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

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26 views

Maximum principle easy proof

Is there an easy proof of the maximum principle from the variational formulation in $\Bbb R^d$, without using Green functions? Variational formulation: $$ \forall v\, \text{ smooth, }\, \int \nabla ...
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1answer
93 views

How do I apply this maximum principle?

I have the maximum principle: $$\text{If } \psi\geq 0 \text{ on }\Gamma. \text{Then }L\psi\geq0\text{ implies } \psi\geq 0 \text{ in } \bar{D},$$ where $D=(0,1)\times (0,T], \Gamma$ is parabolic ...
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1answer
56 views

Why isn't there a general comparison principle for higher order equations?

I am studying elliptic equations. For second order equations (linear and nonlinear), comparison principle has many applications, e.g. to show uniqueness of the (weak) solutions, to construct super- ...
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2answers
20 views

Maximum value estimation

Let $f$ be an analytic function that is not zero at $\{z:|z|<2\}$. Show that for every natural number $n$: $$\max_{|z|=1}|f(z)-\frac{1}{z^n}|>1$$ I know that ...
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0answers
24 views

Maximum principle in PDE [duplicate]

I was told Maximum principle is a common method in proving uniqueness of the solution to certain PDE. Could anyone explain 1) How does Maximum principle work in the context of PDE theory? 2) How ...
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1answer
52 views

How to show that $u(x,t)\le\pi^3-1+\sin(x)$ (heat equation)

Let $u(x,y)$ be a continuous solution of \begin{cases} u_t=u_{xx}+\sin(x) & 0<x<\pi,&t>0, \\[3ex] u(0,t)=u(\pi,t)=0 & t\ge0, \\[3ex] u(x,0)=4x(\pi-x)\quad&0\le x ...
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0answers
14 views

Maximum Principle - Proof

We want to show the maximum principle for a function $f = f(x,t)$ on a n-dimensional hypersurface $M,$ that is, (Corollary) Let $f = f(X,t)$ be a function on M, let $\vec{a}$ be a vector field on ...
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1answer
33 views

Maximum principle type problem

Suppose $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ is sufficiently smooth and is such that $(1/2) u_{xx} + u_{xy} + 2u_{yy} = 0$ in a ball $B$ centered at the origin. Must $u$ attain a maximum inside ...
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1answer
36 views

derivative of expected value of maximum of two stochastics variables (iid)

I need to optimize an expected value of a maximum value for $q$. The problem has three variables, $q$ is a constant and $D_1$ and $D_2$ are stochastic variables with pdf $f(x)$ and cdf $F(x)$. The ...
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1answer
66 views

$L^\infty$ bound on solution of $u_t -\Delta u =f$ where $f \in L^\infty$ and $u_0 \in L^\infty$??

Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega)$ and consider $$u_t - \Delta u = f$$ $$u(0) = u_0$$ with some boundary conditions. Consider the weak form ...
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0answers
24 views

Maximum and Minimum principle of Laplace equation

I want to prove that if $u=u(x, y, t)$ is a solution of the equation: $$\frac{\partial u}{\partial t} = \Delta u,\;\;\;\;\;\;\ where \; \Delta u = \frac{\partial ^2}{\partial x^2} + \frac{\partial ...
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2answers
261 views

Time Derivative of PDE solution

Consider the PDE $$ \frac{\partial \psi}{\partial t}(t,x)=\psi''(t,x)+2\psi\psi'(t,x)-\tilde V'(x)~ on~ [0,D/2]\times(0,\infty)\\ \psi(0,t)=0, \psi(D/2,t)=-k, \\\psi(\cdot,0)=\psi_0 $$ for some ...
1
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1answer
32 views

Constant function with maximum modulus [duplicate]

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
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1answer
65 views

Minimum Modulus Principle for a constant fuction in a simple closed curve

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
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2answers
38 views

Usage of Maximum Modulus Theorem?

I have a function $f(z)=ze^z$, I want to find the maximum value of $|f(z)|$ as $z$ varies over the region $D=\{x+iy: x^2+y^2\leq 4, x,y\geq 0\}$. I was thinking that this is case where I can use the ...
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1answer
66 views

Suppose $u \in C^2(\Omega)\cap C^1(\bar\Omega)$ satisfies $ -\Delta u+u^3=0$ in $\Omega $

Suppose $u \in C^2(\Omega)\cap C^1(\bar\Omega)$ satisfies $$ -\Delta u+u^3=0 \quad\text{ in } \Omega $$ $$ \frac{\partial}{\partial\eta}u+\alpha u=\varphi \quad\text{ on }\partial \Omega $$ for ...
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1answer
31 views

Proving zeros inside a disk using Maximum Principle?

Let $f$ be non-constant analytic in a neighborhood of the closed unit disk such that $|f|=1$ on the unit circle. Show that $f$ has a zero inside the unit disk. It has been suggested to argue this by ...
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0answers
38 views

Question about the maximum principle for the Laplace's equation

Maximum principle for the Laplace's equation: $$\nabla^2 u= \Delta u=u_{xx}+u_{yy}$$ $$\Delta u=f(x,y) \text{ Poisson }$$ Problem with boundary values of the form Dirichlet: $$\left.\begin{matrix} ...
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1answer
48 views

Find the maximum possible value

Help me to find the maximum value of $T$ with $x, y, z \in \Bbb{R_+}$ $$T=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}$$ Thanks :D
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2answers
73 views

Maximum value question of three $xyz+xy+yz+zx$.

I have tried am-gm inequality,i am getting that $xyz$ is greater than $36.9$. I tried hit and trial,but it is of no use also. Could anyone give a definite process?
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1answer
36 views

maximising profit chi-square distribution

A bakery sells rolls in units of a dozen. The demand X (in 1000 units) for rolls has a gamma distribution with parameters alpha=3 and theta=0.5. It cost 40 cents to make a unit that sells for $1 on ...
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1answer
38 views

Showing an entire function f(z) is monomial

Given f is entire and satisfying $|f(z)| \leq 3|z|^{\alpha}$, show that $f(z) = cz^{\alpha}$ for some constant $c$ if $\alpha$ is a positive integer, and $f(z) = 0$ if $\alpha$ is not an integer. I ...
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3answers
137 views

Find the maximum value of $abc$

$a,b,c$ are three positive real numbers such that $ab+bc+ca=12$. Then find the maximum value of $abc$
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0answers
12 views

Weak version of Max. Mod. principle

Let $f$ be holomorphic in a domain $U$. Let $a\in U$. Show there cannot exist $\varepsilon > 0$ such that $D(a,\varepsilon) \subset U$ and $|f(z)|<|f(a)|$ for all $z\in D(a,\varepsilon)$ ...
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0answers
12 views

Holomorphic function extended to entire polydisc

How can I show that every holomorphic function in the border of a polydisc $\Delta \subset \mathbb C^n, n>1,$ has a extention to entire $\Delta$? I know this is just a consequence of Maximum ...
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1answer
76 views

Elliptic PDE - max principle

The maximum principle for elliptic PDEs is established for the nondivergence form as in http://www.ann.jussieu.fr/~frey/cours/UdC/ma691/ma691_ch3.pdf. But what if we are dealing with the divergence ...
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1answer
39 views

Strong maximum principle

Let $S^{n-1}$ denote sphere in $\mathbb{R}^n$ and let $D$ denote open unit disk in $\mathbb{R}^n$. Let $f$ be homeomorphism of $S^{n-1}$ onto itself. Let $F$ be its harmonic extension given by Poisson ...
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2answers
322 views

Intuition Behind Maximum Principle (Complex Analysis)

Let $D$ be an open set in the complex plane and $f(z)$ be a non-constant holomorphic function on D. Then $|f(z)|$ has no local maximum on D. I can follow the proof fine - usually if I don't ...
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0answers
105 views

How to prove comparison principle for parabolic PDE (nonlinear)

Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for $x > 0$ and $F:(0,\infty) \to (0,\infty)$ continuous and increasing with $F(0) = 0$. Consider the PDE $$u_t = \Delta F(u) ...
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1answer
43 views

A maximum principle

Suppose that $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary $\partial\Omega$. Consider the elliptic boundary value problem for $\phi=\phi(x)$, $x\in\mathbb{R}^n$: ...
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0answers
17 views

need help: find a maximum value of matrix-based equation using differential

I have a problem to derive this equation to find the matrix that gives its original equation a maximum value. here is my equation: *find matrix Y that could maximize the F function. the function F is ...
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2answers
63 views

Maximum of linear combination

I have an range like this: $$x + 2y \leq 40$$ $$4x + 3y \leq 120$$ $$x \geq 0, y \geq 0 $$ I made an plot using wolfram alpha. Now I have a linear combination $$4x+5y$$ and I want to find the maximum ...
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0answers
67 views

strong maximum principle for $u$ such that $u'' \geq c(x)u$

I want to prove that if $I$ is an interval in $R$, $c(x) \geq 0$ is continuous and $u \leq 0$ is $C^2$ then if for all $x$ $$u''(x) \geq c(x)u(x)$$ then the strong maximum principle holds for $u$, ...
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1answer
74 views

sequence uniformly convergent on the boundary of a bounded set in $\mathbb{C}$

Let $(f_n)$ be a sequence of functions which are analytic on a bounded region $A\subset\mathbb{C}$ and continuous on the closure $Cl(A)$. Suppose that the sequence is uniformly convergent on the ...
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2answers
66 views

Generalization of the Maximum Principle for elliptic linear PDEs

This question came up when I was trying to solve linear elliptic PDEs. Let $R$ be an open domain and $L$ be a linear elliptic operator such that $$ L \; u = 0 \; \mathrm{on \; R}, \;\; B \; u = f \; ...
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2answers
33 views

Showing $ u =0 $ for a particular laplace equation.

Given open $\Omega\subset \mathbb{R}^n $ with smooth enough boundary, if for $x_0 \in \Omega$ there exists $ u \in C^2(\Omega\setminus\{x_0\}) $( we may as well take $u$ as smooth as we want) that ...
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1answer
35 views

Using the Maximum Principle to prove things about solutions to inhomogeneous Diffusion Equations

I am trying to teach myself PDEs, and I am working on the following problem from Strauss: (a) If $u_{t}-ku_{xx}=f$, $v_{t}-kv_{xx}=g$, $f \leq g$, and $u\leq v$ at $x=0$, $x = l$, and $t = 0$, prove ...
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1answer
26 views

Average temperature of an object with a non-uniform temperature distribution [closed]

We place some solid three-dimensional object on a planar surface that heats the object until some equilibrium situation is reached (the object is also cooling and radiating heat in the air) where any ...
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2answers
58 views

Inequality involving max is confusing me.

I am trying to understand one line in a derivation here. Simply put, the statement is that: $$ max\{\theta \ f_1(x) + (1-\theta) \ f_1(y) \ , \ \theta \ f_2(x) + (1-\theta) \ f_2(y)\} \leq \theta \ ...
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0answers
71 views

Short time existence and Maximum Principle for an explicit parabolic PDE

Consider the following PDE: $\frac{\delta}{\delta t}\psi=\psi^{''}-2\psi^{'}\psi+V' \ on \ [0,D]\times(0,\infty) \\ \psi(0,t)=0, \ \psi(D,t)=-k, \ (k\in\mathbb{N}) , \ \textrm{given } t>0 ...
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2answers
34 views

Finding maximum and minimim of function on an interval. Are there multiple ones?

I found myself stuck at such basic problem. If you're to calculate local maximum and minimum on closed interval, $\langle a, b\rangle$, the $a$ and $b$ may as well be the maximum and minimum points of ...
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1answer
47 views

How to ensure extreme? — using Extreme Value Theorem

I think it's a simple question. How can I ensure the existence of an extreme (maximum or minimum) using the Extreme Value Theorem / Weierstrass theorem? For example, this multivariate case: $$ ...
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1answer
108 views

Evans PDE chapter 2 problem 4

Problem is Give a direct proof that if $u \in C^{2}(U) \cap C(\overline{U})$ is harmonic within a bounded open set $U$, then $\max_{\overline{U}} u =\max_{\partial U} u$. What I think is ...
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1answer
40 views

Uniqueness of variant laplace pre

Here's problem. Let $U$ be a bounded domain in $\mathbb{R}^{n}$ and $\vec{b} : \mathbb{R}^{n} \to \mathbb{R}^{n}$ and $g: \mathbb{R}^{n} \to \mathbb{R}$ be continuous. Show that there can be at ...
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0answers
120 views

Maximum principle question in partial differential equation

Problem is Let $U$ be a bounded domain in $\mathbb{R}^{n}$ and $\vec{b} : \mathbb{R}^{n} \to \mathbb{R}^{n}$ and $g: \mathbb{R}^{n} \to \mathbb{R}$ be continuous. Show that there can be at most ...
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1answer
80 views

maximum of a holomorphic function over a compact subset of a bounded region

If $\Omega$ is a bounded region in $\mathbb{C}$ such that $\mathbb{C}\setminus\Omega$ is connected and $K$ is a compact subset of $\Omega$, show that $\hat{K}_{\Omega} = \hat{K}_{\mathbb{C}}$. ...
3
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1answer
146 views

zeros of a function holomorphic in the closed unit disc

Let $f$ be a holomorphic function in a neighborhood of the closed unit disc $\{z \in \mathbb{C} : |z| \leq 1\}$, and suppose that $\Re{(\bar{z}f(z))} > 0 $ when $|z| = 1$. Prove that $f$ has ...
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1answer
299 views

Problem 6.6-12 of Evans' PDE

Can please somebody tell me, how solve this problem ? We say that the uniformly elliptic operator $$Lu\ =\ -\sum_{i,j}a^{ij}u_{x_ix_j}\ +\ \sum_{i=1}^nb^iu_{x_i}\ +\ cu$$ satisfies the weak maximum ...
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0answers
54 views

Pontryagin principle: does the abnormal multiplier define a minimum

The Pontryagin principle PM provides the necessary condition for a local minimum of the functional $ J(u)=\int L(x(t),u(t))dt \\$ subject to: $\dot x = f(x(t),u(t)) \ \ \ \ x(t0)=x0, \ \ ...
0
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1answer
80 views

Diffusion Equation Maximum Principle

Does the general diffusion equation (http://en.wikipedia.org/wiki/Diffusion_equation) satisfy the maximum principle i.e. is it's maximum value obtained on teh boundry of the region?