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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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Prove that $T$ is a bounded operator on a disk algebra and prove the existence of a Borel measure on a boundary of an open unit disk.

Let $A(D)$ be the space of holomorphic functions on the open unit disk $D$ and continuous on the closed disk $\bar{D}$. Then $A(D)$ is a Banach space if we set $\|f\|=\sup\{|f(z)|:z\in\bar{D}\}$. For $...
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21 views

Cauchy estimates and the maximum principle

I am stuck at a step in a problem where I've been given hints. The hints confuse me, so I'm hoping for some help. Assuming that $f$ is entire and that $$|f(z)| \leq |z| + 1/|z|, $$ for $z \in \mathbb{...
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47 views

$f$ analytic and $|f|\leq1$ on a strip.

Let $E$ be the strip $\{z\in\mathbb{C}:0<\Re z <1\}$. Let $f$ be analytic on $E$ and continuous on $\bar{E}$. Show that if $f$ is bounded on $E$ and $|f|\leq1$ on the boundary of $E$, then $|f|\...
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Maximum Modulus of Complex Function

here's a question I'm working on that I'm a bit stuck on. Let: $f(z) = \frac{z^2}{z + 2}$ Find the maximum value of $|f(z)|$ as $z$ varies over the unit disc. Since $f(z)$ is analytic $\forall z$ in ...
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21 views

Holomorphic function with zero imaginary part over the unit circle is constant [duplicate]

I have this problem I don't know how to approach: let $f$ be continuous on the closed unit disk of the complex plane $\bar{D}(0,1)$ and holomorphic on its interior $D(0,1)$ s.t. its imaginary part is ...
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1answer
33 views

Proving a certain holomorphic function is polinomic

Suppose we have an holomorphic function $f$ on the open unit disk $D(0,1)$ s.t.: $$\forall r\in (0,1) \exists n\in \mathbb{N}| \max_{C(0,r)}|f|=r^n$$ prove that $f$ is polinomic. Honestly I don't ...
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204 views

Function holomorphic in the units disk with different bound

Suppose $f$ is continuous in the closed unit disk $\bar{D}(0,1)$ and holomorphic over its interior $D(0,1)$. Moreover suppose that for $|z|=1$ we have: $\Re(z)\leq0\Rightarrow |f(z)|\leq\ 1$ $\Re(z)&...
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1answer
69 views

Holomorphic function on an annulus

Let $f$ be a holomorphic function on the set $U=\{z \in \mathbb{C}: 1 \leq |z| \leq \pi \}$. Assume that $max_{|z|=1}|f(z)| \leq 1$ and $max_{|z|=\pi}|f(z)| \leq \pi^{\pi}$. How to prove that $...
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1answer
65 views

A maximum principle for bounded functions in unbounded domain

Let $U \subsetneq \mathbb{R}^2$ be a domain. Suppose that $u \in C^2(U) \cap C(\bar{U})$ is a bounded harmonic function such that $u \leq 0$ on $\partial U$. If $U$ is bounded, then the maximum ...
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69 views

Parabolic Maximum Principle for weak solutions

Let $B_1(0)\subset\mathbb{R}^d$ be the unit ball of $\mathbb{R}^d$. Let $u: \overline{B_1(0)}\times [0,T]\to \mathbb{R}$ be a function such that: i) $u\in C(\overline{B_1(0)}\times [0,T])$ ii) $u\...
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27 views

Closed curve for an analityc function

If I have an analityc function $f(z)$ on a domain $B \subset \mathbb{C}$ and a simple and closed curve $C$ that encloses $B$, and if $|f|$ is constant over C, the $f$ is constant on $B$? I haven’t ...
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Harmonic functions and Maximum Modulus Principle. (Real harmonic function implies holomorphic function?)

Theorem. (Maximum Modulus Principle) $\;$If $f$ is a non-constant holomorphic function in a region $\Omega$, then $f$ cannot attains a maximum in $\Omega$. Problem. Prove the maximum principle ...
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1answer
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Use maximum modules principle to prove that if $\sup_{|z| = R}|f(z)|\leq AR^{k} + B$ with $f$ entire, then $f$ is a polynomial

Use Cauchy inequality or maximum modules principle to prove that if $f$ is entire function that satisfies $$\sup_{|z| = R}|f(z)|\leq AR^{k} + B$$ for all $R >0$, for some $k \in \mathbb{Z}$ and ...
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Weak Maximum Principle For Parabolic Equations With Neumann Boundary Condition

Consider the problem: \begin{equation} Lu\equiv \sum^n_{i, j=1} a_{ij}(x, t)\frac{\partial^2 u}{\partial x_i\partial x_j}+\sum^n_{i=1} b_i(x, t)\frac{\partial u}{\partial x_i}+c(x, t)-\frac{\partial u}...
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1answer
23 views

Question on Laplace-Beltrami operator and the maximum principle

So I have this problem on my mind: Let $\Gamma$ be a $2d-$compact manifold and suppose that $u$ is a solution of: $-\Delta u=0 \;\forall x\in \Gamma$ ( $\Delta $ here denotes the Laplace-...
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30 views

maximum principle for multi-component heat equation

Consider the $N$-component coupled linear heat equation in an open subset of $\mathbb{R}\times\mathbb{R}^d$: $\forall \alpha \in \{1,...,N\}$ $$\partial_t u_{\alpha} = \sum_{j,k=1}^d\sum_{\beta=1}^N \...
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20 views

Find harmonic function vanishing on the boundary and with a specific bound

Find all harmonic functions $u$ in a half-plane $H$ so that $u=0$ on $\partial H$ and $\vert u(x)\vert \le \vert x\vert$ in $H$. This domain is not bounded. If it's bounded, then after using the ...
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1answer
111 views

Bounds for Poisson solution with Dirichlet B.C using Maximum Principle

Let $U$ be a bounded, open set in $\mathbb{R}^n$. For any $u \in \mathcal{C}^2(\overline{U})$ show that $0\leq u \leq 1$ if it satisfies the Poisson problem with Dirichlet B.C. as \begin{equation} \...
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2answers
60 views

Lindelöf maximum principle for sets of measure zero

I have searched all around for an answer to this, what I would imagine to be a rather natural question, but to no avail. How does one extend the Lindelöf maximum principle to sets of measure 0? Here's ...
2
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1answer
63 views

Application of the Maximum Principle for the squared norm of the second fundamental form

Firstly, I would like to say that I didn't do a PDE's course and I'm trying study by myself Mean Curvature Flow and consult references for PDEs theory when needed, so I would appreciate if someone ...
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1answer
32 views

Maximum Principle For Gradient Of Poisson Equation

Ran across this problem for a poisson equation. Looks like some sort ofaximum principle for the gradient. Can't quite wrap my head around it. Let $\Omega$ be an open subset of $\mathbb{R}^n$. Suppose ...
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If integral of square of modulus of $f$ over $\mathbb C$ is finite, then $f \equiv 0$ [duplicate]

Assume $f$ is entire and $\int \int |f(x+iy)|^2 dx dy < \infty$ , then show that $f \equiv 0$. My approach was to notice that $|f|^2$ is a subharmonic function and then we can use the mean value ...
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1answer
56 views

Prove inequality $|u| \leq \max_{\partial{\Omega}}{|f|} + \frac{1}{4} \max_{\Omega}{|F|} \ R^2$

Consider the problem $$\cases{ -\Delta{u}=F, \ in \ \Omega \\ u=f \ \ ,on \ \partial{\Omega} } $$ $$$$ The maximum principle says that if $u$ is a solution of $-\Delta{u}=F$ with $F\leq0$,...
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53 views

Discrete Maximum principle for variable coefficient Poisson equation

Consider the standard 5 point different approximation (centered difference for both the gradient and divergence operators) for the variable coefficient Poisson equation $$-\nabla \cdot(a\nabla v)=f$$ ...
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2answers
53 views

Lagrange Multiplier, maximum value to show a equation

I was doing a exercise and I don't understand a part of it. Calculate the maximum value of $f (x, y, z) = \ln x + \ln y + 3 \ln z$ on the portion of the sphere $x^2+y^2+z^2=5r^2$ in the first octant. ...
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32 views

Local Maximun Principle

Let $\Omega \subset \mathbf{R^2}$ be open and $a,b,c \in \mathbf{C(\Omega)}$ with $c<0$ on $\Omega$.Show that if $v\in \mathbf{C(\Omega)}$, and $\Delta u + a(x,y)u_x + b(x,y)u_y+c(x,y)=0$ then $u$ ...
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50 views

Maximum principle application

Assume $f(z)=\dfrac{z^2}{z+2}.$ I want to find: $$\begin{equation} \max_{|z|\leq1}\bigg|\frac{z^2}{z+2}\bigg| \end{equation}$$ Taking the maximum principle into consideration, it must be true that: $$...
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32 views

Find the maximum of $g(ξ)=(e^{πξi}-1)\sum\limits_{k=0}^na_ke^{2πξik}$

Let $\xi\in\mathbb R$, $n\in\mathbb N$, $\{a_k\}_{k=0}^n\in\mathbb R$. I wonder if it is possible to calculate or estimate the maximum of $|g(\xi)|$, where $$g(\xi)=(e^{\pi \xi i}-1)\sum_{k=0}^n a_k e^...
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35 views

optimal control with a constant choice

I am trying to solve an optimal control problem. On top of state and control variables, the objective function depends on a constant scalar that I can choose. I have thought about defining the scalar ...
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Solving Pontryagin's maximum principle for a multi-dimensional Hamiltonian function

System: $$ f(x,u)=\dot{x_i}=u_i, $$ where dim(x)=dim(u)=8X2 Input Constraint: $$ ||u_i(t)||\leq1 $$ Cost: $$ J=\sum_{n=1}^{8} \int_{0}^{t_f} ||x_i(t)-x_{i-1}(t)||^2 dt +\sum_{n=1}^{7}\int_{0}^{...
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2answers
63 views

Proving $\rvert 1+\cos(z)+\cos(2z)+ \dotsb +\cos(nz)\lvert>n+1$

Let $n\in \Bbb N$. Show that for each $r>0$ there exists $z\in D(0,r)$ such that $\rvert 1+\cos(z)+\cos(2z)+ \dotsb +\cos(nz)\lvert>n+1$. I think it could work with the Maximum Principle ...
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1answer
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Give an example of a function $u$ and a space $\Omega$ which fails the weak maximum principles

Let $\Omega \in \Re^2$ be open and bounded. Let $u \in C^2(\Omega) \cap C(\bar \Omega)$ satisfy $$-\Delta u = f$$ with $f>0$. Find an example of $\Omega \in \Re^2$ , $u \in C^2(\Omega) \cap C(\...
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1answer
42 views

Harmonic majoration proof. Why does domain need to be relatively compact?

I am having trouble understanding why $D$ needs to be relatively compact (compact closure) in the following theorem: Theorem Let $U$ be an open subset of $\mathbb{C}$, and let $u: \, U \to [-\infty,...
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2answers
55 views

Showing that $f(z)$ is a constant function in the open ball $B(0;2)$

Let $f(z)$ be an analytic function in the open ball $B(0;2)$. Show that if $f(z)$ is imaginary $\forall z\in C(0;1)$ (a circle centered at $0$ of radius $1$), then $f(z)$ is a constant function in $B(...
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64 views

A holomorphic function which is zero only on a part of the boundary

We have the following question : If $D$ denotes the unit ball in $\mathbb C$, then let $f : \bar D \to \mathbb C$ such that $f$ is continuous in $\bar D$ and analytic on $D$. If $f(e^{it}) = 0$ for ...
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77 views

Maximum Principle for a Diffusion Equation

Let $u$ be defined on $\overline{U_T} = \overline{(U \times (0,T])}$ where $U$ is a bounded open set. Let $u$ solve $$ u_t + b(x,t) \cdot \nabla u= \Delta u $$ where $b$ is an arbitrary, bounded ...
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1answer
40 views

Maximum Modulus Principle and proving that Re(f) does not achieve a local maximum

Let $f:U\rightarrow\mathbb{C}$ be an analytic and nonconstant function and $U$ be open. I want to prove that $Re(f)$ does not achieve a local maximum on $U$. I started by supposing for contradiction ...
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67 views

Finding maximum value of $|f(z)|$ using Maximum modulus theorem?

In the question asked here→ Maximum Modulus Exercise I want to know, if we just want to find maximum value of $|f(z)|$, why 'Marlu' Sir in his answer (here https://math.stackexchange.com/a/325832/...
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1answer
55 views

Maximum Principle for Elliptic PDE

Let $u$ solve the following elliptic equation (where $c$ is continuous, non-negative): $$ \sum_{i,j = 1, ... ,d}a_{ij}\partial_{ij}u(x) - c(x)u(x) \geq 0 $$ So that the matrix $a_{ij}$ is positive-...
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1answer
133 views

What is the name of this “Hopf's theorem”?

I am reading Agricola & Friedrich's Global Analysis. On page 85 they prove this corollary of the Stokes' Theorem: Let $\mathcal{M}$ be a compact, connected, oriented manifold without boundary and ...
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1answer
34 views

Weak maximum principle - Schrödinger operator

I think I'm missing something very basic here. I'm considering the problem given by \begin{cases} Lu > 0, & \text{in } \Omega, \\ u \ge 0, & \text{on }\partial \Omega, \end{cases} where $...
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1answer
37 views

Optimal bound on a problem similar to Schwarz lemma

The complex analysis problem I came up with is the following: Let $\mathbb{D}=\{z\in \mathbb{C} : |z|<1\}$ be the open unit disk and $f:\mathbb{D} \to \mathbb{D}$ be a holomorphic function ...
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2answers
29 views

Proof of extremum principle, real analysis

In his book "Analysis I", Terence provides the proof of extremum principle, saying that each function with bounded domain must obtain maxima/minima. In previuos theorem it is already shown that each ...
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1answer
49 views

$|f(z)| + \ln|z| \le 0$, is $f = 0$?

Let $D = \{z\in \mathbb{C}:0<|z|<1 \}$ , $f:D\rightarrow \mathbb{C}$ holomorphic. Given $$|f(z)| + \ln|z| \le 0$$ for all $z\in D$, where $\ln$ denotes the real natural logarithm, is it true ...
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1answer
122 views

Why can we apply the strong maximum principle?

I am considering the following section of Peter Grindrods „Pattern and Waves“. We begin by considering a scalar equation $$ u_t=\Delta u+f(u,x,t),\quad x\in\Omega\subseteq\mathbb{R}^n, t>0. $$ ...
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1answer
45 views

Can anyone tell me which book is cited?

in the attached section from Peter Grindrod‘s book Pattern and waves from 1991, the authors refers to a book with the number [15] which contains some strong maximum principle for linear parabolic ...
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1answer
72 views

Maximum principle and a differential inequality

Let $z:[0,T]\to \mathbb{R}$ satisfy the following differential inequality$$ \frac{d}{dt}z(t)\leq f(z(t)) $$ where $f$ is a continuous real-valued function. Now, let $y$ be a solution to the equality ...
3
votes
1answer
43 views

How does this follows from the maximum principle?

Suppose $u$ is a holomorphic function in the unit disk and $\lvert u(z)\rvert\leq 1$. It follows from Cauchy's inequality that $$ \left\lvert u(z) - \sum_0^{2N-1}\frac{u^{(k)}(0)z^k}{k!}\right\rvert \...
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0answers
20 views

Uniqueness of policy function for optimal control problems

Let $t \in T \subset \mathbb R_+$ denote time, $x \in X \subset \mathbb R$ the state and $u \in U \subset \mathbb R$ the control. Instantaneous payoffs are $F : X \times U \to \mathbb R$. The ...
1
vote
1answer
110 views

Proof of Weak Maximum Principle for Parabolic Equations

I'm trying understand the proof of the Weak Maximum Principle for Parabolic Equations in this presentation (the proof starts on the slide $11$) for the operator $$Lu \equiv \sum_{i,j = 1}^n a_{ij}(x,...