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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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If $f:D\to D$ is analytic, bijective and $f(0)=0$, then $f(z)=cz$ for some $|c|=1$?

Let $D=\{z\in\mathbb{C}:|z|<1\}$ and $f:D\to D$ an analytic, bijective function with $f(0)=0$. Is it true that $f(z)=cz$ for some $c\in\mathbb{C}$ with $|c|=1$? I think it can be proved using ...
Nah's user avatar
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Convergence in comparison principles of parabolic pdes

I assume that the following equations all have sufficiently smooth strong solutions. I have demonstrated that the solution to the (imaginary-time) Schrödinger equation: \begin{equation*} \begin{...
J.J.Zou's user avatar
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Theorem 2.12 in Lieberman's Second Order Parabolic Differential Equations

In Theorem 2.12 of Liberman's textbook the following result is established. Consider a space-time domain $\Omega\subseteq \mathbb{R}^n \times \mathbb{R}$ ($\Omega$ is not necessarily of cylinder shape ...
Tibeku's user avatar
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The function $u$ reaches its maximum on the boundary [duplicate]

my problem is: Let $\Omega \subset \mathbb{R}^2$ be a bounded domain and $v \in C^2(\Omega) \cap C^1(\overline{\Omega})$ be a solution of $$ v_{xx}+v_{yy}=-2 \quad \text{in} \quad \Omega $$ $$ v = 0 \...
the topological beast's user avatar
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Numerically solving Pontryagin's Maximum Principle

I am trying to understand why when solving this boundary problem in matlab is significantly impacted by the $A_0$ matrix. The optimal control problem I am trying to solve reads: $$ \begin{aligned} ...
zzgsam's user avatar
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Maximum principle for reaction diffusion equation?

Consider the heat equation $$u_t = u_{xx} \;\; \text{ for } \;\; x\in \Omega, t \in [0,+\infty) \,. $$ The strong maximum principle states that if the solution $u$ attains its maximum in the interior ...
900edges's user avatar
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Bi-analytic bijections on $D(0,1)$

Let $f: D(0,1) \rightarrow D(0,1)$ be a bianalytic bijection such that $f(0)=0$. Show that exists some $\omega \in S^1$ such that $$(\forall z \in D(0,1)) : f(z)=\omega z$$ As $f$ is analytic, $$g(z)...
J P's user avatar
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2 votes
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31 views

Phragmen-Lindelof type argument to show entire function identically zero [duplicate]

I want to show that if an entire function $f$ satisfies the bound $|f(z)| \leq 1/|\operatorname{Im}(z)|$ for all $z \in \mathbb{C}$, then it must be identically zero. I know this sort of result is ...
Ari Krishna's user avatar
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Parabolic maximum principle on the whole space

I am aware of the following maximum principle for the heat equation: Let $T \in (0, \infty]$ and $u$ satisfying \begin{align} \partial_t u - \Delta u = 0 &\quad \text{on }\mathbb R^n \times (0, T]...
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How to prove the maximum principle about the first eigenvalue?

Let $\Omega\subset\mathbb{R}^{n}$ is a bounded domain. Let $u\in C^{2}(\Omega)\cap C(\bar{\Omega})$ satisfy $$-\Delta u+c(x)u\leqslant 0\ \ \text{in}\ \ \Omega.$$ We can assume $c(x)$ is sufficiently ...
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Proving $\frac{\sin\pi z}{z}$ is bounded in the punctured unit disk.

I want to prove that $f(z)=\frac{\sin \pi z}{z}$ is bounded in the unit disk. It has a removable singularity in $z=0$. This is because $\lim_{z\rightarrow 0}\frac{\sin \pi z}{z}=\frac{1}{\pi}$. From ...
muhammed gunes's user avatar
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A maximum principle for a linear operator with singular term

It is well know that if $\Omega$ is a smooth boundend domain in $\mathbb{R}^N$ and if we have $$\mathcal{L} u = -\Delta u + c(x) u$$ with $c(x) \ge 0$ in $\Omega$ and $c \in L^\infty(\Omega)$ (or $C^{...
Rodolfo Ferreira de Oliveira's user avatar
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2 answers
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Maximum principle of strong solution of linear parabolic equation in $\mathbb{R}\times [0,T]$

Suppose that there is a strong solution $u(x,t)\in W^{2,1}_{2,loc}(\mathbb{R}\times [0,T])$ solving the linear parabolic equation $$-\partial_t u +\partial_x^2 u +b(x,t)\partial_x u+c(x,t) u =0\quad\...
mnmn1993's user avatar
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$f:\mathbb{C} \to \mathbb{C}$ analytic with $f(z)=f(z+a)=f(z+b)$

Let $a,b \in \mathbb{C}$ linearly independent over $\mathbb{R}$ and $f: \mathbb{C} \to \mathbb{C}$ analytic with $f(z)=f(z+a)=f(z+b)$ for all $z \in \mathbb{C}$. Show that f is constant. WLOG, I'll ...
J P's user avatar
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2 answers
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Non-constant holomorphic function defined on open disk $D(0,1)$

Let $f:D(0,1) \to \mathbb{C}$ be a non-constant holomorphic function. For each $r \in [0,1[$, define: $$h(r)=\max\{|f(z)| : |z| = r\}$$ Show that h is $\textbf{strictly}$ increasing. By Maximum ...
J P's user avatar
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Prove $f$ attains its maximum on boundary

Let $f$ be holomorphic, bounded on $|z|>1$ and continuous on $|z|\geq 1$. Prove that $|f|$ attains maximum on $|z|=1$, i.e $$\sup\limits_{|z|\geq 1}|f(z)|=\max\limits_{|z|=1}|f(z)|.$$ My attempt: ...
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Application of weak maximum principle.

Fix any open, bounded set $U \subset \mathbb{R}^n$ and suppose that $u \in C^2(U) \cap C(\bar{U})$ is a solution of $$-\Delta u = f \,\,\,\text{in}\,U$$ $$u=g \,\,\,\,\,\,\text{on}\,\,\,\partial U,$$ ...
Lilili123's user avatar
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complex analysis - f(z)^d+g(z)^d=1 for all complex number z, where d>=2 [duplicate]

I would like to show that "if holomorphic function $f,g$ satisfies $f(z)^d+g(z)^d=1$ for all $z\in\mathbb{C}$ (where $d\in\mathbb{Z}_{\geq3}$), then both $f$ and $g$ is constant." I know ...
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Applying maximum principle for the heat equation

I'm stuck with the following problem Let $U\subseteq \mathbb{R}$ be a bounded domain, $T>0$ and $u\in C^2(U_T) \cap C(\overline{U_T})$, where $U_T = U \times (0,T]$. If $f=f(x)$ is a continuous ...
Yauset Cabrera's user avatar
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1 answer
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Maximum Principle for Operator $L$

I'm having some trouble showing the following: Let $u\subset\mathbb{R}^n$ be open and bounded and assume that $u\in C^{1,2}(U_T)$. Consider the partial differential operator $$Lu:=u_t-\Delta u+b\cdot\...
JackpotWizard 180's user avatar
4 votes
1 answer
232 views

Polynomial identity : $|P(z)|^2=|Q(z)|^2-|R(z)|^2$ for $z \in \mathbb{D}$

Let us denote by $\mathbb{D}=\{z : |z| <1\}$ and $\mathbb{T}=\{z: |z|=1\}$. Suppose that $P, Q$ and $R$ are polynomials that satisfy the following: $|Q(z)| \geq |R(z)| $ for all $z \in \overline{\...
Curious's user avatar
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$\Delta u = f(u,x)$ in $\Omega \subseteq \mathbb{R}^n$ has at most one solution

I want to show that the PDE $\Delta u = f(u,x)$ in $\Omega \subseteq \mathbb{R}^n$ with the boundary value $u = \varphi$ on $\partial \Omega$ has at most one solution under the following conditions: $...
AlexH's user avatar
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Prove that $u = |\nabla v|^2$ reaches its maximum in $\partial \Omega$, $v$ being a solution to the given Poisson pde

Let $\Omega \subset \mathbb{R}^2$ and $v: \mathbb{R}^2 \to \mathbb{R}$ be a solution to: $$\begin{cases} v_{xx}+v_{yy} = -2 &\text{in $\Omega$}\\ v = 0 &\text{in $\partial \Omega$} \end{cases}$...
nicoyanovsky's user avatar
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Polyharmonic system

In the problem, with $\Omega \subset \mathbb{R}^3$ a smooth bounded domain, and the given functions $h_1, h_2 \in C(\partial\Omega)$ \begin{equation}\nonumber%\label{eq:Pe}\tag{$P_{\varepsilon}$} \...
Bruno's user avatar
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2 votes
1 answer
189 views

Maximum entropy distribution with given mean, variance and skewness?

Is there a way to find the maximum entropy distribution with certain values for the three first moments, when the support is the set of real numbers? Or, without loss of generality, with mean zero, ...
HelloGoodbye's user avatar
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Strong maximum principle for a local maximum

Suppose $L$ is a strictly elliptic operator in the non-divergence form with $c \equiv 0$, $u \in C^2(\Omega)\cap C(\Omega)$ and $Lu \le 0$ in $\Omega$. Prove that if $u$ attains a local maximum at an ...
M. Rubick's user avatar
1 vote
0 answers
28 views

Complex solutions of ordinary BVP with real/complex coefficients

The classic Protter and Weinberger: Maximum principles in Differential Equations states the following uniqueness theorem (theorem 8): Let $u_1$, $u_2$ be solutions of the BVP $u''(x) + g(x) u'(x) + h(...
das_blob's user avatar
1 vote
0 answers
251 views

What hypothesis can I assume in order to obtain infinity solutions for this problem?

It is related to the classic theory of PDE. Suppose that $U \subset \mathbb{R}^m$ is an open, connected and bounded set, $f : U \to \mathbb{R}$ and $g:\partial U\rightarrow \mathbb{R}$ and $c:\...
Silvinha's user avatar
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3 votes
1 answer
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Showing uniqueness of solution to a non-linear Poisson problem

I'm trying to prove that a non-linear Poisson problem has a unique solution. The context is the following: Let $\Omega \subset \mathbb{R}^n$ be a bounded open subset of class $C^2$. Consider the ...
Matheus Andrade's user avatar
1 vote
1 answer
63 views

A property about distance between open subsets and boundary of open set used in a proof of the strong maximum principle

I'm going through a proof of the following statement: Let $\Omega \subset \mathbb{R}^n$ be a connected open set, $L$ be an uniformly elliptic operator in $\Omega$ with $c \equiv 0$ and $u \in C^2(\...
Matheus Andrade's user avatar
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61 views

Solving this optimisation problem that contains integrals

Assume I have the following objective, which consists of choosing the bundle of goods that will maximise the intertemporal utility between date 0 and date T: $$ \max_{C1(t), C2(t), C3(t)} U(C(t))= \...
Meg's user avatar
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1 vote
1 answer
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Function cannot attain a minimum inside the domain

Question is that to prove if $$u'' + e^u = −x \text{ for } 0< x <1$$, then u cannot attain a minimum in $(0,1)$ Step I tried: First, $u'' + e^u <0.$ I assume exists a $c \in (a,b)$ s.t. $u(c)=...
Apple's user avatar
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1 answer
127 views

If $f(z)$ is analytic for $|z|<1$ and $f(z)≤\frac{1}{1-|z|}$,find the best estimate of $|f^{(n)}(0)|$ that Cauchy inequality will yield

There have a solution that Fix $n≥0$ and let $R=\frac{n}{n+1}$ so that $f(z)$ is analytic on$|z|=R$. Then by Cauchy inequality $$|f^{(n)}(0)|≤\frac{n!}{R^n} \max_{z=R}|f(z)|=\frac{n!}{R^n(1-R)}=\frac{...
tianhaowu's user avatar
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1 answer
60 views

weak version of maximum principle for not-quite-subharmonic functions

For smooth functions $f(x,y)$ in a disk, if $f$ is subharmonic then it satisfies the maximum principle. What happens if we relax the subharmonic condition, by requiring only certain bounds on $\Delta ...
Mikhail Katz's user avatar
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Proving $u(x,t) \leq \alpha x(1-x) e^{-\beta t}$ for Heat Diffusion using the Maximum Principle

I have $u(x,t)$ to be defined as the solution to the following partial differential equation for heat diffusion over the domain $S = (0, 1) \times (0, \infty)$. $$ \begin{cases} u_t - u_{xx} &= 0 ...
Talmsmen's user avatar
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1 vote
0 answers
130 views

maximum principle for compact manifolds

Are the maximum/minimum principles on $\mathbb{R}^n$ available for both local or nonlocal operators can be modified for a compact Riemannian manifold? For instance, we have a nonlocal maximum ...
am_11235...'s user avatar
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Variant of strong maximum principle

i have a question about strong maximum principle and mean value formula for heat equuation i saw two versions of these theorems one (In Evans Book) with hypothesis that $u \in C^{2,1}(U\times (0,T])...
RIM's user avatar
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0 answers
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How to prove the comparison principle of this parabolic partial differential equation?

Here $H$ is a positive uniformly convex function. I came across this comparison principle while studying the paper. Maybe the author thought it was easy to prove so he omitted the proof, but I failed ...
Serge's user avatar
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1 answer
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Limsup on boundary implies bound on modulus

I am working on the following problem: Suppose $f$ is analytic on a bounded domain U and that $$\limsup_{U\ni z\to w} |f(z)| \leq K \quad (\dagger)$$ for each $w \in \partial U$. Show that $|f(z)| \...
blomp's user avatar
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Maximum Principle Theorem

I am reading the mean curvature flow from Mantegazza. The maximum principle is stated as follows: Two questions: Why does $ u_{\text{max}}:=\max u(p,t) $ exist? Why is there $T'$ instead of $T$ in ...
Seurat's user avatar
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0 answers
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Complex Analysis - Proving a function is constant

let:$\text{U}\subseteq \mathbb{C}$ as $U$ open and path connected. $f$ analytic in $U$ and does: $\forall z_0\in U,\forall \epsilon >0,\exists z_1,z_2\in Ball_{\epsilon }\left(z_0\right)st:f\left(...
LearningToCode's user avatar
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0 answers
78 views

Product of two convex combinations is a harmonic function

Let two convex combinations $\lambda = \sum_{i = 1}^N \lambda_i \theta_i$ and $\mu = \sum_{i = 1}^N \mu_i \theta_i$ of real constants $\lambda_i$ and $\mu_i$, and variables $\theta_i$ such that $\sum_{...
Bridi's user avatar
  • 63
0 votes
0 answers
32 views

Estimation of heat flux at the boundary

So for heat equation $$T_t - k \Delta T = f(t)1_{r <= R}(r)$$ with initial condition $T(r,0) = 0$. where $r = ||(x,y,z)||$. $f(t)$ could be any positive function that it's integral over time is ...
yuanming luo's user avatar
0 votes
1 answer
87 views

Herman rings for polynomials

I am reading this link on complex dynamics and in Problem 12-1 it asks the reader to prove, using the Maximum Modulus Principle, that Herman rings cannot occur for polynomials. I have seen this ...
Uri Toti's user avatar
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0 answers
102 views

Minimum for a nonlinear heat equation

I'm trying to solve question 4.6 using maximum principle Suppose that $u(x, t)>0$ solves $$ \frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(u \frac{\partial u}{\partial x}\right)+u $...
hbghlyj's user avatar
  • 3,047
0 votes
1 answer
154 views

Perron's method for the Dirichlet problem: Necessity of subharmonic barrier

I am not seeing how to adapt my argument to produce a proof for the following statement: Let $D \subset \mathbb{C}_{\infty}$ omit at least two points, and suppose that $\infty \notin \partial D$ and $\...
porridgemathematics's user avatar
1 vote
0 answers
75 views

How to prove the uniqueness and stability of solution to Poisson equation using the maximum principle?

I came across this problem in an exam. Given a Poisson equation on a bounded region $\Omega$, $$\begin{cases} -\Delta u=f& u\in\Omega\\ \alpha u+\beta\dfrac{\partial u}{\partial n}=g&u\in\...
Nekomiya Kasane's user avatar
0 votes
1 answer
127 views

Comparison principle for p-harmonic functions.

The proof begins by considering the open set $D_{\varepsilon} = \{ x \in \Omega : u(x) > v(x) + \varepsilon \}$ for some $\varepsilon > 0. $ Then it says that $D_{\varepsilon} = \emptyset$ or $...
BillyBOB1's user avatar
1 vote
0 answers
132 views

Maximum principle or comparison principle

the equation $\varphi_t + D_p H(x, Du ^\varepsilon)\cdot D \varphi = \varepsilon\Delta \varphi$ is a linear parabolic equation. Thus, by the comparison principle for parabolic equations, we have for ...
Lilileaf's user avatar
1 vote
1 answer
62 views

Usual maximum principle

We assume that $max_{\mathbb{R}^n \times[0,T]} f(x,t) = f(x_0,t_0)\quad\text{for some }(x_0,t_0)\in\mathbb{R}^n \times[0,T]$ Then, my book says that If $t_0>0$, then, by the usual maximum principle,...
Lilileaf's user avatar

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