# Recent MathOverflow Questions

### Approximation of logarithms (explicit Kronecker)

Math Overflow Recent Questions - Tue, 11/21/2017 - 08:05

Let $\epsilon > 0$ and $N$ be some positive integer. By a theorem of Kronecker we can always find a real number $t ≥ 1$ such that for all integers $1 ≤ k ≤ N$ there is an integer $h_k$ with

$$|t \log(k) - h_k| < \epsilon.$$

(Actually, the theorem deals with numbers that are linearly independent over $\mathbb{Q}$, but we can apply this theorem to the values $\log(p)$ where $p ≤ N$ are primes and choose $\epsilon$ arbitrary small.) When $t \in \mathbb{R}$ this is trivial, then just choose $t = 0$, but we shall be interested in the case as $t$ gets very large. If we now define

$$T(\epsilon, N) = \inf \{ t \in \mathbb{R}_{≥1} \mid \forall 1 ≤ k ≤ N \ \exists h_k \in \mathbb{Z} : |t \log(k) - h_k| < \epsilon\}.$$

My question is: what can we say about the growing of $T(\epsilon, N)$ as $\epsilon \to 0$ and $N \to \infty$? Is there a theorem or some literature dealing with this kind of question? (I assume that for fixed $\epsilon > 0$ (very small) we should have something like $t$ grows as $N!$ as $N \to \infty$.)

### Matchings in graphs with infinite chromatic number

Math Overflow Recent Questions - Tue, 11/21/2017 - 07:35

Is there a simple, undirected graph $G= (V,E)$ with $\chi(G) \geq \aleph_0$, and if $M\subseteq E$ is a matching then $|M|<\chi(G)$?

### Internal Hom of Deligne' tensor product

Math Overflow Recent Questions - Tue, 11/21/2017 - 07:08

I read the following statement (equation 22) in "Monoidal 2-structure of bimodule categories" by Justin Greenough:

Let $\mathcal{C}$ be a finite tensor category (abelian k-linear rigid monoidal category with simple unit and finite dimensional Hom spaces). Let $\mathcal{M}$ and $\mathcal{N}$ be exact left module categories over $\mathcal{C}$.

We introduce the left $\mathcal{C}$-module structure in $\mathcal{M} \boxtimes \mathcal{N}$ (the Deligne' tensor product of $\mathcal{M}$ and $\mathcal{N}$) by:

$$X \otimes (M \boxtimes N) = (X \otimes M) \boxtimes N,$$ where $X \in \mathcal{C}$.

Then the equation 22 tells us that $$\underline{Hom}_{\mathcal{M} \boxtimes \mathcal{N}}(M \boxtimes N, S \boxtimes T) = \underline{Hom}_{\mathcal{M}}(M, S) \otimes \underline{Hom}_{\mathcal{N}}(N,T),$$ where $\underline{Hom}_{*}$ are internal hom for left $\mathcal{C}$ structure in $\mathcal{M} \boxtimes \mathcal{N}$, $\mathcal{M}$ and $\mathcal{N}$.

Now, let us consider the simple case:

let $\mathcal{C}$ be a unitary fusion category and $\mathcal{M} = \mathcal{N} = \mathcal{C}$. Then by the definition of internal Hom and the equation above, we have

$$Hom_{\mathcal{C} \boxtimes \mathcal{C}}(1 \boxtimes 1, X \boxtimes X^*) \cong Hom_{\mathcal{C}}(1, \underline{Hom}_{\mathcal{C} \boxtimes \mathcal{C}}(1 \boxtimes 1, X \boxtimes X^*))\\ \cong Hom_{\mathcal{C}}(1, \underline{Hom}_{\mathcal{C}}(1, X) \otimes \underline{Hom}_{\mathcal{C}}(1,X^*)),$$ where $1$ is the unit of $\mathcal{C}$ and $X$ is a simple object in $\mathcal{C}$ such that $X \ncong 1$ and $X^*$ is the left (or right) adjont of $X$.

Since $\underline{Hom}_{\mathcal{C}}(1, X) = X$ and $\underline{Hom}_{\mathcal{C}}(1, X^*) = X^*$, we have

$$\{0\} = Hom_{\mathcal{C}}(1,X) \otimes Hom_{\mathcal{C}}(1, X^*) \cong Hom_{\mathcal{C} \boxtimes \mathcal{C}}(1 \boxtimes 1, X \boxtimes X^*) \cong Hom_{\mathcal{C}}(1, X \otimes X^*) \neq \{0\}.$$

So it seems that we have a contradiction here. Can anyone tell me if I made a mistake somewhere? Does the equation (22) in "Monoidal 2-structure of bimodule categories" hold? Thank you in advance!

### Is there any "extra regularity" to the solution to Poisson's equation posed on a 3-dimensional polyhedron?

Math Overflow Recent Questions - Tue, 11/21/2017 - 07:00

I am trying to write a proof and I am out of my depth. I need an elliptic regularity result of the form $$\|u\|_{H^{1+\epsilon}(\Omega)} \le C \|f\|_{L^2(\Omega)}$$

for some $\epsilon >0$ where $u$ is the weak solution to either of the following PDEs.

\begin{align*} \nabla\cdot\nabla u &= f\quad x\in \Omega\\ u &= u_D\quad x \in \partial \Omega_D\\ \nabla u\cdot n& = 0\quad x\in \partial\Omega_N \end{align*}

or the pure Nuemann problem with the further restriction that $\int_\Omega f \mathrm{d}x = 0$, \begin{align*} \nabla \cdot \nabla u &= f\quad x\in \Omega,\\ \nabla u \cdot n &= 0 \quad x\in \partial\Omega,\\ \int_\Omega u\, \mathrm{d}x &= 0. \end{align*}

This result is known for the case of two dimensional polygons (I am interested in 3-dimensional polyhedra), and the largest $\epsilon$ depends on the measure of the interior angles.

I have looked into a few promising papers with "Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra" being among them. I suspect that Theorem 1.4, in that paper (which references theorem 2 in On the Agmon-Miranda Maximum Principle for Solutions of Elliptic Equations in Polyhedral and Polygonal Domains), implies what I need, but, like I said, I am out of my depth here and quickly get bogged down, and completely lost.

### Classifications of cubic surfaces

Math Overflow Recent Questions - Tue, 11/21/2017 - 06:46

Is there a known classification of singular cubic surfaces over finite fields?

### Is there an explicit expression for Chebyshev polynomials modulo $x^r-1$?

Math Overflow Recent Questions - Tue, 11/21/2017 - 06:25

This is an immediate successor of Chebyshev polynomials of the first kind and primality testing and does not have any other motivation - although original motivation seems to be huge since a positive answer (if not too complicated) would give a very efficient primality test (see the linked question for details).

Recall that the Chebyshev polynomials $T_n(x)$ are defined by $T_0(x)=1$, $T_1(x)=x$ and $T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$, and there are several explicit expressions for their coefficients. Rather than writing them down (you can find them at the Wikipedia link anyway), let me just give a couple of examples: $$T_{15}(x)=-15x(1-4\frac{7\cdot8}{2\cdot3}x^2+4^2\frac{6\cdot7\cdot8\cdot9}{2\cdot3\cdot4\cdot5}x^4-4^3\frac{8\cdot9\cdot10}{2\cdot3\cdot4}x^6+4^4\frac{10\cdot11}{2\cdot3}x^8-4^5\frac{12}{2}x^{10}+4^6x^{12})+4^7x^{15}$$ $$T_{17}(x)=17x(1-4\frac{8\cdot9}{2\cdot3}x^2+4^2\frac{7\cdot8\cdot9\cdot10}{2\cdot3\cdot4\cdot5}x^4-4^3\frac{8\cdot9\cdot10\cdot11}{2\cdot3\cdot4\cdot5}x^6+4^4\frac{10\cdot11\cdot12}{2\cdot3\cdot4}x^8-4^5\frac{12\cdot13}{2\cdot3}x^{10}+4^6\frac{14}{2}x^{12}-4^7x^{14})+4^8x^{17}$$ It seems that $n$ is a prime if and only if all the ratios in the parentheses are integers; this is most likely well known and easy to show.

The algorithm described in the above question requires determining whether, for an odd $n$, coefficients of the remainder from dividing $T_n(x)-x^n$ by $x^r-1$, for some fairly small prime $r$ (roughly $\sim\log n$) are all divisible by $n$. In other words, denoting by $a_j$, $j=0,1,2,...$ the coefficients of $T_n(x)-x^n$, we have to find out whether the sum $s_j:=a_j+a_{j+r}+a_{j+2r}+...$ is divisible by $n$ for each $j=0,1,...,r-1$.

The question then is: given $r$ and $n$ as above ($n$ odd, $r$ a prime much smaller than $n$), is there an efficient method to find these sums $s_j$ without calculating all $a_j$? I. e., can one compute $T_n(x)$ modulo $x^r-1$ (i. e. in a ring where $x^r=1$) essentially easier than first computing the whole $T_n(x)$ and then dividing by $x^r-1$ in the ring of polynomials?

(As already said, only the question of divisibility of the result by $n$ is required; also $r$ is explicitly given (it is the smallest prime with $n$ not $\pm1$ modulo $r$). This might be easier to answer than computing the whole polynomials mod $x^r-1$.)

### Solutions to a certain Birkhoff-interpolation problem

Math Overflow Recent Questions - Tue, 11/21/2017 - 05:58

$\newcommand{\CC}{\mathbb{C}}$

Let for $n > 1$ and $m = n-1$ $$p = x^n + a_1 x^{n-1} + \cdots + a_m x$$ be a polynomial with $a_i \in \CC$. Call $p^{(i)}(x) = \frac{d^ip}{dx^i}(x)$.

The following question would have an impact on the problem, if sequences of resultants $\mathrm{res}_x(p(x), p^{(n-i)}(x))$, $i = 1,\ldots,s$ are regular sequences:

Take $s < m$ and consider the space $X$ of monic polynomials $p$ as above such that $\alpha_1,\ldots,\alpha_s \in \CC$ (not necessarily distinct, could be even all equal) exist so that

$$p^{(n-1)}(\alpha_1) = p^{(n-2)}(\alpha_2) = \cdots = p^{(n-s)}(\alpha_s) = 0$$

and $$p(\alpha_1) = \cdots = p(\alpha_s) = 0$$

Question:

Can one show that for a certain solution $p(x) \in X$ with $a_{s+1} = 0$ there exists in every neighbourhood (identifying $X \cong \CC^m$) a solution in $X$ with $a_{s+1} \neq 0$?

This is obviously the case for $2 s + 1 \leqslant m$ where one can think of the $\alpha_i$ as fixed and consider the problem as a statement of linear algebra. If $2 s + 1 > m$ it may become necessary to "move the $\alpha_i$ a bit", to get $a_{s+1} \neq 0$.

I did some computer calculations with Maple and it always seemed to be possible to do such a small move, leading to $a_{s+1} \neq 0$, but I could not find a general proof.

Is the answer to the above question maybe already known to specialists in (Birkhoff-)interpolation? What could be helpful to read for making progress on this problem?

### Extracting a divergent subsequence [on hold]

Math Overflow Recent Questions - Tue, 11/21/2017 - 05:41

Let $(x_n)$ be a real sequence in $l^p$ for a fixed $p>1$ but not in $l^1$.

Is there a way to extract a sequence $(x_{\phi(n)})$ which is $l^{1+\varepsilon}$ and not $l^{1-\eta}$? (or a counter example)

### Do prime ideals in polynomial ring generate prime ideals in the ring of holomorphic functions?

Math Overflow Recent Questions - Tue, 11/21/2017 - 05:35

Suppose that $I \subset \mathbb C[z_1,\dots, z_n]$ is a prime ideal. Consider the ideal $I_{hol}$ in the ring of holomorphic functions $f: \mathbb C^n\to \mathbb C$ generated by polynomials from $I$.

Is $I_{hol}$ prime?

### Sharp constant for inequality with convex functions

Math Overflow Recent Questions - Tue, 11/21/2017 - 05:04

This is a follow up to this question, where the optimal constant was left open.

Let $P \subset \mathbb{R}^n$ be bounded, convex, and open. Let $$\mathcal{H} := \{f : P \rightarrow \mathbb{R} : f\text{ is convex and }\int_P f d\lambda = 0\}$$ Question:

What is the largest possible constant $\alpha > 0$ purely depending on $n$ and $P$ such that $$\forall f \in \mathcal{H}: \int_P \left| f \right| d\lambda \geq -\alpha \inf_{P} f$$

Notes:

The question is the same if one restricts $\mathcal{H}$ to functions with $\inf_P f = -1$.

This answer provides a lower bound of $\alpha \geq 4^{-n-1} |P|$. @fedja already mentions that a similar argument can be used to obtain a sharp constant, but I haven't been able to work it out. Hence this question.

### Degree of vertex boundary on Hamming Cube

Math Overflow Recent Questions - Tue, 11/21/2017 - 04:37

Let's take the Hamming graph $Q_n=(V,E)$, and take some subset $S \subseteq V$. Now, an inequality of Bobkov says that the vertex boundary of $S$ (number of neighbours of vertices in $S$) and the edge boundary of $S$ (number of edges from vertices of $S$ to vertices outside $S$) can't both be too small. I was wondering if we can include somewhere there the number of edges between the vertices of the vertex boundary. For instance, for a given size $2^k$, we know that among all sets of such size $k$-dimensional subcubes are the ones that minimize the edge boundary. For such a subcube, though, not only is the vertex boundary large, $(n-k)|S|$, but there are many edges between the vertices of the vertex boundary.

On the other hand, we can take a ball of radius $r$, which has a small edge boundary and large vertex boundary, and the vertex boundary forms an independent set.

So I guess my question is: can we say that for some subset $S$ of a given size, the vertex boundary and the number of edges between vertices of the vertex boundary (maybe normalized somehow) can not both be too small? what if we assume that $S$ is connected? monotone?

### A good reference to the general Chinese Remainder Theorem

Math Overflow Recent Questions - Tue, 11/21/2017 - 03:55

I am writing a paper on the topology of the Golomb space and need a good (standard) reference to the following

General Chinese Remainder Theorem. For integer numbers $a_1,\dots,a_n$ and positive integers $b_1,\dots,b_n$ the intersection $\bigcap_{i=1}^n(a_i+b_i\mathbb Z)$ is not empty if and only if $a_i-a_j\in d_{i,j}\mathbb Z$ for any $1\le i<j\le n$, where $d_{ij}$ denotes the largest common divisor of the numbers $b_i,b_j$.

Looking at the internet, I found this paper in which the General Chinese Remainder Theorem is formulated as an exercise and another paper in which this theorem is proved.

But I am suspecting that such General Chinese Remainder Theorem should be proved in some standard (undergraduate) textbook in Number Theory. I need it for a proper reference. Please help!

I understand that this is rather a reference request and not a problem of research level. In case of downvotes I will delete it as soon as will get a proper answer from experts.

### $q$-analog of an integral from quantum field theory?

Math Overflow Recent Questions - Tue, 11/21/2017 - 03:48

Let the function $f_q(x,y,z)$ completely symmetric in $x,y,z$ depend on $e^x,e^y,e^z$ only ($q$ is considered as a parameter). It is known that $f_q$ is a finite combination of elementary functions and theta functions

$$\theta_q(z)=(q;q)_\infty(z;q)_\infty(q/z;q)_\infty$$

and has the following properties \begin{align} &f_q(x+c,y+c,z+c)=e^cf_q(x,y,z),\tag{1}\\ \\ &f_q\left(\frac{2\pi i}{3},0,z\right)=\left(e^z-1\right) \frac{\theta_q\left(e^{\frac{4 i \pi }{3}-z}\right)}{\theta_q\left(e^{-z}\right)},\tag{2}\\ \\ &f_q(x,y,z)=e^x+e^y+e^z\\ &+\frac32\left(e^x+e^y+e^z\right)\left(1+e^{x-y}+e^{y-x}+e^{y-z}+e^{z-y}+e^{z-x}+e^{x-z}\right)q\\ &+i\frac{\sqrt{3}}{2}\left(e^{-x}+e^{-y}+e^{-z}\right)\left(e^{2x}+e^{2y}+e^{2z}\right)q+O(q^2) \tag{3} \\ \end{align}

Q: Is it possible to find $f_q(x,y,z)$ from these data?

Note: According to comment by fedja the above conditions were not enough to determine the function $f$. Nevertheless is it possible to make further conjectures for the function $f$ from its definition in eq. $(5)$ or find it by calculating integral $(5)$ directly?

Motivation:

This question is related to $q$-analog of the following integral from quantum field theory due to F.A. Smirnov (also see this MSE post): \begin{align} &\int_{-\infty}^{\infty}\prod_{j=1}^3\Gamma\Bigl(\frac 1 3 -\frac {\alpha-\beta_j}{2\pi i}\Bigr) \Gamma\Bigl(\frac 1 3 +\frac {\alpha-\beta_j}{2\pi i}\Bigr)(3\alpha-\sum\beta_m)e^{-\frac{\alpha}2}d\alpha\nonumber\\ &=\frac{(2\pi \Gamma(\frac 2 3))^2}{\Gamma(\frac 4 3)} \prod_{k\neq j}\Gamma\Bigl(\frac 2 3 -\frac {\beta_k-\beta_j}{2\pi i}\Bigr) e^{-\frac12\sum\beta_m}\sum e^{\beta_m},\quad |\text{Im}~\beta_j|<2\pi/3.\tag{4} \end{align}

Exact definition of $f_q$ is as follows

\begin{align} &\int_{-\infty}^\infty\prod_{j=1}^3\frac{\Gamma_q\left(\frac13+\frac{x-\beta_j}{2\pi i}\right)}{\Gamma_q\left(\frac23+\frac{x-\beta_j}{2\pi i}\right)}\frac{\sum_{m}q^{-\frac{\beta_m}{2 \pi i}}-\left(1+q^\frac13+q^{-\frac13}\right) q^{-\frac{x}{2 \pi i}}}{\prod_{m}\sin\left(\frac{\pi}3-\frac{x-\beta_m}{2 i}\right)}e^{-\frac{x}2}dx\\ &=\frac{-2\pi i}{(q;q)_{\infty }^9}\frac{\Gamma_q^2\left(\frac23\right)}{\Gamma_q\left(\frac43\right)}\frac{q^{5/9}}{(1-q)^2}e^{-\frac{1}{2} (\beta_1+\beta_2+\beta_3)} q^{-\frac{\beta_1+\beta_2+\beta_3}{6 \pi i}}\prod_{k\neq j}\Gamma_q\left(\frac23+\frac{\beta_j-\beta_k}{2\pi i}\right)\\ &\times\frac{\theta_q(q^{\frac{\beta_1-\beta_2}{2\pi i}})}{\sin\frac{\beta_1-\beta_2}{2 i}}\frac{\theta_q(q^{\frac{\beta_2-\beta_3}{2\pi i}})}{\sin\frac{\beta_2-\beta_3}{2 i}} \frac{\theta_q(q^{\frac{\beta_3-\beta_1}{2\pi i}})}{\sin\frac{\beta_3-\beta_1}{2 i}} \cdot f_{q_1}(\beta_1,\beta_2,\beta_3).\tag{5} \end{align} where $q_1=e^{-\frac{4 \pi ^2}{\ln(1/q)}}$ and $\Gamma_q$ is the $q$-Gamma function.

I have found strong evidence that $f_q$ has a relatively simple closed form expression.

Eq. $(2)$ is a condition that the difference of LHS and RHS of $(5)$ is an entire function. $(3)$ has been extracted from numerical calculations.

I could't find how to express $(5)$ in terms of q-series in a practical way (in principle it is possible to express $(5)$ in terms of $q$-series, but it contains a sum of $9$ $q$-hypergeometric functions ${}_3\phi_2$'s and $9$ of its derivatives plus $27$ Lambert series) and decided instead to guess the RHS and then prove it using Liouville's theorem.

### Smooth proper fibration of complex projective varieties

Math Overflow Recent Questions - Tue, 11/21/2017 - 03:35

Let $X$ be a smooth projective algebraic variety over the complex numbers.

(a) Do there exist:

a smooth proper map $\pi : \mathcal{X}\to S$ of algebraic varieties over the complex numbers, such that $\mathcal{X}$ and $S$ are smooth, $S$ is connected, $X$ is isomorphic to the fiber of $\pi$ over some $\mathbf{C}$-point of $S$, and such that there exists $s\in S(\mathbf{C})$, a $K$-scheme $X_0$, and an isomorphism of $\mathbf{C}$-schemes: $$(X_0)_{\mathbf{C}} \simeq \mathcal{X}_s$$

with $K$ either:

(a.1) $K = \mathbf{Q}$, (a.2) $K/\mathbf{Q}$ a finite extension, (a.3) $K = \bar{\mathbf{Q}}$?

(b) What can be reasonably called an "obstruction" to (a.i), $i = 1,2,3$?

### Jacobian and configuration space and massey products

Math Overflow Recent Questions - Tue, 11/21/2017 - 03:22

Let $X$ be a comapct Riemann surface of genus $g$ and let $J\: : \: X\to \mathbb{C}^{g}/\Lambda$ be the Abel-Jacobi map. This map is a smooth embedding. Let $p\in X$ such that $J(p)=\Lambda$ and consider $$J\: : \: X\setminus p \to (\mathbb{C}-\Lambda)/\Lambda.$$ Then $J^*$ induces a surjection in complex de Rham cohomology. $J$ is injective and hence is defined at the level of the (ordered) configuration space $$\operatorname{Conf}_{l}(J)\: : \: \operatorname{Conf}_{l}(X\setminus p) \to \operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda).$$ Let $V_{1}\subset H^{2}(\operatorname{Conf}_{l}(X\setminus p), \mathbb{C})$ , resp. $V_{2}\subset H^{2}(\operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})$ be the space generated by Massey products $(a_{1}, \dots, a_{n})$ between degree $1$ elements, for $n\geq 2$. Does $\operatorname{Conf}_{l}(J)^{*}$ induces a surjection $$\operatorname{Conf}_{l}(J)^{*}\: : \: \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{l}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})\oplus V_{2}\to \mathbb{C}\oplus H^{1}(\operatorname{Conf}_{l}(X\setminus p), \mathbb{C})\oplus V_{1}?$$ Notice that $\mathbb{C}$ represents the $0$-th cohomology group.

### Pushforward maps for cohomology of coherent sheaves

Math Overflow Recent Questions - Tue, 11/21/2017 - 03:01

Let $X$ be a smooth projective algebraic variety over a field $k$, of dimension $n$, and let $Z$ be a smooth closed subvariety of dimension $m$, with $i: Z \hookrightarrow X$ the inclusion map.

For any locally free coherent sheaf $\mathcal{F}$ on $X$, there is a pullback map $$\imath^*: H^i(X, \mathcal{F}) \to H^i(Z, \iota^* \mathcal{F});$$ and via Serre duality we have isomorphisms $H^i(X, \mathcal{F})^\vee = H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)$ and $H^i(Z, \iota^* \mathcal{F})^\vee = H^{m-i}(Z, \iota^* \mathcal{F}^\vee \otimes \omega_Z)$, where $\omega_X$ and $\omega_Z$ are the dualising sheaves. Setting $j=m-i$ and $\mathcal{G} = \mathcal{F}^\vee$, we conclude that there is a pushforward map $$\imath_*: H^j(Z, \iota^* \mathcal{G} \otimes \omega_Z) \to H^{j + c}(X, \mathcal{G} \otimes \omega_X),$$ for any $j$ and any locally free coherent sheaf $\mathcal{G}$ on $X$, where $c = n-m$ is the codimension of $Z$ in $X$.

Does this map have an intrinsic description (not using Serre duality)? Can it be defined without assuming that $X$ be projective, or that $\mathcal{G}$ be locally free?

### Reference request for a binomial identity

Math Overflow Recent Questions - Tue, 11/21/2017 - 02:02

I stumbled upon the following (perhaps well-known) identity for a positive integer $k$: $$\sum_{j=0}^n\frac{1}{(k-1)j+1}\binom{kj}{j}\binom{k(n-j)}{n-j}=\frac{1+kn}{1+(k-1)n}\binom{kn}{n}.$$

Could you please give me references where I can find a proof?

### Could we solve the vector value ODE in a approximation way?

Math Overflow Recent Questions - Tue, 11/21/2017 - 01:57

It seems I am too fast to ask the question without thinking carefully.

The equation I consider is find $w:\Omega \to R$ satisfied : $$\frac{Du}{\sqrt{1+|\nabla u|^2}}=Dw. \tag{*}$$ Where $u:\Omega \to R$ is given and $u=0$ on $\partial \Omega$.

It is obvious this equation could not be solved for general $u$, the central obstacle is the frobenius condition(integrable condition).But I still wish to get some more information about this equation, in particular my eager could divide into two part:

1.Could the $u$ for which $(*)$ is solvable is dense in some suitable space, maybe $C^{2}(\Omega)$?

2.For the $u$ which make $(*)$, what information of $w$ could we get from $u$?

Background: I am try to get some geometric explanation for certain PDE similar to mean curvature equation.

I will be appreciate to any related answer and remark.

### Am I allowed to say "first-order Vopěnka cardinal"?

Math Overflow Recent Questions - Tue, 11/21/2017 - 01:11

For a cardinal $\kappa$ such that $V_{\kappa}$ satisfies Vopěnka's principle as a first-order axiom schema, am I allowed to say "first-order Vopěnka cardinal", or is there any kind of standard term for it?

### Is there a $W^{2,2}$ isometric embedding of the flat torus into $\mathbb{R}^3$?

Math Overflow Recent Questions - Mon, 11/20/2017 - 22:21

It is well known that there exists a $C^1$ isometric embedding of flat torus into $\mathbb{R}^3$, and that this embedding cannot be $C^2$.

Is there a $W^{2,2}$ isometric embedding? (i.e an isometric map $f \in W^{2,2}(\mathbb{T}^2,\mathbb{R}^3)$?)