Questions tagged [maximum-principle]
The maximum-principle tag has no summary.
26 questions
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Cotlar type inequality for Riesz transforms
Let $R^s_\mu(x)= \int \frac{y-x}{|y-x|^{s+1}}d\mu(y), x,y \in \mathbb{R}^d, 0<s<d$ be the Riesz transform (of index $s$).
I would like to understand the proof of the following inequality. There ...
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1
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489
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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 ...
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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)\...
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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....
- 3,960
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138
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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 ...
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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
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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
$$...
- 1,795
1
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1
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316
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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 ...
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2
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441
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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.
...
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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\...
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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$. ...
- 29.1k
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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 ...
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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.$$
Can we use the ...
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Maximum principle for poly-harmonic equations
If $u_1\geq 0$ and $u_1\neq 0$, and satisfies
$$-\Delta u_1=|u_1|^{\frac{4}{n-2}} u_1\quad \text { on }\, \mathbb{R}^n,\quad n\geq 3,$$
it follows from maximum principle that $u_1>0$. My question ...
- 569
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Strong maximum principle for weak solutions still holds?
By De Giorge, Nash and Moser solutions of
\begin{equation}
\operatorname{div} (A(x) Du) = 0
\end{equation}
where $Du$ denotes the gradient of $u$ and $A$ is a $\lambda,\Lambda$ elliptic matrix. ...
- 161
2
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1
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281
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Is there a maximum principle for CR functions over domains inside CR manifolds?
I am new to this area and I am a bit confused by the literature. Is there a strong maximum principle for CR functions over domains in a CR manifold, please? If so, could someone please state it (...
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Does the real part of the cross ratio satisfy a maximum principle on a domain in any real submanifold?
Let $C(p_1, p_2; p_3, p_4)$ denote the cross-ratio of the $4$ points $p_i$, for $i = 1, \ldots, 4$, thought of as a holomorphic function on
$$ \Omega = \{ (p_1, p_2, p_3, p_4) \in \mathbb{C}P^1 \times ...
- 5,377
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181
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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
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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_\...
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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
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173
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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' $\...
- 5,183
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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 ...
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271
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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
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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{...
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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 ...
- 1,041
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1
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180
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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:
$$...
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