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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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finding maximum value of BVP

Find the maximum value of $u$ in the following BVP \begin{align*} u_t &= u_{xx}, \; \; t,x \in (0,1) \\ u(0,t) &= 2t^2-t \\ u(1,t) &= \sin \pi t \\ u(x,0) &= x(...
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91 views

Converse of the one-dimensional maximum principle

If a positive function $u:[0,1]\to \mathbb{R}^+$ is known to be continuous and twice differentiable, and to have no local maxima in $(0,1)$ (i.e. $u$ is maximised on the boundary $\{0,1\}$), is it ...
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Equivalence of the Weak Parabolic Maximum Principle

I'm study by myself parabolic PDEs by Avner Friedman's book. Initially, Friedman starts the first section of chapter $2$ with the following: Consider the operator $$(1.1) \ Lu \equiv \sum_{i,j=...
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Chapter $2$ - Section $1$ - Lemma $4$ in Friedman's book

I'm studying by myself Parabolic PDEs by Friedman's book and I have a doubt concerning the proof of the lemma $4$ on section $1$ of chapter $2$: $\textbf{Lemma 4.}$ Let $R$ be a rectangle $$ ...
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Maximum principle for discretized ODE

I discretized the following ODE using central finite differences for 1st and 2nd derivatives: $$u''-bu'=f(u), x\in (0,1)\\u(0)=1, u'(1)=0\\ b>0, f:\mathbb{R_{\ge 0}}\to \mathbb{R}_{\ge 0}$$ The ...
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1answer
61 views

Maximum principle at the border

Let $Ω⊂⊂R^{2}$ a bounded regular domain and $u: \overline{Ω} → R$ a function such that $det(∇^{2}u)≤0$ in $Ω$. I'm trying to proof that you have reached your maximum and minimum in $∂Ω$. In my opinion,...
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Maximum principle for first-order quasilinear pde

Let's suppose that $u\in C^{1}(K[0,1])$ solves $$a(x,y)u_{x}+b(x,y)u_{y}=-u$$ in $K[0,1]$. I need to show that if $$a(x,y)x+b(x,y)y>0$$ for all $(x,y)\in\partial K[0,1]$ then $u\equiv0$ in $K[0,1]$....
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45 views

Uniqueness of Non-Linear Heat Equation

The following problem comes from an old exam: Consider \begin{cases} u_t - \Delta u + |u_{x_1}| = 0 \text{ in } \mathbb{R}^{n} \times (0,\infty) \\ u(x,0)=g(x) \text{ in } \mathbb{R}^n \end{cases} ...
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Fastest structured way to get max(abc) if a+b+c=30

What is the fastest and structured way to get maximum of abc if a+b+c=n, say n=30? a,b,c are non-negative and can be non-integer.
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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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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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57 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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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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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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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
76 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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102 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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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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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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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
31 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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31 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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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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127 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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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 ...
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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
38 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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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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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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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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34 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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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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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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65 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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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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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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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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107 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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106 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
41 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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79 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
68 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
153 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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45 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 $...