# Questions tagged [maximum-principle]

The maximum-principle tag has no usage guidance.

21
questions

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 ...

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
$$...

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 ...

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.
...

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\...

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 $\mathbf{0}\not\in C$. ...

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 ...

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 ...

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 ...

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. ...

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 (...

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 ...

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 $\Omega$...

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_\...

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-...

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' $\...

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 ...

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, ...

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{...

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 ...

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:
$$...