Mathematics

Questions tagged [maximum-principle]

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

In the context of partial differential equations, the maximum principle states that

If $f$ is harmonic, then $f$ cannot attain a local maximum without being constant. That is, if $D$ is a connected open subset of $\mathbb{R}^n$ and there exists $x_0 \in D$ with $$|f(x)| \ge |f(x_0)|$$ for every $x$ in a neighborhood of $x_0$, then $f$ is identically a constant function.

In complex analysis, the maximum principle (usually called the maximum modulus principle) states an analogous result:

If $f$ is holomorphic, then $|f|$ cannot attain a local maximum without being constant. That is, if $D$ is a connected open subset of $\mathbb{C}$ and there exists $z_0 \in D$ with $$|f(z)| \ge |f(z_0)|$$ for every $z$ in a neighborhood of $z_0$, then $f$ is identically a constant function.

References: Maximum principle (PDE), maximum modulus principle.

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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 understand a theorem intuitively beforehand, the proof will…
Sachin Valera
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Maximum Modulus Exercise

Using the maximum modulus theorem in complex analysis, what is a good technique for finding the maximum of $|f(z)|$ on $|z|\le 1$, when $f(z)=z^2-3z+2$? Got some really nice answers below, so I thought I'd share an image showing some contours of…
David
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Maximum value of a complex polynomial on the unit disk

The polynomial is $p(z)=\sum^n_{k=0} a_kz^k$. And I want to prove the following inequality on the unit disk$$\max_{B_1(0)}|p(z)|\geq |a_n|+|a_0|$$ By the maximum modulus principle, the maximum must be on the unit circle and greater than $|a_0|$ by…
Simple Mistake
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Maximum principle for harmonic functions on unbounded domain

I am learning PDE by myself now. I am considering converted the problem to bounded domain to use the strong maximum principle. My attempt: Using the $\lim u(x)=0$ , then exists $\epsilon$ and $N$ , such that if $x>N$ , $|u(x)|<\epsilon$ . Now…
Tony
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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 $|f|\leq 1$ whenever $|x|\leq 1$. I think this is…
Extremal
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Non-linear elliptic PDE uniqueness of solution

I want to show that the following PDE has at most one solution \begin{equation} -\Delta u +c(u)=f \text{ in } \Omega \end{equation} \begin{equation} u=0 \text{ on } \partial\Omega \end{equation} where $\Omega\subset\mathbb{R}^n$ is open and bounded…
Azamat Bagatov
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Determine complex polynomial

Problem Let $P(z) = z^n + a_{n−1}z^{n−1} + \cdots + a_1z + a_0$ be a polynomial of degree $n > 0$. Show that if $\lvert P(z) \lvert \le 1$ whenever $\lvert z \rvert = 1$ then $P(z) = z^n$. I have tried to see $\dfrac{P(z)}{z^n}$, but nothing…
Yimin
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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 imply that there will be an analytic…
user41481
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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 $u\in C^2(U)\cap C(\bar{U})$ $$\left\{\begin{array}{rl} Lu \leq 0 & \mbox{in }…
Cookie
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All compact, connected, complex Lie groups are Abelian.

Here I found one interesting result about connected, complex Lie groups. In the 1 answer, the author claims that this fact is a consequence of the maximus modulus principle, but I can not understand why this is so.
AndrewGap
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Entire + periodic in imaginary direction + bounded on the real line implies constant?

I was reading some slides from a lecture. In a proof, there arose the need to show a certain function $f : \mathbb{C} \to \mathbb{C}$ was constant. The argument proceeded by checking that $f$ was entire $f(z+i) = f(z)$ for all $z$ $f$ was bounded on…
Mike F
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Unexpected hanging paradox maxmin strategies

I have a question about strategies of the players of Unexpected hanging paradox (I am very sorry for a long topic, topic exist already for a while, during this time I try to develop idea how to solve the problem, recently I decided to seek solution…
com
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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 boundary $Bd(A)$. Prove that $(f_n)$ is uniformly…
Chilote
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Solution of $-\Delta u=f(u)$ is positive

Let $\Omega$ be a bounded domain. Assume that $f:\mathbb{R}\to\mathbb{R}$ is a function with $f(0)>0, f^{\prime}(0)\geq0$ and $f^{\prime\prime}(u)>0$ for $u>0$ . Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a solution of the following problem:…
john_psl1298
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Proof of Maximum Principle for Harmonic Functions

A harmonic function is one which solves Laplace's Equation $\Delta u=0$ . The maximum principle states that over some domain $D$ , $u$ achieves a maximum and minimum on $\partial D$ , and nowhere inside $D$ . I am struggling to understand the proof…
rb3652
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