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

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3
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1answer
48 views

Maximum principle-estimation

Let $S=\{x \in \mathbb{R}^2 \mid |x| <1\}$. Using the maximum principle I have to show that the solution of the problem $$-\Delta u(x)=f(x), x \in S \\ u(x)=0, x \in \partial{S}$$ satisfies the ...
0
votes
1answer
28 views

How could we continue to show the inequality?

Let $\Omega$ a bounded space. Let $u_1$ the solution of the problem $$-\Delta u_1(x)=f(x), x \in \Omega \\ u_1(x)=g_1(x), x \in \partial{\Omega}$$ and $u_2$ is the solution of the problem $$-\Delta ...
1
vote
1answer
34 views

How could we continue to get a contradiction?

Let $\Omega$ a bounded space. Using the maximum principle I have to show that the following problem has an unique solution. $$-\Delta u(x)=f(x), x \in \Omega \\ u(x)=g(x), x \in \partial{\Omega}$$ ...
0
votes
0answers
17 views

Heat equation boundedness of solutions

Consider $$u_t-u_{xx}=1,\ 0\leq x \leq 1,\ t>0,$$ with zero Dirichlet boundary conditions and vanishing initial conditions. The solution to the stationary problem, i.e. $$u^s_{xx}=1,\ 0\leq x \leq ...
1
vote
1answer
23 views

Laplace Equations with Neumann boudary-value problem

The problem is that, Assume U is connected, use the maximum principle to show that the only smooth solutions of $-\Delta u=0$ in U and $\frac{\partial u}{\partial \nu}=0$ on $\partial U$ are $ u ...
2
votes
1answer
33 views

Find all holomorphic functions with the following property

Let $D$ be the unit disc. Find all holomorphic functions $f:D\to D$ such that $f(\frac14)=\frac14$, and $f'(\frac14)=\frac7{15}$. I guess that we should use Schwarz lemma. And I guess that the only ...
0
votes
0answers
10 views

maximum principles for beam and higher order equations

It is well known that the inequality $u_t<u_{xx}$ meets the maximum principle: Maximum of function $u(x,t)$ is attained at $t=0$ or on the boundary $x=0$ or $x=L$. Do we have maximum principles or ...
0
votes
0answers
33 views

Proving a version of maximum modulus principle elementarly.

There is this version of maximum modulus principle: If $P$ is a non-constant polynomial, then $|P|$ doesn't have a local maximum. I know that if $P$ is non-constant, then $|P(z)| ...
0
votes
0answers
18 views

Mixture of Maximum Entropy and Minimum Cross Entropy?

Assume you have a discrete prior distribution on a set of points $ P(X=\{0,3,5,6\}) = (.40,.30,.20,.10)$ $E[X]=5/2$ And you want to create a new distribution, $Y$, on $\{0,1,2,3,4,...\}$ using the ...
1
vote
1answer
35 views

Complex entire functions without taking values on a segment are constant!

Let $a,b$ be two distinct complex numbers and $f$ be an entire complex function, i.e. a complex function which is analytic on the whole complex plane, and $$R(f)\subset\mathbb C-\{\lambda ...
0
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0answers
32 views

Finite Difference Method for Heat Equation with Neumann Boundary

I have read the book of Morton and Mayers. In chapter6, it said that the explicit finite difference scheme of a heat equation, $\frac{U^{n+1}_j-U^{n}_{j}}{\Delta ...
0
votes
1answer
24 views

To show that $f = f_1 − f_2$ for $f \in \mathcal{H}(A(0; r_1 ; r_2 ))$

I need to prove the foloowing. Let $A(0; r_1 ; r_2 ) = \lbrace z \in \mathbb{C} : r_1 < |z_1| < r_2 \rbrace $ where $r_1 < r_2$ Show that if $f \in \mathcal{H}(A(0; r_1 ; r_2 ))$ such ...
2
votes
1answer
49 views

Proving that a holomorphic function is constant

I am attempting to prove the following: Let $X$ be a connected complex manifold, and $f\in \mathcal{O}(X)$. For any $x\in X$, there is a complex submanifold of $X$ which is biholomorphic to ...
0
votes
2answers
49 views

Finding max and min on a disk

The problem I'm having trouble with is this; The temperature at all points in the disk $x^2 + y^2 {\leq} 1$ is given by $T = (x+y)e^{-x^2-y^2}$. Fine the minimum and maximum temperatures at the ...
1
vote
0answers
29 views

Maximum Principle for the Derivatives of Parabolic PDE Solution

Is there a maximum principle for the spatial derivatives of the solution of a parabolic PDE with coefficients in front of the spatial partial derivatives depending on spatial variables only, similar ...
1
vote
1answer
31 views

Bounded holomorphic function on an unbounded region

Let $\Omega\subset\mathbb C$ is an unbounded region, with boundary $\Gamma$. Let $f$ be a function holomorphic in $\Omega$ and continuous on $\Omega\cup\Gamma$. If $f$ is bounded on ...
0
votes
1answer
29 views

Question on maximal value of $f(0.4 + 0.5i)$ subject to certain constraints. Use of maximum principle?

The function $f(z)$ is analytic in the unit disk $U = {z:|z|<1}$ and continuous in the closed unit disk. Suppose that $\frac{f(z)}{z^2}$ can be extended to be analytic in the (open) unit disk U ...
0
votes
1answer
23 views

Prove: for $f,g \in Hol(G\subset \mathbb{C})$ $\implies$ max $(|f|+|g|)$ is on the boundary of $G$.

Prove: for $f,g \in Hol(G\subset \mathbb{C})$ $\implies$ max $(|f|+|g|)$ is on the boundary of $G$. I don't really have a direction for this. I know it's got to do with the maximum modulus principle, ...
2
votes
1answer
74 views

Maximal Principle: Why using the new transition matrix $\tilde{P}$?

First some notation: Let $(X,E,P)$ denote a finite, irreducible Markov chain with finite state space $E$ and transition matrix $P$. Choose and fix a subset $E^°$ of $E$, which will be called ...
0
votes
0answers
23 views

Pricing of Binary or Digital Options or Feynman-Kac Equation for $\mathbb E f(X_T)$ with diffusion $X$ and discontinuous function $f$.

I am trying to find references (books, papers, etc.) for calculating $\mathbb E f(X_T)$, where $X_T$ is a diffusion and $f$ is a real function that is not continuous by means of solving a PDE or ...
1
vote
2answers
35 views

Calculate the maximum points where the distance between every two points is within a certain range

The predefined boundary is defined as $ (len, width) $. (i)How to find the maximum points where the distance between every two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, is within a certain range ...
4
votes
0answers
95 views

Weak maximum principle

We say that the uniformly elliptic operator $$Lu\ =\ -\sum_{i,j=1}^na^{ij}u_{x_ix_j}\ +\ \sum_{i=1}^nb^iu_{x_i}\ +\ cu$$ satisfies the weak maximum principle if for all $u\in C^2(U)\cap ...
1
vote
4answers
45 views

Practical and traditional way to get the maximum of $\frac{x^2}{(x^4+1)}$?

The maximum seems to be $\dfrac{1}{2}$, but how do you get this value and why? Does it have anything to do with the graph of the function?
0
votes
1answer
48 views

Prove that every convex polygon with area $1$ is contained in a parallelogram of area $\frac{4}{3}$

Prove that every convex polygon with area $1$ is contained in a parallelogram of area $\frac{4}{3}$ I can only show that polygon is contained in a rectangle of area $2$.
0
votes
4answers
70 views

Find maximum and minimum of $f(x,y)=x+y$

I would appreciate if somebody could help me with the following problem: Question: Find maximum and minimum of $f(x,y)=x+y$ when $$x^3+y^3=1,x,y \geq 0$$ I tried but couldn’t get it that way.
1
vote
1answer
48 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
37 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
69 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
39 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
57 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
1answer
16 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
94 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
20 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
31 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
22 views

Given the PDE $u_t = u_{xx}$, how does one show that $u(x, t) \leq \mathrm{max}_x\;u_0(x)$ for all $x$ and $t$?

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
53 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 ...
6
votes
2answers
162 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 ...
1
vote
0answers
25 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
24 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
145 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
25 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
226 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
vote
1answer
98 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
115 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
78 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
36 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
89 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
81 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
58 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 ...
1
vote
0answers
75 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, ...