Skip to main content

All Questions

Filter by
Sorted by
Tagged with
1 vote
0 answers
46 views

Help with a surface of delay differential equations

This question is difficult for me to phrase, as it's very much outside of my mathematical purview. This is a question which intersects directly with my research, but as I work predominantly in ...
1 vote
0 answers
145 views

Integrability conditions imply existence of potential

I'm looking for a proof of the following well-known theorem: If $f$ is a continuously differentiable vector field in a simply connected region $G\subset \mathbb{R}^n$ which satisfies the ...
2 votes
0 answers
141 views

For a 1-parameter family of metrics, how do we compute the derivative of the intrinsic geometrical objects like curvature, Hessian, etc

Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying $$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$ where $g_0$, $...
1 vote
0 answers
55 views

Projection of a real analytic manifold onto subspace is union of real analytic submanifolds

Let $M$ be a compact connected real analytic submanifold of the Euclidean space $\mathbb{R}^{n} \times \mathbb{R}$ and denote by $\pi : \mathbb{R}^{n} \times \mathbb{R} \rightarrow \mathbb{R}^{n}$ the ...
5 votes
1 answer
329 views

Reference for the rectifiablity of the boundary hypersurface of convex open set

The boundary of any convex open set $X$ is $\mathbb R^n$ is a rectifiable hypersurface. To see this, intuitively, simply take a sphere $S_d$ with diameter $d\in(0,+\infty]$ that contains $X$. The ...
2 votes
0 answers
269 views

Extending Green's theorem from very special regions to more general regions

Green's theorem Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ ...
1 vote
1 answer
1k views

Approximation of a continuous function by a smooth one on an open set

I'm interested in the following kind of theorems : Let $M$ be a real analytic manifold and $U$ an open set of $M$. Let $f : U \to \mathbb{R}$ a continuous function. Then, there is a $C_{\infty}$ ...
2 votes
1 answer
287 views

Regularity of the reparametrization map between curves [closed]

I am looking for a reference for the following kind of results. Let $\Gamma$ be the space of Lipschitz curves $\text{Lip}([0,1]; \mathbb R^d)$ equipped with the sup norm. Let $B$ be a Borel subset of ...
2 votes
0 answers
67 views

On two functions with isodirectional gradients

Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$ \begin{equation} (\...
3 votes
1 answer
459 views

Are Carnot groups (as Carnot Caratheodory metric spaces) doubling?

I need to use the Lebesgue differentiation theorem for doubling metric measure spaces and was wondering if Carnot groups are doubling. If yes, is there any reference you can point me to? Thank you.
7 votes
2 answers
518 views

Morse lemma with least amount of regularity.

I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across ...
0 votes
1 answer
182 views

Surjectivity of "nice maps" from local properties

What tools are available from real algebraic geometry, analysis and topology to check surjectivity of a map $f:M_{1}\rightarrow\mathbb{R}^{d}$ from local properties and maybe function values? ...
5 votes
0 answers
310 views

Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...