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

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18 views

Maxima and minima with Hessian matrix

I was looking for maxima and minima conditions and came across this Wikipedia link But I have a problem in my problem sheet where it is asked to prove that if $ f\in C^2 ,grad f(a,b)=0, ...
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1answer
22 views

Zero of holomorphic function

Let $\Omega \subset \mathbb{C}$ be an open set that contains the unit ball $D$ and let $f \in \mathcal{O}(\Omega)$ a non constant map s.t. $|f(z)| = 1$ for all $z \in \partial D$. Show that $f$ has a ...
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1answer
22 views

How does the maximum principle apply to show that $v = 0$ in $U$?

My question concerns one part of the proof of Theorem 3 in §6.5 in PDE Evans. We consider in this section the boundary-value problem $$\begin{cases}Lw=\lambda w & \text{in }U \\ \, \, \, \, ...
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7answers
146 views

If $|f| \le |g|$, does analytic continuation of $g$ imply analytic continuation of $f$?

Let $f,g$ be two holomorphic functions on a domain $D$ such that $|f(z)| \le |g(z)|$ for all $z \in D$. Further suppose that there is an analytic continuation of $g$ to a bigger domain $D'$. Does that ...
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2answers
54 views

Max Mod Principle

I'm stuck with the following exercise: Let $f$ be holomorphic on an open set containing $\bar{D}$, the closed unit disk. Prove that there exists a $z_0 \in \partial D$ such that ...
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1answer
53 views

Is there a proof for the maximum principle without the Cauchy integral theorem?

All the theorems about holomorphic functions seem to rely on the Cauchy integral theorem: Liouvilles theorem about bounded whole functions, the maximum principle, the open mapping theorem for ...
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1answer
21 views

Maximum Principle for the PDE $\Delta u - a^2u=a^2$

I have this Dirichlet probem \begin{align*} \Delta u - a^2u&=a^2\quad\text{on}\;\, \Omega \subset \mathbb{R}^n \\ u&\equiv 0\quad \, \text{ on }\partial \Omega, \end{align*} where $a^2$ is ...
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1answer
41 views

An application of strong maximum principle for harmonic functions

This should be an easy application, since it is given as a remark without proof in Evan's pde book. Let U be connected and u harmonic in U. Then u is positive everywhere in U if u is positive ...
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1answer
16 views

Maximum principle, quantitative version

Is it true that $$ \|f\|_{L^\infty(\Omega)}\leq C\|\Delta f\|_{L^\infty(\Omega)}+C\|f\|_{L^\infty(\partial\Omega)} $$ for all $f\in C^2(\Omega)$ and smooth $\Omega$?
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1answer
55 views

Showing $f$ is constant using (?) the mean value theorem

So I'm working through a packet of old problems and I was wondering if any one could lend me a hand with this one. Let $D$ be an open domain in $\mathbb{C},$ containing the unit disc. Let $f: ...
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1answer
42 views

Maximum principle for the heat equation with Dirichlet conditions

Let us consider the Laplacian operator in a domain $\Omega\subset \mathbb{R}^n$, with Dirichlet boundary conditions. For all $f\in L^2(\Omega),$ we denote by $S(t)f$ the solution of the equation $$ ...
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1answer
55 views

Prelim problem $\sup_{\bar\Omega}|\nabla u|^2\leq C(\sup_{\Omega}u^2+\sup_{\Omega}f^2+\sup_{\Omega}|\nabla f|^2)+\sup_{\partial\Omega}|\nabla u|^2$

This is an old prelim problem. Let $\Omega \subset \mathbb{R}^n$ open and bounded with smooth boundary. If $u\in C^3(\bar\Omega)$ solves $$ -\sum_{i,j=1}^n a_{ij}(x)\,u_{x_ix_j}(x)=f(x)\quad ...
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1answer
19 views

How to show that the Poisson equation has its maximum on the boundary?

$$\Delta f=h, x\in\Omega$$ $$f=F, x\in\partial \Omega$$ where $h$ is $C^1$ function s.t. $0\le h$, $0\le h'$ and the domain is in 3D. then, I want to show that $f$ has its max on boundary. please ...
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1answer
22 views

Why does the inequality hold?

Let $u(x,y), x^2+y^2 \leq 1$, a solution of $$u_{xx}(x,y)+2u_{yy}(x,y)+e^{u(x,y)}=0, x^2+y^2\leq 1$$ Show that $\min_{x^2+y^2 \leq 1} u(x,y)= \min_{x^2+y^2=1} u(x,y) $. We suppose that ...
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0answers
38 views

Maximum principle for weak sub- and supersolutions

This is PDE Evans, 2nd edition: Chapter 9, Exercise 6: Assume that $\underline{u},\bar{u}$ are smooth, sub- and supersolutions of the boundary-value problem $(1)$ in §9.3. Use the maximum ...
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1answer
38 views

Given the holomorphic maximum modulus principle, prove Hopf's lemma

To smooth out my lecture notes, I'm looking for a derivation of Hopf's lemma for harmonic functions $u \colon D \subset \mathbb{R}^2 \to \mathbb{R}$ from the maximum modulus principle (and mean value ...
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1answer
23 views

Maximum modulus of a holomorphic function on a disc within a certain sector

Given the polynomial $$f(z) = az^n + b \qquad (n \geq 2)$$ and a modulus $0 < \rho < 1$, can one find a modulus $0 < r < \rho$ such that there is a point $$w \in \{ |z| \leq r \} \cap \{ ...
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0answers
11 views

Harmonic functions locally null on connected open set

Let $u$ be a harmonic function on $U$ connected open set of $\mathbb{R}^n$ and suppose there is a open set $V\subset U$, such that $u(x)=0$ for every $x\in V.$ Show that $u=0$ in $U$. So, I tried to ...
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1answer
53 views

Proof of the Lindelöf theorem related to the radial limit of an analytic function in the unit disc

Hi I am looking for the proof of this theorem here by Lindelöf: "Suppose $\Gamma$ is a curve with parameter interval $[0,1]$, such that $|\Gamma(t)| < 1$ if $t < 1$ and $\Gamma(1)=1$. If $g \in ...
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0answers
29 views

Interior of a closed curve

I'm working through a proof that contains this particular argument which I think is highly non-trivial but no justification is given - the context is complex analysis and the proof is of Lindelof's ...
2
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1answer
65 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 ...
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1answer
32 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 ...
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1answer
37 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. $$-\Delta u(x)=f(x), x \in \Omega \\ u(x)=g(x), x \in \partial{\Omega}$$ ...
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0answers
23 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 ...
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1answer
28 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
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1answer
36 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 ...
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0answers
13 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 ...
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0answers
35 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)| ...
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1answer
42 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 ...
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0answers
64 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 ...
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1answer
28 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
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1answer
54 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 ...
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2answers
52 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 ...
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0answers
38 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 ...
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1answer
80 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 ...
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1answer
32 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 ...
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1answer
25 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
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1answer
76 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 ...
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0answers
29 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 ...
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2answers
43 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 ...
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0answers
113 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 ...
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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?
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4answers
72 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.
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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 ...
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1answer
40 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 ...
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1answer
74 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. $$\forall k=0,...,N. \ f^{(k)}(0)=0 \implies\exists M=const . \ \forall z.|z|\lt1:|f(z)|\le M|z|^{N+1}$$
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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 $4$ ...
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2answers
61 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
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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
108 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, ...