ in $U$. Show that $u$ attains it's minimum on the boundary of $U$. ...
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### How to prove that $|u(x)|\leq 1$ for all $x \in \Omega$, given u solves $-\vec \nabla^2 u = (1-u^2)\cdot u$ and $u=0$ on $\partial \Omega$.

Let $\Omega$ be a bounded domain and let $u$ be a solution to the PDE $-\vec \nabla^2 u = (1-u^2)\cdot u$, which satisfies $u=0$ on $\partial \Omega$. Show that $|u(x)|\leq 1$ for all $x \in \Omega$. ...
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### The modulus of a holomorphic function $f$ on a domain $D$ split into two subdomains $D_1$ and $D_2$.

Let $D$ be a simply connected domain on the complex plain $\mathbb{C}$ with analytic boundary. Let $\Gamma$ be a simple curve that connects two discrete points $a$ and $b$ on the boundary of $D$, so ...
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### Three-mass rotational system with Lagrange multipliers and PMP

I have been dealing with this problem for a while and ran into the following difficulty, which confused me a little and I need help to clarify. Using the Lagrangian ...
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Consider a smooth solution that satisfies the elliptic equation $$-\Delta u=(u-1)(u+1)\ \text{ in }\ \{|x|<1\},\ \ u=f(x)\ \text{ on }\ |x|=1,$$ where $f$ is a continuous function such that $-1<... • 5,814 3 votes 1 answer 157 views ### An exercise from Stein's complex analysis - Phragmen-Lindelof principle I am considering exercise 9 from chapter 4 of Stein and Sharkarchi's complex analysis: (a). Let$F$be a holomorphic function in the right half-plane that extends continuously to the boundary, that is,... • 873 1 vote 0 answers 40 views ### Uniqueness and maximum principle The following is from O. Zeitouni's "Lecture Notes on Random Walks in Random Environments": Suppose I have a function$V_{a,b,c}:\mathbb{Z}\longrightarrow\mathbb{R}$(for$a,b\in\mathbb{Z}$... • 1,422 1 vote 0 answers 68 views ### Failure of the maximum principle What are easy examples of some operators that don't satisfy the maximum principle? For the double derivative operator$u\mapsto u''$with Dirichlet boundary conditions on$L^2(0,1)$, we have$-u''\...
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I have a purely theoretical doubt, because the references I read didn't make it so clear to me. When we are analyzing the maximum principle, two ways are suggested: $\bullet$ (Weak maximum principle): ...
Suppose u is a non constant harmonic function in $\mathbb{R}^n$, how can I show that the functions $F(r) = \sup_{z \in B(0,r)}|u(z)|$ and $G(r)= \int_{z \in B(0,1)}|u|^2(rz)dS_z$ are both strictly ...