# 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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### If $f:D\to D$ is analytic, bijective and $f(0)=0$, then $f(z)=cz$ for some $|c|=1$?

Let $D=\{z\in\mathbb{C}:|z|<1\}$ and $f:D\to D$ an analytic, bijective function with $f(0)=0$. Is it true that $f(z)=cz$ for some $c\in\mathbb{C}$ with $|c|=1$? I think it can be proved using ...
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### Convergence in comparison principles of parabolic pdes

I assume that the following equations all have sufficiently smooth strong solutions. I have demonstrated that the solution to the (imaginary-time) Schrödinger equation: \begin{equation*} \begin{...
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### Theorem 2.12 in Lieberman's Second Order Parabolic Differential Equations

In Theorem 2.12 of Liberman's textbook the following result is established. Consider a space-time domain $\Omega\subseteq \mathbb{R}^n \times \mathbb{R}$ ($\Omega$ is not necessarily of cylinder shape ...
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### Maximum principle for reaction diffusion equation?

Consider the heat equation $$u_t = u_{xx} \;\; \text{ for } \;\; x\in \Omega, t \in [0,+\infty) \,.$$ The strong maximum principle states that if the solution $u$ attains its maximum in the interior ...
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### $f:\mathbb{C} \to \mathbb{C}$ analytic with $f(z)=f(z+a)=f(z+b)$

Let $a,b \in \mathbb{C}$ linearly independent over $\mathbb{R}$ and $f: \mathbb{C} \to \mathbb{C}$ analytic with $f(z)=f(z+a)=f(z+b)$ for all $z \in \mathbb{C}$. Show that f is constant. WLOG, I'll ...
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### Non-constant holomorphic function defined on open disk $D(0,1)$

Let $f:D(0,1) \to \mathbb{C}$ be a non-constant holomorphic function. For each $r \in [0,1[$, define: $$h(r)=\max\{|f(z)| : |z| = r\}$$ Show that h is $\textbf{strictly}$ increasing. By Maximum ...
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### Prove $f$ attains its maximum on boundary

Let $f$ be holomorphic, bounded on $|z|>1$ and continuous on $|z|\geq 1$. Prove that $|f|$ attains maximum on $|z|=1$, i.e $$\sup\limits_{|z|\geq 1}|f(z)|=\max\limits_{|z|=1}|f(z)|.$$ My attempt: ...
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### Application of weak maximum principle.

Fix any open, bounded set $U \subset \mathbb{R}^n$ and suppose that $u \in C^2(U) \cap C(\bar{U})$ is a solution of $$-\Delta u = f \,\,\,\text{in}\,U$$ $$u=g \,\,\,\,\,\,\text{on}\,\,\,\partial U,$$ ...
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### complex analysis - f(z)^d+g(z)^d=1 for all complex number z, where d>=2 [duplicate]

I would like to show that "if holomorphic function $f,g$ satisfies $f(z)^d+g(z)^d=1$ for all $z\in\mathbb{C}$ (where $d\in\mathbb{Z}_{\geq3}$), then both $f$ and $g$ is constant." I know ...
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### Applying maximum principle for the heat equation

I'm stuck with the following problem Let $U\subseteq \mathbb{R}$ be a bounded domain, $T>0$ and $u\in C^2(U_T) \cap C(\overline{U_T})$, where $U_T = U \times (0,T]$. If $f=f(x)$ is a continuous ...
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### Maximum principle or comparison principle

the equation $\varphi_t + D_p H(x, Du ^\varepsilon)\cdot D \varphi = \varepsilon\Delta \varphi$ is a linear parabolic equation. Thus, by the comparison principle for parabolic equations, we have for ...
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We assume that $max_{\mathbb{R}^n \times[0,T]} f(x,t) = f(x_0,t_0)\quad\text{for some }(x_0,t_0)\in\mathbb{R}^n \times[0,T]$ Then, my book says that If $t_0>0$, then, by the usual maximum principle,...