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

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32 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
43 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 ...
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0answers
28 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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25 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
11 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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3answers
99 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 ...
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1answer
35 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
33 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 ...
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1answer
28 views

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, ...
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4answers
50 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
59 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
32 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
36 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 ...
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2answers
42 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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0answers
18 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
22 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
15 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
45 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
72 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
28 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 ...
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1answer
23 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
52 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 ...
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1answer
21 views

Inequality between Laplacians at a point of local maximum

I never before worked with maximum principle. Can someone tell me how I proceed with this: Let $v\in C^{\infty}$ $u$ is continuous and $\Omega\in \mathbf{R}^n$ and suppose that $u − v$ has a ...
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1answer
53 views

Prove $\max_{\bar \Omega} |\nabla u|^2=\max_{\partial \Omega} |\nabla u|^2$.

Let $\Omega \in \mathbb R^n$, $n>1$, be a bounded domain with smooth boundary. Let $u \in C^1 (\bar \Omega)$ be harmonic in $\Omega$. (a) Prove $\max_{\bar \Omega} |\nabla u|^2=\max_{\partial ...
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1answer
35 views

If $\Delta u - u \geq 0$ in $D$, show that $u\leq \max\{\ \max\limits_{\partial D}(u), \ 0 \ \}$

Let $D$ be a bounded domain, with boundary $\partial D$. Suppose $u \in C^{2}(D)\cap C^{0}(D\cup\partial D)$ satisfies $\Delta u - u \geq 0$ in all of $D$. I am supposed to prove then that $u\leq ...
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0answers
39 views

Is this just a direct consequence of the Maximum-Modulus Theorem?

Let $D$ be a bounded region and $f$ is an analytic function on $D$. Show that if there is a constant $c ≥ 0$ such that $|f(z)| = c$ for all $z$ in the boundary of $D$ then either $f$ is a constant ...
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0answers
27 views

Discrete maximum principle for finite difference to the heat equation

Given a continuous-time, discrete space approximation of $u_t = u_{xx}$ with homogeneous Dirichlet boundary condition. The discrete ODE version for the nodal values is: $u'_j = \frac{u_{j-1}+u_{j+1} - ...
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0answers
24 views

If two holomorphic function whose maximal on compact set is the same, then are they equal?

I tried to prove that if equality holds for Hadamard's three circle theorem, then the function is form of $f(z) =Cz^{\lambda}$ where $C \in \mathbb{C}$, $\lambda \in \mathbb{R}$. I know one of the ...
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2answers
52 views

Does there exist a holomorphic function with the following property?

Does there exist a holomorphic function $f$ defined over $D = \{ z : |z| < 1 \}$ such that $|f| \rightarrow \infty$ when $|z| \rightarrow 1$? My approach: If such an $f$ exists, then for a given ...
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0answers
40 views

Prove solution to heat equation is bounded if initial data is bounded

Heat equation: $$\begin{cases} u_t-ku_{xx}=0, x \in \mathbb{R}, t \gt 0 \\ u(x,0) = u_0(x), x \in \mathbb{R} \\ \end{cases} $$ I'm given that for constants $M, N \in \mathbb{R}$, we have $N \leq ...
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2answers
34 views

Finding the maximum of a function with two variables

I started with f(x,y)=20x-x^2+30y-y^2+xy I need to find the maximum values for both x and y. I have differentiated it using implicit differentiation and have gotten to an answer of: ...
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4answers
80 views

Is there a Maximum Principle for Biharmonic eigenvalue problem?

Let $$\Delta^2 u-\lambda u =0$$ where $\lambda>0$ and $$\Delta^2 u = \frac{\partial ^4 u }{\partial x^4} + 2 \frac{\partial^4 u }{\partial x^2 \partial y^2} + \frac{\partial ^4 u }{\partial y^4}$$ ...
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0answers
31 views

Maximum principle for heat equation with dissipation

Suppose that $u$ satisfies $$u_t - ku_{xx} + cu=0$$ $$u(x,0) = \phi(x)$$ for $t >0 $ and $x \in [a,b]$. Show that u satisfies the maximum principle My attempt If $v =e^{ct}u$, then $v$ satisfies ...
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1answer
38 views

Maximum Modulus principle on Rudin's theorem.

I have studied rudin's Real and complex analysis, and have a question on the proof that why the level set $E$ is compact. Could you give me some hint? 11.13 Theorem If a continuous function $u$ ...
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0answers
32 views

PDE: Laplace equation Maximum Principle

The maximum priciple for Laplace equation assumes, in both PDE textbooks by Fritz John and Lawrence Evans, that the domain of the harmonic solution be bounded. Is the maximum priciple still valid if ...
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0answers
51 views

Existence of solution at $t = 0$ of the heat equation with Neumann condition

Suppose $x_{t}-\triangle{x} = 0$ on $\omega\times (0,T)$ with $\frac{\partial{x}}{\partial{v}} = 0$ on $\partial{\omega}$. Prove that $x$ achieves its max and min at initial times $t = 0$. My ...
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1answer
117 views

Show that the wave equation does not satisfy the maximum principle

It is asked to show that the wave equation does not satisfy the wave equation The wave equation is given by $$\left\{\begin{matrix} u_{tt}=c^2u_{xx} \\ u(x,0)=\phi(x) \\ u_t(x,0)=\psi(x) ...
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1answer
38 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
34 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
42 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
169 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
63 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
59 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
26 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
59 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
19 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 $\Omega$?
3
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1answer
61 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
54 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 $$ ...
3
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1answer
71 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 ...