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

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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 $s=60$, ...
3
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1answer
72 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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0answers
12 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
79 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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76 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
35 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
22 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 ...
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0answers
26 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
43 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
40 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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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
62 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
162 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
29 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
41 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
100 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
67 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
23 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 ...
2
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1answer
60 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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19 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
36 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
43 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
77 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
43 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
271 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
39 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
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1answer
108 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
49 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
73 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
42 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
101 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
57 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
47 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
157 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
18 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 ...
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1answer
79 views

Elliptic PDE - max principle

The maximum principle for elliptic PDEs is established for the nondivergence form as in http://www.ann.jussieu.fr/~frey/cours/UdC/ma691/ma691_ch3.pdf. But what if we are dealing with the divergence ...
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1answer
46 views

Strong maximum principle

Let $S^{n-1}$ denote sphere in $\mathbb{R}^n$ and let $D$ denote open unit disk in $\mathbb{R}^n$. Let $f$ be homeomorphism of $S^{n-1}$ onto itself. Let $F$ be its harmonic extension given by Poisson ...
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363 views

Intuition Behind Maximum Principle (Complex Analysis)

Let $D$ be an open set in the complex plane and $f(z)$ be a non-constant holomorphic function on D. Then $|f(z)|$ has no local maximum on D. I can follow the proof fine - usually if I don't ...
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0answers
143 views

How to prove comparison principle for parabolic PDE (nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$. Consider the PDE $$u_t = \Delta F(u) ...
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1answer
49 views

A maximum principle

Suppose that $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary $\partial\Omega$. Consider the elliptic boundary value problem for $\phi=\phi(x)$, $x\in\mathbb{R}^n$: ...
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2answers
112 views

Maximum of linear combination

I have an range like this: $$x + 2y \leq 40$$ $$4x + 3y \leq 120$$ $$x \geq 0, y \geq 0 $$ I made an plot using wolfram alpha. Now I have a linear combination $$4x+5y$$ and I want to find the maximum ...
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0answers
69 views

strong maximum principle for $u$ such that $u'' \geq c(x)u$

I want to prove that if $I$ is an interval in $R$, $c(x) \geq 0$ is continuous and $u \leq 0$ is $C^2$ then if for all $x$ $$u''(x) \geq c(x)u(x)$$ then the strong maximum principle holds for $u$, ...
2
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1answer
93 views

sequence uniformly convergent on the boundary of a bounded set in $\mathbb{C}$

Let $(f_n)$ be a sequence of functions which are analytic on a bounded region $A\subset\mathbb{C}$ and continuous on the closure $Cl(A)$. Suppose that the sequence is uniformly convergent on the ...
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2answers
82 views

Generalization of the Maximum Principle for elliptic linear PDEs

This question came up when I was trying to solve linear elliptic PDEs. Let $R$ be an open domain and $L$ be a linear elliptic operator such that $$ L \; u = 0 \; \mathrm{on \; R}, \;\; B \; u = f \; ...
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2answers
37 views

Showing $ u =0 $ for a particular laplace equation.

Given open $\Omega\subset \mathbb{R}^n $ with smooth enough boundary, if for $x_0 \in \Omega$ there exists $ u \in C^2(\Omega\setminus\{x_0\}) $( we may as well take $u$ as smooth as we want) that ...