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

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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 ...
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1answer
12 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 ...
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1answer
58 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, ...
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0answers
16 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 ...
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0answers
25 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 ...
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0answers
10 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 ...
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0answers
29 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 ...
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0answers
56 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 ...
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4answers
54 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 ...
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0answers
19 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 ...
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1answer
19 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 $s=60$, ...
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1answer
77 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 $N$, ...
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1answer
19 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 $-\triangle u = \lambda u$, ...
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1answer
36 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 ...
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1answer
86 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.
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0answers
80 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 ...
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2answers
42 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 ...
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1answer
26 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
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0answers
35 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 ...
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1answer
45 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 ...
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1answer
44 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 ...
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0answers
29 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 ...
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0answers
65 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, ...
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3answers
165 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 ...
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0answers
30 views

multivarable optimization problem, what is the procedure?

Sorry for this obvious question. I am trying to maximize an objective function that consist of 5 variables (a,b,c,d,e) over a and b. That is , $max _{a,b}f(a,b,c,d,e).$ So I procedure I took is ...
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1answer
57 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 ...
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1answer
101 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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1answer
71 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- ...
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2answers
24 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 ...
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0answers
24 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 ...
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1answer
61 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 ...
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0answers
20 views

Maximum Principle - Proof

We want to show the maximum principle for a function $f = f(x,t)$ on a n-dimensional hypersurface $M,$ that is, (Corollary) Let $f = f(X,t)$ be a function on M, let $\vec{a}$ be a vector field on ...
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1answer
38 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 ...
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1answer
44 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 ...
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1answer
79 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 ...
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0answers
50 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 ...
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2answers
273 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 ...
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1answer
40 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 ...
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1answer
170 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 ...
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2answers
51 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 ...
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1answer
75 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 ...
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1answer
57 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 ...
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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} ...
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1answer
48 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
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2answers
111 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?
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1answer
107 views

maximising profit chi-square distribution

A bakery sells rolls in units of a dozen. The demand X (in 1000 units) for rolls has a gamma distribution with parameters alpha=3 and theta=0.5. It cost 40 cents to make a unit that sells for $1 on ...
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1answer
53 views

Showing an entire function f(z) is monomial

Given f is entire and satisfying $|f(z)| \leq 3|z|^{\alpha}$, show that $f(z) = cz^{\alpha}$ for some constant $c$ if $\alpha$ is a positive integer, and $f(z) = 0$ if $\alpha$ is not an integer. I ...
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3answers
165 views

Find the maximum value of $abc$

$a,b,c$ are three positive real numbers such that $ab+bc+ca=12$. Then find the maximum value of $abc$
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0answers
14 views

Weak version of Max. Mod. principle

Let $f$ be holomorphic in a domain $U$. Let $a\in U$. Show there cannot exist $\varepsilon > 0$ such that $D(a,\varepsilon) \subset U$ and $|f(z)|<|f(a)|$ for all $z\in D(a,\varepsilon)$ ...
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0answers
21 views

Holomorphic function extended to entire polydisc

How can I show that every holomorphic function in the border of a polydisc $\Delta \subset \mathbb C^n, n>1,$ has a extention to entire $\Delta$? I know this is just a consequence of Maximum ...