# Tagged Questions

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### $C^{2, \alpha}$ regularity for elliptic equations with Neumann boundary conditons

Say $\Omega\subseteq \mathbb{R}^n$ is a bounded open set and $0<\alpha<1$. I need some $C^{2, \alpha}(\overline\Omega)$ regularity result for elliptic equations with Neumann boundary conditions ...
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### Why do functions with compact support include those that vanish at infinity?

The support of a function is defined in Wikipedia as "the set of points where the function is not zero-valued, or the closure of that set". Functions with compact support in $X$ are defined in ...
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### Meaning of quasiperiodicity in classical KAM

I'm learning about the classical KAM theorem, and I can't quite infer precisely what the term "quasi-periodic solution" means in the theorem's statement. I'm reading the following introductory note: ...
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### Why this functional isn't differentiable?

I saw in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. That the functional ...
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### What is the meaning of $f(x) \rightarrow a$ as $g(x) \rightarrow b$?

The motivating example was the case: $$f(x, y)\rightarrow0\mathrm{\ \ as\ \ }\sqrt{x^2+y^2}\rightarrow\infty$$ What exactly does this mean? I might define it as: Any sequence $x_n$ with ...
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### Definition of $C^k$ boundary

Can someone give me a resonable definition of $C^k$ boundary, e.g., to define and after give a brief explain about the definition. I need this 'cause I'm not understanding what the Evan's book said. ...
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### question about Morse theory in Hilbert space

This is sade to be the Morse theory in Hilbert space ,and i want to know the definition (or where i can find it ) of : The qth singular relative homology groupe The qth critical group Please; ...
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### Definition of metastability

I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
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### Problem of understanding

Please , can someone help me to understand this part of text please What it means "... for some $k\in \mathbb{N} \cup \lbrace 0 \rbrace$ and uniformly fore a.e $t\in [0,2\pi]$" and please how to ...
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### Is a function always a monotonically increasing function

For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function? Alternatively, how does the definition of a limit guarantee that ...
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this is the lemma of Whitehead And i really don't understand the proof How to see that $k$ is well defined (i.e how to write an element from $X\cup_{\varphi_i}e^{\lambda} , i=0,1$ ) and how to ...
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### The definition of $p$ capacity of a set $A\subset\mathbb{R}^n$

I am having a bit of difficult understanding the definition of the $p$-capacity of a set $A\subset\mathbb{R}^n$ and I was wondering if anyone would be able to clarify whether I have the right idea or ...
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### How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$

Let $u(x,y)\to l$ as $(x,y)\to z_0=(x_0,y_0).$ How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$ Choose $\epsilon>0.$ Then ...
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### Direct sums and direct products

This question has been in my head for a while. And today it appears again when I am reading Arveson's book on $C^*$-algebras. He says Countable direct products of Polish spaces are Polish. ...
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### what does it mean to say a space is norm separable?

I came across in my textbook the term: norm separable. I looked in the textbook and online and could not find a definition.
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### Definition of matrix derivative

Let $A$ be an $m \times n$ matrix. Let $a_{ij}$ be an element of $A$. What does the notation $\frac{\partial A}{\partial a_{ij}}$ mean?
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### Help in Analysis

I'm studying the article "On W1;p Estimates for Elliptic Equations in Divergence Form" of L. A. CAFFARELLI and I. PERAL. There, you can find We use the classical Hardy-Littlewood maximal operator, ...
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### Rate of convergence of double sequences

Suppose $\{X_{n,m} \}$ be a double sequence of real numbers and suppose $\lim_{n}\lim_{m}X_{n,m}=X$. What is the definition and reference for the rate of convergence of double sequences?
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### Differentiability of a matrix function

To prove the differentiability of a function defined on $M\in M_n$, how should I adapt the definition of differentiability? $h^{-1}[f(x+h)-f(x)-hL(x)]\to 0$  How do I take the reciprocal of a ...
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### Definition clarifications on functions of vectors and their derivatives

Definition clarifications would be appreciated: How do I interpret the following ? For $f: R^n\to R^m$, $Df(\vec\xi)(\vec{x})$ in differentiation of a vector function. I know it as a function that ...
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### Definition of $\ell_\infty$

What does the $\ell_\infty$ space stand for? Thank you.