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

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24 views

Periodic boundary conditions Poisson

Given the Poisson eq. $\Delta u = -f$ , $u \in H^{1}(\Omega)$ , where in 1d we have $\Omega = [0,1]= S^{1}$ that has periodic boundaries and given we have mean value $\int_{S^{1}} u dy= u(x)$, what ...
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26 views

Maximum Principle variation.

Let $f$ be a non-constant holomorphic function in a bounded open connected set $\Omega$ in $\mathbb{C}$. Let $M:=\lim \sup_{n \to \infty} |f(z_n)|$ for every sequence ${z_n}$ in $\Omega$ which ...
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2answers
48 views

Uniform convergence of n-fold composition using Schwarz lemma

Let $f$ be an analytic function mapping the unit disk $\mathbb D$ to itself with $f(0) = 0$ and $|f'(0)| < 1$. Let $f^{n} = f \circ f \circ \dots \circ f$ be the function obtained by composing $f$ ...
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0answers
14 views

Bound on Rayleigh quotient on $H^{1}_{\text{per}}[0,1]^{n}$

Let $Q=[0,1]^{n}$. Define a $Q$-periodic function as follows: A function $f:\mathbb{R}^{n}\to\mathbb{C}$ is said to be $Q$-periodic if for any $x\in\mathbb{R}^{n}$ and any $z\in\mathbb{Z}^{n}$, it ...
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1answer
57 views

Exponential boundedness for PDE

Suppose we have $\kappa$ bounded and the following equation: $${\partial^2\kappa\over\partial t\partial\theta}=\kappa^2{\partial^3\kappa\over\partial\theta^3}+2\kappa{\partial\kappa\over\partial\...
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1answer
43 views

Maximum and minimum modulus principle

Let $U\subset \mathbb C$ be a bounded domain and $f:\overline{U}\to\mathbb C$ continuous and holomorphic $U$. Show that $|f(z)|\leq\max\{|f(w)|:w\in\partial U\}$ for all $z\in U$. Show that ...
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1answer
36 views

Maximum principle and open mapping theorem

Let $f:B_2(0)\to\mathbb C$ be holomorphic with $f(1)=1$ and $f(-1)=-1$. Show that there is a $z\in B_2(0)$ and a $\varepsilon>0$ s.t. $f(z)=1-\varepsilon$. Is it reasonable to say that since $|f|$...
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26 views

A comparison principle for a nonlinear parabolic PDE

We know the following comparison principle holds for the diffusion equation: Suppose that $u(x,t)$ and $v(x,t)$ satisfy \begin{equation} \begin{cases} u_t\ge \Delta u+F(x,t,u), \ \ &x\in\Omega,\ \...
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22 views

Exercise for proving maximum modulus principle

I have this exercise: let $D$ be a domain and $a \in D$ such that $D'(a,r)$ is a subset of $D$ (here $'$ for closure). suppose that $f$ is holomorphic on $D$ and let $A = \max|f(z)|$ with $|z - a| = r$...
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57 views

Optimal control with state-dependnet solution

I'm trying to solve the following control problem $$ \begin{eqnarray*} \max & & \int_{0}^{T}\sum_{i=1}^{2}-c_{i}(x_{i}-u_{i})^{+}\\ s.t. & & \dot{x}_{i}=\alpha_{i}-\beta_{i}\min(u_{...
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1answer
104 views

Prove an inequality using complex analysis

If $f:\mathbb{D}\rightarrow\mathbb{D}$ is holomorphic then prove that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq|f(z)| \leq\frac{|f(0)| + |z|}{1 - |f(0)||z|} $$ I have been wracking my brain for ...
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47 views

The Maximum Modulus Principle Applied to the Proof of Schwarz Lemma

I am using the following statement of the Maximum Modulus Principle: Theorem: Let $G$ be a region and let $f$ be holomorphic on $G$. Suppose $\exists~ a \in G$ such that $|f(z)| \leq |f(a)| ~ ~\...
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22 views

Taylor expansion in proof of weak maximum principle

Picture below is part of proof of weak maximum principle. On the red line ,I don't know how to use the Taylor expansion to get $-u''(x_0) \le 0$. I think the Taylor expansion of $u(x)$ at $x_0$ is $$ ...
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17 views

Diffusion/Heat equation, weak maximum principle

Consider the problem: $$u_t -div(A(x) \nabla u) +a(x) u = f $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) $$ and its variational formulation $$ \lt \dot{u(t)},v \gt_* + B(u(t),v;t) =(f,v)_{L^2} \quad \forall ...
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Strong maximum principle for heat equation

Let $M$ be a closed Riemannian manifold. If $u \in H^1(0,T;L^2) \cap L^2(0,T;H^1)$ is a weak solution of $$u_t - \Delta u = f$$ $$u(0) =u_0$$ where $f \in L^2(0,T;L^2)$ with $f(t,x) \geq 0$ a.e. and $...
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1answer
14 views

Sign of coefficient in the weak maximum principle

( Weak maximum principle) Let $u \in C^{2}(\Omega)\cap C(\bar{\Omega})$ satisfies $Lu \geq 0$ with $c(x) \leq 0$ Em $\Omega$. Then $u$ attains in $\partial \Omega$ its nonnegative maximum in $\bar{\...
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1answer
53 views

On the curve $y =\frac{ 2x - 1} {x - 1}$ , $x > 1$ drag tangent such that area of right triangle bounded by this tangent and the axes is minimal .

On the curve $y =\frac{ 2x - 1} {x - 1}$ , $x > 1$ drag tangent such that area of right triangle bounded by this tangent and the axes is minimal . What is the minimum value of this area ? My work:...
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48 views

Dirichlet problem of second-order elliptic PDE on unbounded open set

Set $\Omega$ an unbounded open set in $\mathbb{R^n}$, $L$ is the elliptic operator $Lu=-a_{ij}D_{ij}u+b_i D_i u+cu$ which satisfies the uniform elliptic condition. Also $c \ge 0$, $u \in C^2(\Omega) \...
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1answer
32 views

$L^\infty$ bound on solution of $u_t - \Delta u + cu = f$ and dependence on $c > 0$

Let $u$ be the weak solution to $$u_t - \Delta u + cu = f$$ $$u(0) = 0$$ with zero Neumann data, on a bounded domain and time interval $[0,T]$. Here $c > 0$. If $f$ is in $L^\infty$ in time and ...
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1answer
30 views

Why $\frac{\partial u}{\partial \nu} \ge 0$ at the maximum point?

In Evans' book "Partial Differential Equations", on page 348, it is mentioned that the outer normal derivative at the maximum point (if it exists) $\displaystyle \frac{\partial u}{\partial \nu}(x_0) \...
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1answer
15 views

Analytic function on the annulus $A = \{z : 1 \le |z| \le 4\}$ onto $B = \{z : 1 \le |z| \le 2\}$ s.t. $C_1 \to C_1$, $C_4 \to C_2$?

Question: Does there exist an analytic function mapping $A = \{z : 1 \le |z| \le 4\}$ onto $B = \{z : 1 \le |z| \le 2\}$ and taking $C_1 \to C_1$, $C_4 \to C_2$, where $C_r$ is the circle of radius $r$...
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54 views

(without Phragmén-Lindelöf) $f$ is of exponential type and bounded on the real axis, then there exists $C>0$ such that $|f(x+iy)|\leq Ce^{\tau |y|}$

Question: Without using Phragmén-Lindelöf, show that if $f$ is of exponential type and uniformly bounded on the real axis, then there exists $C>0$ such that $|f(x+iy)|\leq Ce^{\tau |y|}.$ ...
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1answer
40 views

How can we compute the maximum of $|\zeta(z)|$ on the square with vertices $2,3,3+i$ and $2+i$?

By the maximum modulus principle, since $\zeta(z)=\sum_{n=1}^{\infty}\frac{1}{n^z}$, with $\Re z>1$, is analytic inside the square Q of vertices $2,3,3+i$ and $2+i$, and continuous on Q, $$\max_{z\...
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0answers
21 views

Inequality among Hessians

I know that this could be a stupid question but I would like to be sure on this point. I need to study the Maximum Principle on Petersen's Riemannian Geometry. This is the first Lemma. Let $ (M,...
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50 views

Maximum Principle for Nonlinear Heat Equation

For a function $f(x,t)$ I'm aware of a fairly strong result about the PDE: $\sigma^2 f_{xx} + f_t = 0,$ $f(x,0) = h(x)$, which guarantees that any local maxima of $f$ or any of its derivatives in $...
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1answer
47 views

Inhomogeneous Diffusion Equation - Maximum Principle

Let $u$ be a solution of the inhomogeneous diffusion equation $u_{t} = ku_{xx} + f$ on the rectangle {${0 < x < l, 0 < t < T}$} with initial data $\phi$ and boundary data $g$. Prove that ...
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36 views

Maximum Principle of the Diffusion Equation

Consider a solution of the diffusion equation $u_{t} = u_{xx}$ in {$0\leq x\leq l,0\leq t<\infty$} a) Let $ M(t)$ = the maximum of $u(x,t)$ in the closed rectangle {$0\leq x\leq l,0\leq t<T$}....
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0answers
47 views

How to classify harmonic functions on the punctured disk without the Schwartz reflection principle?

I am working through old qual problems at the University of Minnesota and am trying to find an alternate solution to the following problem. Determine all continuous functions on $\{ z : 0 < \left| ...
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1answer
47 views

Harmonic function zeros on open subset

Let $h$ be an harmonic function on $\Omega\subset\mathbb{C}$. Let $A\neq\emptyset$ an open subset of $\Omega$ such that $h\mid_A\equiv 0$. Prove $h\mid_\Omega\equiv 0$. I thought on taking a point ...
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31 views

Infinite number of points with modulus 1

Let $f:D(0,1)\rightarrow \mathbb{C}$ be analytic such that $f(0)=1$. Prove that there are an infinite number of distinct points $z_{n}$ in $D(0,1)$ such that $|f(z_{n})|=1$ Remark: $D(0,1)$ is the ...
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27 views

maximum principle in a harmonic and positive function

$\begin{equation}u=\frac{1-(x^2+y^2)}{(1-x^2)+y^2}\end{equation}$ is harmonic and positive in $x^2+y^2<1$ because $u=0$ in $x^2+y^2=1$ except in $(1,0)$. Is the maximum principle valid for $u$?. ...
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0answers
18 views

Maximum principle for heat equation with Neumann boundary condition

please help me with this problem. Let $u\in C^2_1([0,l)\times (0,T))\cap C^0([0,l]\times [0,T])$ be a solution of the heat equation $u_t-u_{xx}=0$ such that $u_x(0,t)=0$. Prove that the maximum of $u$ ...
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108 views

Prove that $|f|\leq 1$ whenever $|x|\leq 1$.

Let $f :\mathbb{R}^2\rightarrow \mathbb{R}^2 $ be everywhere differentiable such that the Jacobian is not singular at any point in $\mathbb{R}^2$. Assume $|f|\leq 1$ whenever $|x|=1$. Prove that $|f|\...
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1answer
41 views

Maximum modulus pronciple

The problem I had was: If $f$ is holomorphic on $U \subseteq \mathbb{C}$ and $\exists z_0$ in $D(0,1)$ (which is fully contained in $U$) such that $|f|$ has a local maximum at $z_0$ then $f$ is ...
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1answer
50 views

Some basic questions regarding the Maximum Principle for Harmonic Functions,

I've seen a uniqueness argument come up a few times but I don't really understand it. The argument is that if two harmonic functions $f$ and $g$ agree on the boundary of some domain, then $f-g$ or $g-...
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1answer
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Why does $v(z)=\text{Im}\left[\left(\frac{1+z}{1-z}\right)^2\right]$ not contradict maximum principle?

Since $\left(\frac{1+z}{1-z}\right)^2$ is holomorphic in $\mathbb{D}$, its imaginary part is harmonic, and we have $$\underset{r \uparrow 1}{\lim}v(re^{i\theta})=0 \quad \forall ~ \theta \in [0, 2\pi)$...
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4answers
62 views

Showing $f(z)$ is constant when $Im(f^2(z)-f(z))=0$

I don't have the full question but I'm assuming $f(z)$ must be entire for this to occur. Also note that $u>1$ where $u$ is the real part of $f(z)$. If we solve the given eqn then we are left with ...
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2answers
78 views

An entire function such that $if(z)= \bar f(z)$ for all $z$ is constant

Let $f$ be an entire function such that $if(z)= \bar f(z)$ for all $z\in\Bbb C$. Show that $f$ is the constant function. I didn't understand it but I took some notes which seem like gibberish to me ...
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1answer
34 views

Prove $v$ is harmonic and $\lim_{r \uparrow 1} v(re^{i\theta}) = 0$

Prove that if $v(z) = \mathrm{Im}[(\frac{1+z}{1-z})^2]$, then $v$ is harmonic on the unit disc and $\lim_{r \uparrow 1} v(re^{i\theta}) = 0$ for all $\theta \in [0,2\pi)$. Explain why this does not ...
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0answers
38 views

Prove or disprove the following:$\{p_n\}$ converges uniformly to $f$ on $\{z\in \mathbb{C}: |z|=a\}$

Let $A=\{z\in \mathbb{C}: a<|z|<b\}$. Suppose $f$ is continuous on the boundary of $A$. Suppose there is a sequence of holomorphic polynomials $\{p_n\}$ that converges uniformly to $f$ on $\{z\...
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57 views

If $f$ is a non-constant analytic function on $B$ such that $|f|$ is a constant on $\partial B$, then $f$ must have a zero in $B$ [duplicate]

The question goes like this: Let $B=\{z\in C: |z|<R\}$ and $\partial B$ is the circle of radiu $R$. If $f$ is a non-constant analytic function on $B$ such that $|f|$ is a constant on $\partial ...
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1answer
59 views

Uniqueness for a mixed Dirichlet/Neumann problem

Let $L$ be a uniformly elliptic operator with $c \equiv 0$ on a bounded domain $U \subset \mathbb{R}^n$ with $C^2$-boundary, and let $\partial U = S_1 \cup S_2$. Suppose $u \in C^2 (\bar{U})$ ...
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1answer
27 views

Hyperplane problem by Lagrange multiplier method

Solve the problem using Lagrange multiplier method. Find the point that belongs to both hyperplanes xT c = β and xT d = γ which is closest to the origin.
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1answer
20 views

Holomorphic function vanishes on the boundary

Let $f \in \mathcal{O}(D) \cap \mathcal{C}(\overline{D})$, $D$ - a bounded region in $\mathbb{C}$. Suppose that $f(z)=0$ for $z \in \partial D$. Prove that $f(z)=0$ for all $z \in D$. At first I ...
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2answers
34 views

Maximum minimum of $x^2$ on the open interval $-1<x<1$

The question goes like this: Does the function $f(x)=x^2$ have a maximum on the open interval $(-1,1)$ ? And a minimum? Explain Not exactly sure how to give a proper answer really. For me it makes ...
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1answer
49 views

Expected value of a maximum of two draws compared to expected value of each

I am no mathematician, so I apologise in advance for not explaining myself properly, and for asking something that is probably utterly obvious for most of you. The question has to do with the ...
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0answers
76 views

How can I conclude that f(z) is constant? [duplicate]

I'm struggling a bit to arrive at the conclusion that $f(z)$ is a constant. Suppose $f(z)$ is holomorphic on and inside the unit disk, and that it has no zeroes on the interior. Also, assume that $|...
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1answer
32 views

Prove that if $K$ is polynomially convex, then $\mathbb{C}-K$ is connected.

Prove that if $K$ is polynomially convex, then $\mathbb{C}-K$ is connected. My Try: Suppose $K$ is polynomially convex. Then $K=\hat{K}$. So, given $z\in \mathbb{C}-K$, there is a polynomial $p(z)$ ...
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1answer
31 views

Weak maximum principle for the Telegrapher's equation (?)

I'm struggling with the following: Let a bounded connected domain $U \subset \mathbb{R}^2$ be given. Prove the weak maximum principle for the equation $$u_{xx} + 2u_{yy} + u_{y} =0 \qquad (*)$$on $...
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2answers
53 views

$f$ is a monotone increasing, but not necessarily continuous, on $\mathbb{R}^n$, $A$ is compact. Is $f$ always has a maximum on $A$?

Call a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ nondecreasing if $x,y \in \mathbb{R}^n$ with $x \geq y$ implies $f(x) \geq f(y)$. Suppose $f$ is a nondecreasing, but not necessarily ...