Questions tagged [maximum-principle]

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

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Maximum principle for elliptic equation in exterior domain

I have a question on maximum principle for elliptic equation in exterior domain. Suppose that $u$ is infinitely differentiable in $\mathbb{R}^n$ and bounded in $\mathbb{R}^n$. I want to prove that ...
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1answer
21 views

Use Schwarz' Lemma and/or Maximum Modulus Principle to Prove This Proposition

Suppose the following conditions hold true: (1) The function $f$ is analytic and contractive in the open unit disk. (2) $f(0)=0$. (3) $\exists z_1 \ne z_2 \in B(0,1)$, $|z_1|=|z_2|$, $f(z_1)=f(z_2)$...
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1answer
39 views

Maximum modulus principle problem

I have the function $f$ which is holomorphic on the open unit disc $|z|<1$ and satisfies $|f(z)|\leq cos(\frac{\pi |z|}{2})$. I am asked to find $f$. I know that I am supposed to use maximum ...
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15 views

Dirichlet Problem with Isolated point has no solution

I have found this question on Fraenkel's Maximum Principles: Suppose I have a bounded domain with an isolated point on the boundary. I do not understand how there can be certain functions $g(x)$ for ...
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1answer
33 views

If $f,g$ are holomorphic in an open connected set $U$ and for some $z_0\in U$ we have $|f|+|g|\leq|f(z_0)|+|g(z_0)|$ then $f$ and $g$ are constant [duplicate]

I want to prove the statement of the question. I know we should apply some form of the maximum modulus principle, however, I don't know which function to pick, because the absolute values are quite ...
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11 views

Interior ball condition, outer unit normal vector

Suppose $U$ is bounded and open. If $U$ satisfies the interior ball condition at $x^0$, then there exists an open ball $B\subset U$ with $x^0\in \partial B$. Question: If $x^0\in\partial U$, is the ...
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50 views

Using the maximum modulus principle to show an inequality

Let $f$ be a holomorphic function defined on a neighborhood of $\bar{D}$, where $D$ is the unit disc, and suppose that $f(0) = 0$. (a) Show that $g(z) = f(z)/z$ is holomorphic on a neighborhood of $\...
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1answer
57 views

Invariant interval for ODE implies invariant interval for parabolic PDE

Let $u$ solve $u_t-\Delta u = f(u)$ on $[0,T] \times \Omega$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ and $u=0$ on $\partial{\Omega}$. Also, let $\partial{\Omega}$ be sufficiently smooth ...
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8 views

Discretization error of implicit finite difference scheme with robin boundary by maximum principle

I am handling the following heat equation with Robin condition by implicit finite difference scheme: \begin{cases} u_t = \frac{1}{2}u_{xx}, (t,x) \in [0,T]\times[0,1], \\ u(t,0) = 0 = u(0,...
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42 views

How to show that the limit function fullfills $|F(z)| = 1$ for $|z|=1$ using the maximum principle?

Let $D\subset \mathbb{C}$ be bounded by $|z| = 1$ and some continuum $C$ in $|z| < 1$ such that $0 \not \in D$. The sequence $(f_n)$ of holomorphic and injective functions is defined on $D$ and ...
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12 views

Are there maximum principles related to the third boundary condition?

I am working on this problem \begin{equation} \begin{cases} u''(s)+\frac{2}{s} u'(s)=R^2 f(u) \quad \text{ for } \eta<s<1, \\ u'(\eta)=0, \ u'(1)+\beta R (u(1)-\bar{\sigma})=0, \end{cases} \end{...
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20 views

Avoidance principle for mean curvature flow

I am reading Mean Curvature Flow and Isoperimetric Inequalities by Manuel Ritoré, Carlo Sinestrari, Vicente Miquel and Joan Porti and I am trying understand the use of an hypothesis. The Avoidance ...
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1answer
43 views

What do the functions in the picture mean?

I'm a freelancer coder and I was coding some equations. But for the equation below, I don't know what they mean? can anybody help me so I can model it? $$\large\delta^*=\arg\{\underset{\delta}{\max}E[...
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1answer
26 views

Maximum principle for the heat operator

Let $M$ be a compact Riemannian manifold. Let $u\in M\times[0,T)\to\mathbb R$ be a smooth function. Let $\Delta$ be the Laplace operator on $M$. Suppose there is a constant $C$ such that $$\Delta u-\...
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24 views

Phragmen-Lindelöf limit problem

I am trying to solve this problem : Let S be a closed sector of $\mathbb{C}$ and $f$ a continuous function which is analytic and bounded in the interior of S. We suppose that $f(z) \to a$, when $z$ ...
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35 views

Dynamic Programming and Hamiltonian problem

Consider the following infinite-horizon optimal control problem for a firm in continuous time. At any moment $t \geq 0$, let $s(t) \in [0, 1]$ be the relative size of the market for the firm’s product....
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1answer
53 views

maximum principle shortest distance between 2 points avoiding a circle , Algebra calculations (optimal Control Course)

Hope you are well and safe The following question is to find the shortest distance between 2 points avoiding a circle between them using the Maximum principle (see photo 1 & photo 2). the ...
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1answer
27 views

Why does a local maximum point imply that the second derivative is zero in this problem?

This is from chapter 6.1 in Introduction to Partial Differential equations (Tveito & Winther, 1st edition). The book states roughly: Let $v \in C^2((0,1)\cap C([0,1]))$ be a function satisfying ...
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1answer
32 views

Extension of solution PDE

Let us consider a non-negative function $u \in C^{0,\alpha}(B_1)$ such that $\Delta u=1$ in the set $\{u>0\}\cap B_1$. Is it true that $$ \Delta u = \chi_{\{u>0\}} \quad\mbox{in }B_1? $$ In ...
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52 views

Given analytical functions such that $|f(z)|=|g(z)|$ on all points of a closed curve, is it true that $\left|\frac{f}{h}\right|=1$ on the curve?

Let $f:\mathbb{C}\to\mathbb{C}$ and $g:\mathbb{C}\to\mathbb{C}$ be analytical functions on all $|z|\leq R$, such that $|f(z)|=|g(z)|$ for all $|z|=R$ and $f,g\neq 0$ for all $|z|<R$, prove that ...
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32 views

An upper bound for the modulus of the derivative of an analytic function in the unit disk (from D. Sarason's “Complex Function Theory”)

I'm having some trouble tackling the following, which appears as an exercise after the Schwarz Lemma part. Exercise VII.17.3. Let $f$ be a holomorphic map of the unit disk $D$ into itself. Prove that:...
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1answer
24 views

Intuition from Boundary Point Lemma (Hopf Lemma)

Consider the classical Boundary Point Lemma: Let $L$ be an elliptic operator. Boundary Point Lemma Suppose $\Omega$ has the interior sphere property and that $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ ...
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33 views

Aleksandrov maximum principle for semiconvex function

Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and convex function $w$ (it's equivalent saying that $u$ is semiconvex if exists ...
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33 views

Greatest value of expression

Let x, y, z are real numbers such that $$xyz + x + z = y $$and $ xz \neq 1$. If the greatest value of the expression $$\frac{2}{x^2+1}-\frac{2}{y^2-1}+\frac{3}{1+z^2}$$ is $\frac{p}{q}$ where $p$ ...
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4answers
47 views

How do I prove the function $f(x,y)=(3−x−y)xy$ has a local maximum?

I have calculated the Hessian matrix for such a function, which turns out to be, H$f(x)$ = \begin{bmatrix}-2y&3-2x-2y\\3-2x-2y&-2x\end{bmatrix} How can one prove that such a function ...
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16 views

Inequalities from uniform ellipticity condition ( Trudinger paper)

I am reading the paper by Neil Trudinger called "Comparison Principle and Pointwise Estimates for Viscosity Solutions" and i don't understand some inequalities. Consider the function $F=F(x,z,r,p)$ ...
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1answer
18 views

Bound on holomorphic functions with a prescribed zero of order k

Suppose $f$ is holomorphic on an open disc $D$ centered at $a$ of radius $R$. Suppose that $|f(z)|\leq M$ on $D$. Suppose further that $f^{(n)}(a)=0$ for $0\leq n<k$. Then show that $|f(z)|\leq M\...
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9 views

Show that $\max_{\bar \Omega} u=\max_{K_{T}} u$. Where $K_{T}$ is the parabolic boundary of $ \Omega$.

Let $\Omega \subset \mathbb R^{n}$ be a bounded, smooth domain. Suppose that for some $T>0$, u is continuous on the closure $ \Omega_{T}:= \Omega \times (0,T)$ and is a smooth, non-negative ...
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57 views

prove you have found all such functions

find all possible entire functions f with the property that $|f(z)|\le2|z|+1$ for all $z\in C$. Prove that you have found all such functions. First of all I am self studying complex analysis so sorry ...
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1answer
47 views

harmonic function with Laplace equation

Let be A an open and bounded set in $\mathbb{R}^{n}$ with border $\partial A$. A function $f:A\to \mathbb{R} $ is harmonic in $A$ if $f$ is continuos in $\bar{A}$ and satisfies Laplace ecuation $\...
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1answer
28 views

Maximum norm of functions; real functional analysis

I am stuck on this exercise. We are studying the boundary value problem \begin{equation} \begin{cases} u(x) - u''(x) = f(x),&x \in (0,1) \\ u(0) = u(1) = 0 \\ \end{cases} \label{prob} \end{...
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55 views

Evans PDE, Problem 5, Chapter 2

I'm taking my first theoretical PDE course in a year and am bashing my head against a rock with this problem. Prove that there exists a constant $C$, depending only on $n$, such that $\max_{B(0,1)}|...
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30 views

Strong Maximum Principle: every sequence $z_n\in U$ converging to $\partial U$ or $|z_n|\to\infty$

Suppose $U$ is an open connected subset of the complex plane, and $f(z)$ is a non-constant holomorphic function on $U$. Suppose there exists $M\geq0$ such that $$ \varlimsup_{n\to \infty} |f(z_n)|\...
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22 views

Let $D$ be domain and $f \in H(D)$ is homomorphic function. Let $E$ is bounded contour for which function $|f(z)|=$ const, such that $E \subset D$

Let $D$ be domain and $f \in H(D)$ is homomorphic function. Let $E$ is bounded contour and closed for which function $|f(z)|=$ const, such that $E \subset D .$ Prove, that in $E$ exist point $z,$ ...
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45 views

Checkered plane

In a game we have a checkered plane we have a horizontal line that marks a "height $0$" (I can not explain properly, I will send a drawing along). In the first part of the game you can, as you prefer, ...
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1answer
64 views

Using the maximum principle on a rectangle to find properties on an infinite strip

Let $S$ be the strip $S=\{z\in \mathbb{C}:\operatorname{Re}(z)\in(0,1)\}$ and let $f:\overline{S}\to\mathbb{C}$ be a bounded and continuous function that is holomorphic on $S$. We have been given a ...
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1answer
35 views

Exercises related to Maximum modulus theorem

and the following is a hint from same book I’d like to know why we should consider $f_n(z)$ instead of $f(z)$. Is there an advantage of considering such auxiliary function? And also on the circle $C$...
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1answer
21 views

Bounding a function using maximum principle

Let $$V=\left\{r\mathrm{e}^{i\theta}:r>0,\theta\in\left(\frac{-\pi}9,\frac\pi9\right)\right\},$$ $f:\overline{V}\to\mathbb{C}$ be continuous and holomorphic on $V$. Show that if $|f(z)|\leq\exp\...
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44 views

Lower and upper bounds for harmonic functions

Is there any theorems to obtain Lower and upper bounds for harmonic functions on a complex domain that contains a segment of the real line, especially in that segment? I found some old maximum ...
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1answer
61 views

Question about Maximum Modulus Principle applied to $|f|+|g|$

Let the functions $f$ and $g$ be holomorphic in $U$ and continuous in $\overline{U}$. Show that $|f(z)| + |g(z)|$ attains its maximum on $\{|z| = 1\}$. Hint: consider the function $h = e^{iα}f + e ^{...
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51 views

Sum of squares of harmonic functions

I have to show $w(x)=\sum_ju_j^2(x)$ has its maximum value on the boundary $\partial\Omega$. The functions $u_j$'s are all harmonic. I think I need to show that $w$ is also harmonic and then I can use ...
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14 views

Weak Maximum Principle as a corollary of Strong Maximum Principle?

I am reading http://www.axler.net/HFT.pdf. The author proves the following : $\Omega$ is an open connected set and $u$ is real-valued and harmonic in $\Omega$. If $u$ attains a maximum in $\Omega$ ...
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17 views

maximum principle for perturbed heat equation

consider an open and bounded set $V \subset \mathbb{R}^n$ and a $T>0$, then define the set $V_{T} = V \times [0,T]$ and the parabolic boundary $\Gamma_{T} = \partial V \times [0,T] \cup V \times \{...
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1answer
55 views

If $ u_t=u_{xx} $ and $u(x,0)=4x(1-x),~x\in [0,1],~~u(0,t)=u(1,t)=0,~t\geqslant 0$, prove $0<u<1$ for $x\in (0,1),~t>0$

Let $u$ be the solution of the heat equation initial and boundary value problem: $$\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2},~~x\in (0,1),~t>0,$$ and: $$u(x,0)=4x(1-x),~x\...
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23 views

Maximum Principle of Laplace equations

$u$ is the $C^2$ solution of $$\begin{cases}\Delta u = 0 &\text{in }\mathbb{R}^d\backslash B_R\\ u = 0 & \text{on } \partial B_R\end{cases}$$ So the problem asks to show that ...
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1answer
90 views

pde, initial boundary question

To be honest I do not know how I should define this question. Apologies Question: Let $u(x,t)$ be continuous on $(-\infty, \infty) \times [0,T]$ and suppose that $$u_t = u_{xx}$$ with $u(x,0) = 0,...
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38 views

Estimative for gradient of harmonic function

I have been trying to solve the following Let $f$ be continuous in $\overline{B}_{R}$. Suppose $u\in C^{2}(B_{R})\cap C(\overline{B}_R)$ satisfies $\Delta u = f$ in $B_R$. Prove that $|\nabla u(0)|\...
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28 views

Discrete maximum principle for for a discrete parabolic operator

While reading a paper on the topic 'Numerical solutions for generalized Black-Scholes equation', It is given that their numerical scheme can be executed explicitly by solving a linear system $\mathbf ...
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27 views

Weak maximum principle for the Dirichlet problem

In Brezis' functional analysis he goes through the weak maximum principle for the Dirichlet problem. Brezis assumes $u\in H^{1}(\Omega)\cap C(\overline{\Omega})$ and then fixes a function $G\in C^{1}(\...
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27 views

Maximum Principle and Uniqueness for Diffusion Equation

I'm interested in proving that the following parabolic problem has a unique solution: $$\begin{cases} \dfrac{\partial y}{\partial t} (t;x)=r\cdot y(t;x)+D\cdot\Delta y(t;x)-c(x;t)\cdot y(t;x),\ \...

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