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

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0answers
25 views

What is the maximum area of a trapezium with 3 known sides and unknown angles. [on hold]

The Question: A major company in your city has both new equipment capable of making guttering in the shape of an open top trapezium. The sheet metal used is 22 cm wide and bent such that the base s ...
0
votes
1answer
42 views

How to find maximum value?

Let $R$ be the region $z \leq 1$. Compute the maximum value of $|z^2 + z + 2|$ in $R$ and find out the point where this function reaches this value in $R$. I let $$z = \cos \theta + i \sin ...
1
vote
1answer
32 views

Whether such non constant entire function exists

I need to tell whether $\exists f$ non constant,entire, with $f(0)=e^{i\alpha},|f(z)|={1\over2}\forall z\in\partial\mathbb{D}$ False due to Maximum Modulas Principle $\exists f$ non ...
-2
votes
1answer
42 views

Corollary from Maximum Modulus Principle and Schwarz's Lemma

Need to prove this implication derived from the maximum principle, but have no clue how. $$\forall k=0,...,N. \ f^{(k)}(0)=0 \implies\exists M=const . \ \forall z.|z|\lt1:|f(z)|\le M|z|^{N+1}$$
1
vote
1answer
36 views

How can I find the Pythagorean hypotenuse which gives a maximum Pythagorean triangles?

The following Pythagorean hypotenuses have many possibilities of triangles. $125$ has three triangles $$35, 120, 125$$ $$44, 117, 125$$ $$75, 100, 125$$ the $365$ has $4$ triangles , $85$ has $4$ ...
3
votes
2answers
50 views

Prove that there exists $c \in \mathbb{R}$ such that $f'(c)=0$ if $ \lim_{x\to-\infty}f(x) =\lim_{x\to\infty}f(x) = 0 $.?

Suppose that $f$ is differentiable on $ \mathbb{R} $ and $ \lim_{x\to-\infty}f(x) =\lim_{x\to\infty}f(x) = 0 $. Prove that there exists $c \in \mathbb{R}$ such that $f'(c)=0$ . and can I extending ...
0
votes
0answers
11 views

Bayesian Statistics … Γ(α,β) Posterior Probability and Estimators

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
0
votes
1answer
14 views

Searching for a bound on a Mobius functions defined on $B(0,r)$.

Let $\alpha \in \mathbb C$ such that $|\alpha| \in (0,1)$. Prove that if $z\in \mathbb C$ is such that $|z| \le r < 1$ then : $$ \frac{\alpha+|\alpha| z}{\alpha(1-\overline{\alpha} z)}\le ...
4
votes
1answer
71 views

Green's function of Dirichlet problem

Let $G(y,x)$ be the Green's function for the Dirichlet problem for Laplace equation on domain $\Omega$ with smooth boundary. Show that $$K(y,x):=\frac{\partial}{\partial \textbf{n}_y}G(y,x)\geq 0, ...
0
votes
0answers
17 views

Complex Analysis, showing a function is zero [duplicate]

Let $\Omega$ be the right half plane excluding the imgainary axis and $f\in H(\Omega)$ such that $|f(z)|<1$ for all $z\in\Omega$. If there exists $\alpha\in(-\frac{\pi}{2},\frac{\pi}{2})$ such that ...
0
votes
0answers
29 views

Maximum principle applied to the Delta “function”.

Let $\Omega\in \mathbb{R}^n$ be a bounded domain with a fixed $x\in \Omega$. Let $u$ be a function subject to $$\begin{cases} \Delta u=\delta (y-x), & \forall y \in \Omega\\ u=0, & \forall y ...
1
vote
0answers
13 views

How do I show that the maximum principle holds for the PDE $u_t = u_{xx}$?

We have the PDE $u_t = u_{xx}$ with initial conditions $u(x, 0) = u_0(x)$ given. How does one show that $u(x, t) \leq \mathrm{max}_x\;u_0(x)$ for all $x$ and $t$? I later have to show that a maximum ...
1
vote
0answers
37 views

Complex analysis question, maximum principle application

Let $\Omega=\{z, \text{Re}z>0\}$ Suppose that $f$ is continuous in the closure of $\Omega$ and $f$ is holomoprhic on $\Omega$ and there are constants $A<\infty $ and $\alpha<1$ such that ...
4
votes
0answers
63 views

Complex analysis, showing a function is constant

Let $\Omega$ be the right half plane excluding the imgainary axis and $f\in H(\Omega)$ such that $|f(z)|<1$ for all $z\in\Omega$. If there exists $\alpha\in(-\frac{\pi}{2},\frac{\pi}{2})$ such that ...
5
votes
4answers
59 views

Minima and Maxima of $\left({\frac{\sin 10x}{\sin x}}\right)^2$

It's a question from $BNMO$.that still haunts me a lot. I want to find an answer to this question. Find the minima and maxima ...
1
vote
0answers
20 views

Maximum principle for a “modified” laplacian

Let $\Omega \in \mathbb{R}^{n}$ be bounded dmain, given $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a solution of $$\Delta u+c(x)u=0$$ where $c(x)\leq 0 \in \Omega.$ Shown that $u=0$ on $\partial ...
0
votes
1answer
20 views

Maxima and Minima, If s = 60, what should the side of the cut out be…

A square piece of steel, s cm on a side, is to made into an equipment chassis by cutting equal squares out of the corners, folding up the sides , and welding the seam to form a pan. $A)$ If $s=60$, ...
3
votes
1answer
87 views

Maximum principle for harmonic functions on unbounded domain

I am learning PDE by myself now. I am considering converted the problem to bounded domain to use the strong maximum principle. My attempt: Using the $\lim u(x)=0$, then exists $\epsilon$ and $N$, ...
0
votes
1answer
22 views

Maximum principle of eigenvalue-like problem for harmonic equation

I am trying to identify for which $\lambda$ so that I can obtain the Maximum principle, that is, $$\sup_{\overline{\Omega}}u\leq \sup_{\partial \Omega}u$$ for equation $-\triangle u = \lambda u$, ...
2
votes
1answer
86 views

Optimal Control - difference between indirect/direct approaches

In an indirect method, my understanding is we convert the Continuous Optimal Control problem to a 2-point boundary problem by using initial conditions on the states and terminal conditions on the ...
1
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1answer
90 views

Prove that f is one-to-one on D

Let $f$ be an analytic function on a disc $D$ whose center is the point $z_0$. Assume that $|f'(z)-f'(z_0 )|<|f'(z_0)| $ on D. Prove that $f$ is one-to-one on D.
2
votes
0answers
84 views

Max Principle for Heat Equation with Neumann Boundary Condition

This is a homework problem. I am looking for an additional hint or reference to theorems/ideas that can help. Some of my thought process is presented below. Suppose \begin{align*} u_t - \Delta u ...
1
vote
2answers
50 views

Maximum Likelihood Estimation Question

I'm really struggling with this question. From my understanding in order to find the maximum likelihood estimator for theta, the function needs to be partially differentiated with respect to theta ...
0
votes
1answer
29 views

Maximum Value - Analytic function

I am having a hard time figuring out where to start and what results to use to address the following question: Suppose $f(z)$ is analytic in the unit disc $D=\{z:|z|<1\}$ and continuous in the ...
2
votes
0answers
56 views

Comparison and maximum principle for parabolic pde

I was told the comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the maximum principle. I also know comparison principle ...
1
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1answer
55 views

Strong maximum principle for weak solutions?

For a general linear parabolic equation, is a strong maximum principle possible when the solutions are merely weak solutions (i.e. they lie in a Bochner space)? Is there some proof possible that does ...
1
vote
1answer
46 views

Linear algebra - as applied to the weak maximum principle

Currently I have little background in theory-based linear algebra (I never took an upper-division linear algebra course), so I am asking this question here. Although this is from page 345 of PDE by ...
0
votes
0answers
32 views

An E.O.Q. problem in an university restaurant.

A university restaurant can serve up to 800 meals a day; the number of students using the university restaurant on any given day is a random variable whose probability distribution is given by the ...
2
votes
0answers
71 views

Prove the maximum principle for a solution of the PDE $(-\Delta + \lambda)u=0$

Show that if $u$ satisfies $(-\Delta + \lambda)u=0$ in a smooth region $\Omega$ where $\lambda$ is a nonnegative real number, then u satisfies the maximum principle. We can just assume u is smooth, ...
0
votes
3answers
174 views

How exactly do I prove that I find the maximum of the function

I am currently trying to maximize an objective function $f(a,b,c,d,e)$ over the variable $b$ only. By taking the derviative of f over b, setting it to zero, I can solve b in terms of the other 4 ...
1
vote
1answer
67 views

Maximum principle easy proof

Is there an easy proof of the maximum principle from the variational formulation in $\Bbb R^d$, without using Green functions? Variational formulation: $$ \forall v\, \text{ smooth, }\, \int \nabla ...
2
votes
1answer
103 views

How do I apply this maximum principle?

I have the maximum principle: $$\text{If } \psi\geq 0 \text{ on }\Gamma. \text{Then }L\psi\geq0\text{ implies } \psi\geq 0 \text{ in } \bar{D},$$ where $D=(0,1)\times (0,T], \Gamma$ is parabolic ...
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votes
1answer
80 views

Why isn't there a general comparison principle for higher order equations?

I am studying elliptic equations. For second order equations (linear and nonlinear), comparison principle has many applications, e.g. to show uniqueness of the (weak) solutions, to construct super- ...
0
votes
2answers
26 views

Maximum value estimation

Let $f$ be an analytic function that is not zero at $\{z:|z|<2\}$. Show that for every natural number $n$: $$\max_{|z|=1}|f(z)-\frac{1}{z^n}|>1$$ I know that ...
0
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0answers
25 views

Maximum principle in PDE [duplicate]

I was told Maximum principle is a common method in proving uniqueness of the solution to certain PDE. Could anyone explain 1) How does Maximum principle work in the context of PDE theory? 2) How ...
2
votes
1answer
62 views

How to show that $u(x,t)\le\pi^3-1+\sin(x)$ (heat equation)

Let $u(x,y)$ be a continuous solution of \begin{cases} u_t=u_{xx}+\sin(x) & 0<x<\pi,&t>0, \\[3ex] u(0,t)=u(\pi,t)=0 & t\ge0, \\[3ex] u(x,0)=4x(\pi-x)\quad&0\le x ...
0
votes
0answers
21 views

Maximum Principle for subsolutions of heat equation with drift term

This question is about a proof in the article Mean Curvature Evolution of Entire Graphs by Klaus Ecker and Gerhard Huisken, The Annals of Mathematics, 2nd Ser., Vol. 130 (page 455). You can find the ...
0
votes
1answer
40 views

Maximum principle type problem

Suppose $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ is sufficiently smooth and is such that $(1/2) u_{xx} + u_{xy} + 2u_{yy} = 0$ in a ball $B$ centered at the origin. Must $u$ attain a maximum inside ...
1
vote
1answer
52 views

derivative of expected value of maximum of two stochastics variables (iid)

I need to optimize an expected value of a maximum value for $q$. The problem has three variables, $q$ is a constant and $D_1$ and $D_2$ are stochastic variables with pdf $f(x)$ and cdf $F(x)$. The ...
1
vote
1answer
80 views

$L^\infty$ bound on solution of $u_t -\Delta u =f$ where $f \in L^\infty$ and $u_0 \in L^\infty$??

Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega)$ and consider $$u_t - \Delta u = f$$ $$u(0) = u_0$$ with some boundary conditions. Consider the weak form ...
0
votes
0answers
52 views

Maximum and Minimum principle of Laplace equation

I want to prove that if $u=u(x, y, t)$ is a solution of the equation: $$\frac{\partial u}{\partial t} = \Delta u,\;\;\;\;\;\;\ where \; \Delta u = \frac{\partial ^2}{\partial x^2} + \frac{\partial ...
0
votes
2answers
281 views

Time Derivative of PDE solution

Consider the PDE $$ \frac{\partial \psi}{\partial t}(t,x)=\psi''(t,x)+2\psi\psi'(t,x)-\tilde V'(x)~ on~ [0,D/2]\times(0,\infty)\\ \psi(0,t)=0, \psi(D/2,t)=-k, \\\psi(\cdot,0)=\psi_0 $$ for some ...
1
vote
1answer
50 views

Constant function with maximum modulus [duplicate]

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
2
votes
1answer
233 views

Minimum Modulus Principle for a constant fuction in a simple closed curve

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
0
votes
2answers
52 views

Usage of Maximum Modulus Theorem?

I have a function $f(z)=ze^z$, I want to find the maximum value of $|f(z)|$ as $z$ varies over the region $D=\{x+iy: x^2+y^2\leq 4, x,y\geq 0\}$. I was thinking that this is case where I can use the ...
1
vote
1answer
79 views

Suppose $u \in C^2(\Omega)\cap C^1(\bar\Omega)$ satisfies $ -\Delta u+u^3=0$ in $\Omega $

Suppose $u \in C^2(\Omega)\cap C^1(\bar\Omega)$ satisfies $$ -\Delta u+u^3=0 \quad\text{ in } \Omega $$ $$ \frac{\partial}{\partial\eta}u+\alpha u=\varphi \quad\text{ on }\partial \Omega $$ for ...
0
votes
1answer
61 views

Proving zeros inside a disk using Maximum Principle?

Let $f$ be non-constant analytic in a neighborhood of the closed unit disk such that $|f|=1$ on the unit circle. Show that $f$ has a zero inside the unit disk. It has been suggested to argue this by ...
1
vote
0answers
43 views

Question about the maximum principle for the Laplace's equation

Maximum principle for the Laplace's equation: $$\nabla^2 u= \Delta u=u_{xx}+u_{yy}$$ $$\Delta u=f(x,y) \text{ Poisson }$$ Problem with boundary values of the form Dirichlet: $$\left.\begin{matrix} ...
1
vote
1answer
49 views

Find the maximum possible value

Help me to find the maximum value of $T$ with $x, y, z \in \Bbb{R_+}$ $$T=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}$$ Thanks :D
0
votes
2answers
119 views

Maximum value question of three $xyz+xy+yz+zx$.

I have tried am-gm inequality,i am getting that $xyz$ is greater than $36.9$. I tried hit and trial,but it is of no use also. Could anyone give a definite process?