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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What is the domain in the unit disk such that a holomorphic function f remains biholomorphic? [duplicate]

Let $\mathbb{D}$ denote the unit disk in the complex plane $\mathbb{C}$. Suppose that $f: \mathbb{D}\rightarrow\mathbb{D}$ is holomorphic and $f(0)=0$, $f'(0)=a>0$. How can I prove that $f$ is ...
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What fails in the maximum value principle if the domain is not connected?

We have discussed the maximum value principle: Let $\Omega \subset \Bbb{C}$ be an open connected subset and $f:\Omega \rightarrow \Bbb{C}$ be an analytic function. Assume that there exists $z_0\in \...
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Prove that $\Delta u=F$ with these conditions has at most one solution

Let $\alpha >0$, and let $\Omega\subset \mathbb{R}^N$ be and open domain. I want to prove that the following problem has at most one solution. $$\Delta u=F \quad \text{in } \Omega$$ $$u=f \quad \...
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$u$ be the sol of $\Delta u+ a_i(x)\frac{\partial u}{\partial x_i}= f(x)$ if $f\geq 0$ then u is constant and $f=0$

Let $\Omega$ be a bounded domain with smooth boundary . Let $u$ be a solution of the problem $\Delta u+ a_i(x)\frac{\partial u}{\partial x_i}=f(x)$ and $\frac{\partial u}{\partial n}=0$ . Assume that $...
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Maximum value in an annulus

$0<r<R$ and $f:\bar{A_{r,R}}\to\mathbb{C}$, a function defined on the closure of the annulus. $f$ is holomorphic in the interior and continuous on the boundary. For $s\in(r,R)$ write $\theta=\...
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Maximum principle for a strong solution to non-homogenous Laplace equation

I am searching for a reference for this apparently well known fact (the part below Theorem 1.1 in the picture i.e. the equation $(6)$): This screenshot is from https://math.aalto.fi/~astalak2/files/...
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Proving entire function to be constant

Given $f$ is an entire function and for any $z\neq0$, $f$ satisfies $f(z)=f(\frac{1}{z^2})$. The question asks to prove that $f$ is constant. My approach: For any $|z|>1$, we have $\dfrac{1}{|z|}&...
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"Counterexample" to the Maximum Principle? Strauss example 2.4.2

The solution to the one-dimensional heat equation $u_{t} = ku_{xx}, -\infty < x < +\infty, t > 0$, with initial condition $u(x,0) = e^{-x}$ is given by $u(x,t) = e^{kt-x}$. According to the ...
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How to use maximum principle with an equation similar to the heat equation?

Suppose that $u(t,x)\in C_t^1C_x^2(\Omega_T)\cap C(\overline{\Omega_T})$ satisfying $$ \begin{cases} \partial _tu-\Delta u+c\left( x \right) u\le 0,\left( t,x \right) \in \Omega _T,\\ u\left( t,x \...
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Necessity of boundeness of domain in Strong Maximum Principle

The Strong Maximum Principle for $c \geq 0$ is as fallowing: Assume $u \in C^{2}(U) \cap C(\overline{U})$ and $$ c \geq 0 \text{ in } U. $$ Supose also $U$ is connected. i) If $$ Lu \leq 0 \text{ in } ...
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Quotient of solutions of a semilinear Dirichlet problem is $L^\infty$

I am reading a paper of Brezis and Oswald about existence and uniqueness of positive solutions to sublinear elliptic equations: \begin{equation} - \Delta u = f(x, u), \ u \geq 0, \ u \not\equiv 0 \...
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Case where solution does not obey maximum principle

Find a solution of the equation $$ -\frac{d^2u}{dx^2} - u = -1$$ in the interval $|x| < 1$ which does not obey the maximum principle. The maximum principle: If $u \in C^2(\Omega)$ satisfies $\Delta ...
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Uniqueness of the interpolation function $B_0^{1-s} B_1^s$ appearing in the Hadamard 3-lines theorem/Lindelof theorem

In Terry Tao's notes https://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/#lind, the interpolation $B_0^{1-\theta} B_1^\theta$ appearing in the Hadamard 3-lines theorem ...
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Estimate the error caused by domain perturbation for Dirichlet Laplacian equation

$\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, consider the Dirichlet laplacian equation on $\Omega$. \begin{cases} -\Delta u=f & \mbox{in }\Omega\\ u|_{\partial \Omega}=0 \end{cases} ...
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Interpolation of $L^p$ spaces (from book by Caffarelli-Cabre)

In the book "Fully nonlinear elliptic equations" by L. Caffarelli and X. Cabre, Theorem 4.8 (b) we want to prove the following maximum principle, for any $p>0$, \begin{align*} \sup_{...
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Proving if a certain function has a maximum inside a domain

Let $f :\mathbb{C} \longrightarrow \mathbb{C}$ be an analytic injective function with $f(0)=0$, let $r>0$. Since $\partial B_r(0)$ is a compact set, then $|f(z)|$ assumes a minimum on $\partial B_r(...
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Criteria for when holomorphic functions can be approximated on a compact set

Let $K$ be a compact subset of an open set $G \subseteq \mathbb C$. I am trying to prove that the following are equivalent If $f$ is analytic in a neighborhood of $K$ and $\epsilon > 0$, then ...
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Question about proof of Liouville's theorem [duplicate]

I am trying to work through a proof of Liouville's theorem here and ran into an issue. In his answer (as I understand it) Eremenko first establishes that $|f(0)|\leq \max_{|z|=r}|f(z)|$ for entire ...
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Estimating the $L^{\infty}$-norm of a Laplacian of a function

Considering the problem \begin{equation} \left\{ \begin{array}[c]{11} \Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\ \Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\ \end{...
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Using maximum principle to get this pointwise bound

I'm trying to understand a small statement from Pigati & Stern - 2020 - Minimal submanifolds from the abelian Higgs model. Let $u$ be a section of a complex line bundle $L\to M$ over a closed ...
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Why maximum principle holds for scalar conservation law?

I wonder why the maximum principle holds for scalar conservation law. Consider the PDE below. $u_t+f(u)_x=u_{xx}, \quad u_0(x)=u(0,x)$ (where $f$ is a smooth real function) The paper which I am ...
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How to show that the imaginary part of a function assumes its maximum on the boundary of a compact set [duplicate]

I need to show that the maximum modulus principle also holds for $Im(f)$, not just $|f|$, that is: Let $G$ be a bounded, open, connected subset of $\mathbb{C}$. Let $f: \overline{G} \rightarrow \...
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2 answers
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Holomorphic function on disk with two fixed points is identity [duplicate]

Let $f : \Bbb D \to \Bbb D$ a holomorphic function that has two fixed points : $\exists w_1, w_2$ such that $f(w_1)=w_1$ and $f(w_2)=w_2$. Show that $f(z)=z$ $\forall z \in \Bbb D$. I wanted to do ...
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Bounded harmonic function in the unit disc centered at origin .

Find a bounded harmonic function in unit disc centered at origin and taking value $\sin 2\theta$ on boundary. I tried many ways to solve this question but not getting the answer as $r^2\sin 2\theta$. ...
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Solve an optimization problem (pontryagin's maximum principle)

"Formulate an optimal control problem for a population with an Allee effect growth term, in which the control is the proportion of the population to be harvested. Choose an objective functional ...
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How to continue this proof for the minimum principle for Laplace's equation?

I know we can prove the minimum principle by using the maximum principle, by replacing $u(x,y)$ by $-u(x,y)$, but I keep getting stuck. Can someone help me figure out how to continue? So we have ...
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How come, in the heat equation, the maximum can not be attained on the upper boundary of the rectangle we construct?

First, I want to apologize for this seemingly elementary question, it's just something that has confused me for a very long time and I can't find anything telling me why. So we have the heat equation $...
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Estimate of derivative analytic function on the unit disc

Let $f$ be an analytic function on the closed unit disc $\bar{\mathbb{D}}$, such that for all $z$ with $|z|=1$ we have that $|f(z)-z|<|z|$. Prove that $|f'\left(\frac{1}{2}\right)|\leq8$ My ...
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4 votes
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Maximum principle for parabolic equation with mixed boundary condition

Suppose $u : [0,1] \times [0,T] \to \mathbb{R}$, let's consider a simplifed problem \begin{equation} \left\{\hspace{5pt}\begin{aligned} &\dfrac{\partial u}{\partial t} - \partial_x^2 u - \...
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Does exist an holomorphic function $f$ on $|z|>1$, non-constant, and bounded by the limit at $\infty$?

I have a question in my book that asks to show that if a function $f$ is holomorphic on $\{z:|z|>1\}$, continuous on $\{z: |z|=1\}$and has limit on $\infty$ then its maximum is for some point $w$ ...
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1 answer
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Maximum principle question for heat equation problem

Let $u(t, x) \in C_{t}^{1} C_{x}^{2}\left(\Omega_{T}\right) \cap C(\overline{\Omega_{T}})$ satisfies: $$ \begin{cases}\partial_{t} u-\Delta u+c(x) u \leq 0, & (t, x) \in \Omega_{T} \\ u(t, x) \leq ...
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Show that there exists a unique $z_0\in B(0,1)$ s.t $f(z_0)=z_0$.

Let $\varepsilon >0$ and let $f:B(0,1+\varepsilon )\rightarrow B(0,1)$ be a holomorphic function. Show that there exists a unique $z_0\in B(0,1)$ s.t $f(z_0)=z_0$. For this problem, I don't want to ...
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Optimal control problem with boundaries depending on control

I have a relatively standard optimal quadratic control problem on infinite horizon : $\int_0^\infty (R-u)^\top (R-u) + C(t)^\top u~ dt $ subject to $\dot R = kR - k u $ but with one specific. The ...
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Show that exist a function $u$ continuous in $\overline{\Omega}$ [duplicate]

Show that exist a function $u$ continuous in $\overline{\Omega}$: $$\lim_{k\rightarrow \infty}u_k=u\quad \text{uniformly on }\overline{\Omega}$$. The problem is: Let $\Omega\subset \mathbb R^n$ ...
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1 vote
1 answer
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Heat Equation $u_{t} - \Delta u + u u_x + uu_y = 0$

Let a $\Omega = \{(x,y); -1 < x < 1, -1 < y < 1\}$ and suppose that u is a smooth solution of \begin{equation} \left\{ \begin{aligned} &u_{t} - \Delta u + u u_x + uu_y = 0, \...
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$f$ analytic in $\overline{\mathbb{D}}$ s.t $\int_{|z|=1}|f(z)-z||dz|\leq\frac{1}{100}$. Prove that $f$ has at least one zero in $\mathbb{D}$.

Question: Let $f$ be analytic in a neighborhood of $\overline{\mathbb{D}}$ (the closed unit disk) such that $\int_{|z|=1}|f(z)-z||dz|\leq\frac{1}{100}$. Prove that $f$ has at least one zero in $\...
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If $u\left(0,0\right)$ is a local maximum, then there is $r_0>0$ such that u is a constant function in $D_{r_0}$

I need to solve this problem for my vector calculus class Set $D_{r}=\{x^2+y^2\leq r^2\},\text{ }C_r=\{x^2+y^2=r^2\},\text{ }\forall r>0$ a. Suppose $u\in C^1\left(D_1\right)$, set $I_{u}\left(r\...
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A problem about using the moving plane method to prove radial symmetry of the $C^{2}$ global solution of a elliptic PDE in $R^{2}$

Recently I'm learning using the moving plane method to prove radial symmetry of the $C^{2}$ global solution of a PDE in $R^{2}$. Since the Laplacian operator is invariant under rotations, we only have ...
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4 votes
1 answer
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Proof of weak maximum principle for heat-type equations

Here is the statement of the scalar weak maximum principle on Riemanninan manifolds with boundary. Theorem (Scalar weak maximum principle): Let $M$ be a compact manifold with boundary and $g_t$ a ...
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maximum principle.. Proof check.

Suppose $$0 = - \Delta u+u \quad \text{in } U$$ where $U$ is open connected bounded set and $u(x)$ is $C^2(U)$ and $C(\partial U)$, and that $\max_{\partial U} u > 0$. Prove that $\max_{\partial U}...
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Minimum value on the boundary of U help

Question: Let $U \subset\mathbb{R}^2$ be an open and bounded set and $u\in C^2(U)\cap C(U)$ satisfy $\Delta u(x)=\frac{-1}{1+|x|}$ in $U$. Show that $u$ attains it's minimum on the boundary of $U$. ...
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How to prove that $|u(x)|\leq 1$ for all $x \in \Omega$, given u solves $-\vec \nabla^2 u = (1-u^2)\cdot u$ and $u=0$ on $\partial \Omega$.

Let $\Omega$ be a bounded domain and let $u$ be a solution to the PDE $-\vec \nabla^2 u = (1-u^2)\cdot u$, which satisfies $u=0$ on $\partial \Omega$. Show that $|u(x)|\leq 1$ for all $x \in \Omega$. ...
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The modulus of a holomorphic function $f$ on a domain $D$ split into two subdomains $D_1$ and $D_2$.

Let $D$ be a simply connected domain on the complex plain $\mathbb{C}$ with analytic boundary. Let $\Gamma$ be a simple curve that connects two discrete points $a$ and $b$ on the boundary of $D$, so ...
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Three-mass rotational system with Lagrange multipliers and PMP

I have been dealing with this problem for a while and ran into the following difficulty, which confused me a little and I need help to clarify. Using the Lagrangian ...
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2 votes
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Maximum principle for $-\Delta u=(u-1)(u+1)$

Consider a smooth solution that satisfies the elliptic equation $$-\Delta u=(u-1)(u+1)\ \text{ in }\ \{|x|<1\},\ \ u=f(x)\ \text{ on }\ |x|=1,$$ where $f$ is a continuous function such that $-1<...
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3 votes
1 answer
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An exercise from Stein's complex analysis - Phragmen-Lindelof principle

I am considering exercise 9 from chapter 4 of Stein and Sharkarchi's complex analysis: (a). Let $F$ be a holomorphic function in the right half-plane that extends continuously to the boundary, that is,...
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1 vote
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Uniqueness and maximum principle

The following is from O. Zeitouni's "Lecture Notes on Random Walks in Random Environments": Suppose I have a function $V_{a,b,c}:\mathbb{Z}\longrightarrow\mathbb{R}$ (for $a,b\in\mathbb{Z}$ ...
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1 vote
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Failure of the maximum principle

What are easy examples of some operators that don't satisfy the maximum principle? For the double derivative operator $u\mapsto u''$ with Dirichlet boundary conditions on $L^2(0,1)$, we have $-u''\...
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Doubt about the strong maximum principle on unbounded sets

I have a purely theoretical doubt, because the references I read didn't make it so clear to me. When we are analyzing the maximum principle, two ways are suggested: $\bullet$ (Weak maximum principle): ...
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Application of maximum princple for harmonic function

Suppose u is a non constant harmonic function in $\mathbb{R}^n$, how can I show that the functions $F(r) = \sup_{z \in B(0,r)}|u(z)|$ and $G(r)= \int_{z \in B(0,1)}|u|^2(rz)dS_z$ are both strictly ...
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