Skip to main content

Questions tagged [maximum-principle]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0 votes
0 answers
142 views

Inequalities on the distribution of the maximum of the normalized sum $\max_{k = 1,\dots,n} \frac{|S_k|}{\sqrt{k}}$

Let $\{X_i\}_{i\in\mathbb{N}}$ be i.i.d. random variables with $\mathbb{E}(X) = 0$,$\mathbb{E}(X^2) = \sigma^2$ and finite moments. Let $S_k = \sum_{i = 1}^{k} X_i$ and consider the normalized ...
MathRevenge's user avatar
0 votes
1 answer
62 views

Derive elliptic maximum principle from weak derivatives

Let $U$ is a connected open set, and $a^{ij}, c^i \in L^\infty (U).$ $a^{ij}$ satisfies the uniform ellipticity condition. Suppose that $u\in H^1(U) \cap C(\overline U)$ satisfies the condition that $$...
Ma Joad's user avatar
  • 1,683
0 votes
1 answer
114 views

Well posedness of the Plateau problem under lack of uniqueness

The title of this question may seem an oxymoron, but let me describe it and you'll see that perhaps it is not. Premises I am analysing the following Plateau problem. Let $G\subsetneq\Bbb R^n$ be a ...
Daniele Tampieri's user avatar
1 vote
2 answers
183 views

Problem in understanding maximum principle for subharmonic functions

I am reading subharmonic functions and their properties from the book From Holomorphic Functions to Complex Manifolds by Grauert and Fritzsche. Let me first define what a subharmonic function is. ...
Anacardium's user avatar
7 votes
2 answers
957 views

Polynomials having all their zeros on the unit circle

Let $P(z)=\sum_{k=0}^na_kz^k$ be a polynomial of degree $n$ having all its zeros on the unit circle. Let $M=\max_{0\leq k\leq n}\lvert a_k\rvert$. The polynomial $P(z)=z^n+1$ has $\max_{\lvert z\...
user159888's user avatar
1 vote
0 answers
86 views

Positive semidefinite maximum principle

Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Let $\mu$ be a Borel probability measure on $M_n(K)$ supported on a compact set $C$ of positive semidefinite matrices with $\mathbf{0}\not\in C$. ...
Joseph Van Name's user avatar
6 votes
0 answers
278 views

Models of ZF (without Inaccessible cardinals) where only the full Axiom of Choice fails, but the Axiom of Countable Choice remains true?

Solovay's model (which assumes $I$ = "existence of inaccessible cardinal") will be a well-known construction to produce a model of ZF where only the full Axiom of Choice ($AC$) fails, but ...
Yauhen Yakimovich's user avatar
2 votes
1 answer
165 views

Strong maximum principle in entire space

Let $n\geq 3$, $K$ is a bounded function in $\mathbb{R}^n$. If $u$ is a nonegative solution but not equal to 0 of $$-\Delta u=Ku^{\frac{n+2}{n-2}}\quad \text{in }\,\mathbb{R}^n.$$ Can we use the ...
Davidi Cone's user avatar
2 votes
0 answers
34 views

Maximum principle for poly-harmonic equations

If $u_1\geq 0$ and $u_1\neq 0$, and satisfies $$-\Delta u_1=|u_1|^{\frac{4}{n-2}} u_1\quad \text { on }\, \mathbb{R}^n,\quad n\geq 3,$$ it follows from maximum principle that $u_1>0$. My question ...
Davidi Cone's user avatar
2 votes
0 answers
102 views

Strong maximum principle for weak solutions still holds?

By De Giorge, Nash and Moser solutions of \begin{equation} \operatorname{div} (A(x) Du) = 0 \end{equation} where $Du$ denotes the gradient of $u$ and $A$ is a $\lambda,\Lambda$ elliptic matrix. ...
user29999's user avatar
  • 191
2 votes
1 answer
204 views

Is there a maximum principle for CR functions over domains inside CR manifolds?

I am new to this area and I am a bit confused by the literature. Is there a strong maximum principle for CR functions over domains in a CR manifold, please? If so, could someone please state it (...
Malkoun's user avatar
  • 5,041
1 vote
0 answers
62 views

Does the real part of the cross ratio satisfy a maximum principle on a domain in any real submanifold?

Let $C(p_1, p_2; p_3, p_4)$ denote the cross-ratio of the $4$ points $p_i$, for $i = 1, \ldots, 4$, thought of as a holomorphic function on $$ \Omega = \{ (p_1, p_2, p_3, p_4) \in \mathbb{C}P^1 \times ...
Malkoun's user avatar
  • 5,041
5 votes
0 answers
145 views

weak maximum principle for weighted laplacian

Consider a radial weight $w=|x|^2 \geq 0$ for all $x\in \mathbb{R}^n$ and consider the operator $$Lu= \frac{1}{w}\operatorname{div}(w\nabla u).$$ Then if $-Lu\leq 0$ on a smooth bounded domain $\Omega$...
Student's user avatar
  • 655
2 votes
0 answers
105 views

A maximum principle in $\mathbb{R}^N$

Let $\delta > 0$ and define $$ H_\delta(x) = \prod_{j=1}^{N} \cosh(\delta x_j), \quad \forall x \in \mathbb{R}^N. $$ By straightforward calculations we get $\Delta H_{\delta} (x) = \delta^2 H_\...
Lucas Linhares's user avatar
1 vote
0 answers
66 views

Parabolic PDE: Zero now means zero anytime before

Studying some mathematical models I came across a simple-looking question that I do not know how to handle. If we have the following problem: $$\begin{cases} \dfrac{\partial Z}{\partial t}-\Delta Z=aZ-...
Bogdan's user avatar
  • 1,330
2 votes
0 answers
97 views

How do you construct barriers for minimal surfaces?

There is no comparison principle for minimal surfaces: two minimal surfaces $M_1, M_2 \subset B$ in the unit ball of $\mathbf{R}^3$, with the boundary $\partial M_1 \subset \partial B$ lying 'above' $\...
Leo Moos's user avatar
  • 4,968
2 votes
0 answers
97 views

Maximum principle geometric interpretation

I have heard in one of the lectures I attended that subsolutions cannot touch even tangentially since both the strong maximum principle and the weak maximum principle says that subsolution doesn't ...
User1723's user avatar
  • 307
2 votes
0 answers
197 views

Maximum modulus principle for vector valued functions of several complex variables

In the following paper: Shub and Smale, "On the Existence of Generally Convergent Algorithms", Journal of Complexity 2, 2-11 (1986), trying to understand Lemma 2 on page 4. Paraphrased, ...
user125930's user avatar
2 votes
0 answers
127 views

Parabolic maximum principle for non-compact manifold with boundary

Let $M:=\mathbb{R}^n\setminus \mathbb{D}^n$, where $\mathbb{D}^n$ is the open unit ball in $\mathbb{R}^n$ and $u\colon M\times [0,\infty)\rightarrow \mathbb{R}$ is solution of the following PDE \begin{...
Sumanta's user avatar
  • 632
1 vote
0 answers
120 views

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 the use of moving plane method to prove radial symmetry of $C^{2}$ global solution of a PDE in $R^{2}$, and I'm reading a paper where this method is applied: precisely I'm ...
Elio Li's user avatar
  • 745
1 vote
1 answer
110 views

Phragmén–Lindelöf principle for the critical exponent

Let $f(z)$ be a holomorphic function in the angle $A=\{0<\arg z<\frac{\pi}2\}$, continuous in $\bar A$, satisfying $|f(z)|\le M$ on $\partial A$ and satysfying the following growth condition: $$...
Andrew's user avatar
  • 2,655