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

learn more… | top users | synonyms

1
vote
0answers
18 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) ...
0
votes
0answers
23 views

What is the MAP (Maximum a posterior) hypothesis?

Hello All, I am having some trouble with this question.. ...
0
votes
1answer
38 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 $\phi=\phi(x)$, $x\in\mathbb{R}^n$: ...
0
votes
0answers
14 views

need help: find a maximum value of matrix-based equation using differential

I have a problem to derive this equation to find the matrix that gives its original equation a maximum value. here is my equation: *find matrix Y that could maximize the F function. the function F is ...
2
votes
2answers
39 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 ...
1
vote
0answers
64 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 $u$, ...
1
vote
1answer
41 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 ...
0
votes
1answer
24 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 \; ...
0
votes
0answers
25 views

Simple Problem of Selection.

Mike has N different items. He has M orders of customers and each customer has a set of items they want. Customer will not accept partial order. So Mike can give one item to atmost 1 customer. Find ...
1
vote
2answers
27 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 ...
1
vote
1answer
24 views

Using the Maximum Principle to prove things about solutions to inhomogeneous Diffusion Equations

I am trying to teach myself PDEs, and I am working on the following problem from Strauss: (a) If $u_{t}-ku_{xx}=f$, $v_{t}-kv_{xx}=g$, $f \leq g$, and $u\leq v$ at $x=0$, $x = l$, and $t = 0$, prove ...
0
votes
1answer
22 views

Average temperature of an object with a non-uniform temperature distribution [closed]

We place some solid three-dimensional object on a planar surface that heats the object until some equilibrium situation is reached (the object is also cooling and radiating heat in the air) where any ...
0
votes
2answers
50 views

Inequality involving max is confusing me.

I am trying to understand one line in a derivation here. Simply put, the statement is that: $$ max\{\theta \ f_1(x) + (1-\theta) \ f_1(y) \ , \ \theta \ f_2(x) + (1-\theta) \ f_2(y)\} \leq \theta \ ...
0
votes
0answers
51 views

Short time existence and Maximum Principle for an explicit parabolic PDE

Consider the following PDE: $\frac{\delta}{\delta t}\psi=\psi^{''}-2\psi^{'}\psi+V' \ on \ [0,D]\times(0,\infty) \\ \psi(0,t)=0, \ \psi(D,t)=-k, \ (k\in\mathbb{N}) , \ \textrm{given } t>0 ...
0
votes
2answers
28 views

Finding maximum and minimim of function on an interval. Are there multiple ones?

I found myself stuck at such basic problem. If you're to calculate local maximum and minimum on closed interval, $\langle a, b\rangle$, the $a$ and $b$ may as well be the maximum and minimum points of ...
0
votes
1answer
42 views

How to ensure extreme? — using Extreme Value Theorem

I think it's a simple question. How can I ensure the existence of an extreme (maximum or minimum) using the Extreme Value Theorem / Weierstrass theorem? For example, this multivariate case: $$ ...
1
vote
1answer
57 views

Evans PDE chapter 2 problem 4

Problem is Give a direct proof that if $u \in C^{2}(U) \cap C(\overline{U})$ is harmonic within a bounded open set $U$, then $\max_{\overline{U}} u =\max_{\partial U} u$. What I think is ...
2
votes
1answer
39 views

Uniqueness of variant laplace pre

Here's problem. Let $U$ be a bounded domain in $\mathbb{R}^{n}$ and $\vec{b} : \mathbb{R}^{n} \to \mathbb{R}^{n}$ and $g: \mathbb{R}^{n} \to \mathbb{R}$ be continuous. Show that there can be at ...
2
votes
0answers
92 views

Maximum principle question in partial differential equation

Problem is Let $U$ be a bounded domain in $\mathbb{R}^{n}$ and $\vec{b} : \mathbb{R}^{n} \to \mathbb{R}^{n}$ and $g: \mathbb{R}^{n} \to \mathbb{R}$ be continuous. Show that there can be at most ...
1
vote
1answer
61 views

maximum of a holomorphic function over a compact subset of a bounded region

If $\Omega$ is a bounded region in $\mathbb{C}$ such that $\mathbb{C}\setminus\Omega$ is connected and $K$ is a compact subset of $\Omega$, show that $\hat{K}_{\Omega} = \hat{K}_{\mathbb{C}}$. ...
3
votes
1answer
114 views

zeros of a function holomorphic in the closed unit disc

Let $f$ be a holomorphic function in a neighborhood of the closed unit disc $\{z \in \mathbb{C} : |z| \leq 1\}$, and suppose that $\Re{(\bar{z}f(z))} > 0 $ when $|z| = 1$. Prove that $f$ has ...
0
votes
1answer
248 views

Problem 6.6-12 of Evans' PDE

Can please somebody tell me, how solve this problem ? We say that the uniformly elliptic operator $$Lu\ =\ -\sum_{i,j}a^{ij}u_{x_ix_j}\ +\ \sum_{i=1}^nb^iu_{x_i}\ +\ cu$$ satisfies the weak maximum ...
1
vote
0answers
41 views

Pontryagin principle: does the abnormal multiplier define a minimum

The Pontryagin principle PM provides the necessary condition for a local minimum of the functional $ J(u)=\int L(x(t),u(t))dt \\$ subject to: $\dot x = f(x(t),u(t)) \ \ \ \ x(t0)=x0, \ \ ...
0
votes
1answer
70 views

Diffusion Equation Maximum Principle

Does the general diffusion equation (http://en.wikipedia.org/wiki/Diffusion_equation) satisfy the maximum principle i.e. is it's maximum value obtained on teh boundry of the region?
0
votes
3answers
64 views

Find the max value of a function at a given interval

Trying to determine local max for a function at interval $[-4, 6]$. $$f(x)= x^3 -3x^2-24x + 7$$ Is the proper next step to take the derivative of $f(x)$ and find the roots, set roots = to zero?
1
vote
1answer
72 views

Linearization by freezing the coefficients of the main part of the PDE

Let $\Omega\subset C^0$ a bounded domian in $\mathbb{R}^2$. Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a non negative classical solution of $$ ...
2
votes
1answer
48 views

Do I have to use a maximum principle?

Let $\Omega\subset C^0$ a bounded domian in $\mathbb{R}^2$. Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a non negative classical solution of $$ ...
1
vote
1answer
111 views

Harmonic function takes both positive and negative values

I am a little confused on the following question: Suppose that $u$ is harmonic nonconstant on a $D(z_0,R)$ and $u(z_0)=0$. Is it true that on each circle $C(z_0,r)$, with $0<r<R$, the function ...
0
votes
1answer
278 views

Find the Maximum and Minimum values of $e^z$ when $z\le 1$.

I need help finding the maximum and minimum values of $|e^z|$ on $|z|\le1$. I know we use the maximum modulus theorom but i cant seem to get an answer.
0
votes
1answer
85 views

Maximum likelihood and Bayesian Theorem

In Bayesian theorem, $$p(y|x) = \frac{p(x|y)p(y)}{p(x)}$$, and $p(x|y)$ is called the likelihood, and I assume it's just the conditional probability of $x$ given $y$, right? The maximum likelihood ...
0
votes
0answers
45 views

Discrete Maximum Principle : condition on stock variable

I am reading chapter 10 of "Optimization in Economic Theory" By Avinash Dixit and I am missing something at the point where the Envelope Theorem is applied. Here is the problem. The setting We are ...
0
votes
2answers
37 views

Finding maximum volume (where $x^2+y^2+z^2=r^2, x,y,z>0$)

I would appreciate if somebody could help me with the following problem Q. Finding maximum volume $$(x+y+z)^2(xyz)^2 ~~~(\text{where} ~~x^2+y^2+z^2=r^2, x,y,z>0)$$
5
votes
3answers
305 views

The condition for Y to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$

I would like to know the condition for a random variable Y in order to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$, where $X_1$ and $X_2$ are iid. Any help would be ...
2
votes
1answer
123 views

Maximum principle of holomorphic functions

Show: If $f$ is a non-constant holomorphic function on a domain $G$, then $\lvert f\rvert$ has no local maximum on $G$. Hint: Use the mean value property of holomorphic functions! Suppose ...
2
votes
1answer
79 views

Weak maximum principle for the p-Laplacian

For the equation $\Delta_p u = 0 $ in $U$ ($U$ open and bounded), does a weak maximum principle hold? (The maximum and minimum occur on $\partial U$)? If yes, someone can indicate a book with the ...
-1
votes
1answer
46 views

Minimum/Maximum Extremas in a Plane

Explain one way in which extrema of a function of two variables f(x,y) defined on a closed, bounded region of S of plane differ from extrema of a function of one variable f(x) defined on a closed ...
1
vote
1answer
141 views

Use maximum modulus theorem to control the number of zeros of analytic functions.

Let $f$ be a analytic function on $\bar B(0,R)$,with $\| f(z)\|\leq M$ in $\| z \| \leq R$,suppose $\|f(0)\|=a>0$,the the number of zeros of $f$ in $B(0,\frac{1}{3}R)$ is equal or less than $\log 2 ...
2
votes
1answer
74 views

Method of Moving Planes and Method of Moving Spheres

Under what conditions should I use the methods of moving spheres instead of the method of mobile plans? Under what conditions should I use the method of moving planes, but I can not use the method of ...
1
vote
1answer
46 views

Simple question of maximum value a part can have?

We have to partition n chocolates among m children. Children will be happy if max and min a child has got is less than 2. What is the max a child can get?? For n=6 m=3 ,the partition will be 2 2 2 ...
2
votes
1answer
56 views

Is there a weak maximum principle applicable to this situation?

I am reading a paper and have trouble understanding the following step on page 581: Assume we know that $w: \mathbb{R}^n \to \mathbb{R}$ is smooth and bounded. Furthermore $\Vert w ...
1
vote
0answers
40 views

formulas to maximize the output

Good day, I have a math problem in my game development like this: I have two numbers (football player skills): Attack skill (AS) and Defend skill (DS). They are in range from 1.0 to 8.0... now I want ...
0
votes
1answer
39 views

Need help understanding a transformation

I know this might be an unusual question, but please bear with me. In the book ¨An Introduction to Maximum Principles and Symmetry in Elliptic Problems¨ There is the following example of ...
4
votes
0answers
109 views

Maximum principle for heat equation

$u_1$ and $u_2$ solve $u_t-u_{xx}=f(u)$ for some $C^{\infty}$ function on $(a,b)\times[0,\infty)$ On the vertical boundaries: $u_1(a,t)=u_2(a,t)=u_1(b,t)=u_2(b,t)=0$ and there exists ...
2
votes
1answer
49 views

Let $f$ be defined on $[a,b]$, Prove that if f has a local maximum at a point $x \in (a,b)$, and if $f'(x)$ exists, then $f'(x)=0$

Is this proof correct: Let's choose a $\delta$ to that $a < x - \delta < x < x + \delta < b$ If $ x - \delta < t < x$ then $\frac {f(t) - f(x)} {t-x} \geq 0$ Letting $t ...
0
votes
1answer
57 views

Maximum Modulus theorem applied on mapping

Question: For $|z_0|<R$, I want to show that the mapping $$T(z)=\frac {R(z-z_0)} {R^2-\bar{z_0}z}$$ takes the open disc of radius $R$ $1-1$ and onto the unit disc and $z_0\rightarrow 0$. Hint: ...
0
votes
1answer
276 views

Maximum modulus principle exercise.

I have a maximum modulus principle exercise question and I'm stuck trying to understand the solution at the moment. Here goes: Let $c\in \mathbb{D}= \left\{z \in\mathbb{C}: |z|<1\right\}$ and ...
6
votes
1answer
100 views

Entire + periodic in imaginary direction + bounded on the real line implies constant?

I was reading some slides from a lecture. In a proof, there arose the need to show a certain function $f : \mathbb{C} \to \mathbb{C}$ was constant. The argument proceeded by checking that $f$ was ...
0
votes
1answer
82 views

Understanding the Hamiltonian function

Based on this function: $$\text{max} \int_0^2(-2tx-u^2) \, dt$$ We know that $$(1) \;-1 \leq u \leq 1, \; \; \; (2) \; \dot{x}=2u, \; \; \; (3) \; x(0)=1, \; \; \; \text{x(2) is free}$$ I can ...
3
votes
0answers
85 views

Unexpected hanging paradox maxmin strategies

I have a question about strategies of the players of Unexpected hanging paradox (I am very sorry for a long topic, topic exist already for a while, during this time I try to develop idea how to solve ...
2
votes
0answers
56 views

Is there an exact example which show that the non-negativity in weak maximum principle is necessary?

We know that the Weak Maximum Principle assert that for an uniformly elliptic operator $L=a^{ij}(x)D_{ij}+b^i(x)D_i+c$, if $c\leq0$ and $Lu\geq0$ in a bounded domain $\Omega$, then $u$ attains on ...