Questions tagged [maximum-principle]
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25 questions
3 votes
1 answer
471 views
Reference of a maximum principle used in a paper written by Brezis and Merle
In the paper "Uniform estimates and Blow-up behavior for solutions of $-\Delta u =V(x)e^u$ in two dimensions" in the Theorem 1 (A basic inequality), we have the following result: Let $\Omega ...
- 305
6 votes
1 answer
361 views
Generalization of the "# of zeros cannot increase with time" theorem to parabolic PDEs with inhomogenous term
Various flavors of following result exist: Suppose we have a uniformly parabolic scalar PDE posed on real line, with all coefficients $a(x,t)$ and $b(x,t)$ bounded. $\partial_tu(x,t)=(\sigma^2/2)\...
- 1,831
3 votes
0 answers
60 views
Deciphering Ladyženskaja-Solonnikov-Ural′ceva : maximum principle for parabolic systems?
I used to have in mind the following mantra for parabolic systems as compared to their scalar version : In general, there's no maximum principle for parabolic systems. Let me precise this a bit more....
- 3,940
2 votes
0 answers
138 views
There holds a type of strong maximum principle for p-caloric functions, that is solutions of $u_t -\Delta_p u=0$
That is, there holds strong maximum principle for solutions of $u_t- \Delta_p u=0$? I know that it holds for caloric functions, that is, for solutions of $u_t- \Delta u=0$. A priori, I don't want to ...
- 161
5 votes
1 answer
234 views
Reinforced Maximum Principle
Let $U\subset{\mathbb R}^n$ be a bounded open domain with smooth boundary. I assume that $U$ is diffeomorphic to a ball. You may think of $L=\Delta$ and $U$ is the unit ball. Let $L=\operatorname{div}(...
- 53.1k
0 votes
1 answer
118 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 $$...
- 1,785
1 vote
1 answer
307 views
Well posedness of the Plateau problem under lack of uniqueness
The title of this question may seem an oxymoron, but let me describe it and you'll see that perhaps it is not. Premises I am analysing the following Plateau problem. Let $G\subsetneq\Bbb R^n$ be a ...
- 6,830
1 vote
2 answers
423 views
Problem in understanding maximum principle for subharmonic functions
I am reading subharmonic functions and their properties from the book From Holomorphic Functions to Complex Manifolds by Grauert and Fritzsche. Let me first define what a subharmonic function is. ...
- 71
7 votes
2 answers
1k views
Polynomials having all their zeros on the unit circle
Let $P(z)=\sum_{k=0}^na_kz^k$ be a polynomial of degree $n$ having all its zeros on the unit circle. Let $M=\max_{0\leq k\leq n}\lvert a_k\rvert$. The polynomial $P(z)=z^n+1$ has $\max_{\lvert z\...
- 559
1 vote
0 answers
116 views
Positive semidefinite maximum principle
Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Let $\mu$ be a Borel probability measure on $M_n(K)$ supported on a compact set $C$ of positive semidefinite matrices with $\mathbf{0}\not\in C$. ...
6 votes
0 answers
389 views
Models of ZF (without Inaccessible cardinals) where only the full Axiom of Choice fails, but the Axiom of Countable Choice remains true?
Solovay's model (which assumes $I$ = "existence of inaccessible cardinal") will be a well-known construction to produce a model of ZF where only the full Axiom of Choice ($AC$) fails, but ...
2 votes
1 answer
270 views
Strong maximum principle in entire space
Let $n\geq 3$, $K$ is a bounded function in $\mathbb{R}^n$. If $u$ is a nonegative solution but not equal to 0 of $$-\Delta u=Ku^{\frac{n+2}{n-2}}\quad \text{in }\,\mathbb{R}^n.$$ Can we use the ...
- 531
2 votes
0 answers
62 views
Maximum principle for poly-harmonic equations
If $u_1\geq 0$ and $u_1\neq 0$, and satisfies $$-\Delta u_1=|u_1|^{\frac{4}{n-2}} u_1\quad \text { on }\, \mathbb{R}^n,\quad n\geq 3,$$ it follows from maximum principle that $u_1>0$. My question ...
- 531
2 votes
0 answers
159 views
Strong maximum principle for weak solutions still holds?
By De Giorge, Nash and Moser solutions of \begin{equation} \operatorname{div} (A(x) Du) = 0 \end{equation} where $Du$ denotes the gradient of $u$ and $A$ is a $\lambda,\Lambda$ elliptic matrix. ...
- 161
2 votes
1 answer
281 views
Is there a maximum principle for CR functions over domains inside CR manifolds?
I am new to this area and I am a bit confused by the literature. Is there a strong maximum principle for CR functions over domains in a CR manifold, please? If so, could someone please state it (...
- 5,357
1 vote
0 answers
82 views
Does the real part of the cross ratio satisfy a maximum principle on a domain in any real submanifold?
Let $C(p_1, p_2; p_3, p_4)$ denote the cross-ratio of the $4$ points $p_i$, for $i = 1, \ldots, 4$, thought of as a holomorphic function on $$ \Omega = \{ (p_1, p_2, p_3, p_4) \in \mathbb{C}P^1 \times ...
- 5,357
5 votes
0 answers
178 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 $\Omega$...
- 541
2 votes
0 answers
155 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_\...
- 239
1 vote
0 answers
91 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: $$\begin{cases} \dfrac{\partial Z}{\partial t}-\Delta Z=aZ-...
- 2,029
2 votes
0 answers
171 views
How do you construct barriers for minimal surfaces?
There is no comparison principle for minimal surfaces: two minimal surfaces $M_1, M_2 \subset B$ in the unit ball of $\mathbf{R}^3$, with the boundary $\partial M_1 \subset \partial B$ lying 'above' $\...
- 5,163
2 votes
0 answers
140 views
Maximum principle geometric interpretation
I have heard in one of the lectures I attended that subsolutions cannot touch even tangentially since both the strong maximum principle and the weak maximum principle says that subsolution doesn't ...
- 347
2 votes
0 answers
267 views
Maximum modulus principle for vector valued functions of several complex variables
In the following paper: Shub and Smale, "On the Existence of Generally Convergent Algorithms", Journal of Complexity 2, 2-11 (1986), trying to understand Lemma 2 on page 4. Paraphrased, ...
- 75
2 votes
0 answers
188 views
Parabolic maximum principle for non-compact manifold with boundary
Let $M:=\mathbb{R}^n\setminus \mathbb{D}^n$, where $\mathbb{D}^n$ is the open unit ball in $\mathbb{R}^n$ and $u\colon M\times [0,\infty)\rightarrow \mathbb{R}$ is solution of the following PDE \begin{...
- 632
1 vote
0 answers
143 views
A problem about using the moving plane method to prove radial symmetry of the $C^{2}$ global solution of a elliptic PDE in $R^{2}$
Recently I'm learning the use of moving plane method to prove radial symmetry of $C^{2}$ global solution of a PDE in $R^{2}$, and I'm reading a paper where this method is applied: precisely I'm ...
- 999
1 vote
1 answer
169 views
Phragmén–Lindelöf principle for the critical exponent
Let $f(z)$ be a holomorphic function in the angle $A=\{0<\arg z<\frac{\pi}2\}$, continuous in $\bar A$, satisfying $|f(z)|\le M$ on $\partial A$ and satysfying the following growth condition: $$...
- 2,750