For questions about the use of maximum principle, either in the context of partial differential equations or complex analysis.
-1
votes
1answer
71 views
Complex has became so hard after the min\max modulus principle. Need some proofs and examples. [closed]
1) $f(z)$ being non constant and analytic in a domain $D$
if $f(z)$ continuous on $\overline{D}$ and $|f(z)|$ is constant on the boundary
I need to prove that $f(z)$ must have a zero inside $D$!
2) ...
0
votes
1answer
36 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
55 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
27 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
62 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 ...
1
vote
0answers
38 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 $Lu>0$ in a bounded domain $\Omega$, then $u$ attains on ...
1
vote
1answer
45 views
Maximum modulus theorem
(a) If $f(z)$ is analytic inside and on a simple closed curve $C$ enclosing $z=a$, prove that
$$ \left( f(a)\right)^n = \frac{1}{2 \pi i} \oint_C \frac{f(z)^n}{z-a} dz , \; \; n=1,2,3,...$$
(b) ...
0
votes
1answer
38 views
Maximize $4x_1+x_2+3x_3$
Maximize $4x_1+x_2+3x_3$ given the constraint $x_1+x_2+x_3=x$, I used lagrange multiplier and it gave me 3 different values of $\lambda$, what does this indicate?
2
votes
1answer
71 views
Maximum Modulus Theorem and Annulus
Suppose that $f$ is analytic in the annulus: $1 \leq \vert z \vert \leq 2 $, that $\vert f \vert \leq 1$ for $\vert z \vert = 1$ and that $\vert f \vert \leq 4$ for $\vert z \vert = 2$. Prove $\vert ...
1
vote
2answers
55 views
Maximum Principle for Poisson Equation
For a smooth $u(x)$, $x \in \mathbb{R}^n$, satisfying:
$\Delta u = -f$ for $||x||<1$ , $u=g$ on $||x||=1$
I want to show that there exists a constant $C$ such that:
$$\max\{|u|:||x||\leq 1\} ...
0
votes
2answers
53 views
Please help to make me understand why I cant optimize this function: $U=x^{1/3}*y^{2/3}$ ?
If I want to maximize a production the function of which is given by $$L=-x^2+10x-2y^2+12y$$ I know I have to take the partial derivatives of of the function in respect to X and Y, so $$\frac ...
0
votes
0answers
41 views
Optmizing sum of two vectors
I apologize in advance for the title, but I don't know how to express exactly what I want to do.
So, here's my problem: I have 66 vectors, each one with 8 values, those values can be positive or ...
1
vote
0answers
30 views
Verification for maximum principle
Given optimal control problem
$$
\dot x = f(t,x(t),u(t)), \quad x(0) = x_0,\\
J(u) = \int_0^T f^0(t,x(t),u(t))dt \to \min,
$$
we can apply Pontryagin's maximum principle to get a necessary condition ...
5
votes
3answers
56 views
Determine complex polynomial
Problem
Let $P(z) = z^n + a_{n−1}z^{n−1} + \cdots + a_1z + a_0$ be a polynomial of degree $n > 0$. Show
that if $\lvert P(z) \lvert \le 1$ whenever $\lvert z \rvert = 1$ then $P(z) = z^n$.
I ...
3
votes
1answer
117 views
Maximum Modulus Exercise
Using the maximum modulus theorem in complex analysis, what is a good technique for finding the maximum of $|f(z)|$ on $|z|\le 1$, when $f(z)=z^2-3z+2$?
Got some really nice answers below, so I ...
