Optimal $L_p$-estimate for elliptic operator

Let us start with the Poisson equation
begin{equation}
-Delta u=f
end{equation}
on a domain $Omega$. The classical regularity says if the the boundary of the domain is sufficiently smooth, then there exists some $p>2$, such that for all $fin L^{q}$ for $qin[p^*,p]$, where $p*$ is the conjugate of $p$, implies that $uin W_0^{1,q}$. For a $C^2$ boundary, we even have $pin(0,infty)$. What I want to do now is to apply a similar result for a general elliptic operator $L$ with $Lu=f$. To do this, I want to apply the fixed point idea which was given in Gröger’s paper. But $p>2$ is not enough good for three dimensional space, so in fact we need some $p>3$. To apply the idea given by Gröger, I need in fact an estimate $J_p$ which is given by
begin{equation}
|u_f|_{W_0^{1,p}}leq J_p |f|_{L^p},
end{equation}
where $u_f$ is the solution of the Poisson’s equation. Such a constant exists due to Gröger’s theorem, although we can only know that $J_p=J_{p^*}geq J_2=1$. But to apply results for $p>3$, an optimal estimate for $J_3$ is needed.

The difficulties here are the lacking of test functions. Like in $L^p$ theory, a test function has the form $u^{p-1}$, but then its derivative contains also the term $(p-1)u^{p-2}$ due to chain rule. Another possibility is that we can in fact give the exact solution by using singular kernels, but then the estimates seem to be non optimal.

So my question is, is there any reference giving hope to this question? Any suggestion is very welcome!

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and are thus well-known special functions, and the same holds for some elliptic integrals. But what is the relation between the named elliptic functions and the named elliptic integrals?

It seems that the Jacobi amplitude $varphi=operatorname{am}(u,k)$ is the inverse of the elliptic integral of the first kind, $u=F(varphi,k)$. Or related to this, $x=operatorname{sn}(u,k)$ is the inverse of $u=F(x;k)$. It looks to me as if all of Jacobi’s elliptic functions relate to the elliptic integral of the first kind. For other named elliptic functions listed by Wikipedia, like Jacobi’s $vartheta$ function or Weierstrass’s $wp$ function, it is even harder to see a relation to Legendre’s integrals.

Is there a way to express the inverse of $E$, the elliptic integral of the second kind, in terms of some named elliptic functions? I.e. given $E(varphi,k)=u$, can you write a closed form expression for $varphi$ in terms of $k$ and $u$ using well-known special functions and elementary arithmetic operations?

In this post the author uses the Mathematica function FindRoot to do this kind of inversion, but while reading that post, I couldn’t help wondering whether there is an easier formulation. Even though the computation behind the scenes might in fact boil down to root-finding in any case, it feels like this task should be common enough that someone has come up with a name for the core of this computation.

Does the theorem that genera vanishing on even-dim complex projective bundles are elliptic also apply for…

Ochanine proved in this paper that for genera taking values in $mathbb{Q}$-algebras, vanishing on even-dimensional projective bundles is equivalent to being an elliptic genus (i.e. a specialization of Euler’s formal group law). Does the implication “vanishing on even-dim projective bundles => specialization of Euler” still apply integrally, up to a finite set of primes (probably 2 and 3)? Just from skimming through it, it seems the argument would still go through over $mathbb{Z}[frac{1}{6}]$, but I’m not certain.

How Elliptic-Curve affects the Server Key Exchange parameters

In a finite field DHE, the server sends the following parameters in the server key exchange message:
$p$: prime
$g$: group
$g^b$: the server’s public DH key

In DHE_RSA (non anonymous DHE), the server signs the parameters. The full parameters according to the specs are:

ServerKeyExchnage($p$, $g$, $g^b$, sign(hash($mathit{server_nonce}$,
$mathit{client_nonce}$,$p$,$g$,$g^b$))

The client sends it ClientKeyExchange which contains the client’s public DH key:

ClientKeyExchange($g^a$)

If the key exchange is Elliptic-Curve not finit-field, precisely how this affects the parameters and messages above? What does it changes in terms of the exact parameters the client and server send?

is this explicit linear operator hypo-elliptic

Consider an operator of the form
$$L(phi):=Delta phi + gamma phi_{rr}$$ here the $r$ denotes derivative with respect to the radial variable (we are in $ R^N$ say where $N ge 3$).

I am curious whether I can apply some abstract results to $L$ that require $L$ to be hypo-elliptic (I kinda know what the word means but have absolutely zero experience).

My belief is this is not hypo-elliptic since I can use separation of variables (using spherical harmonics) to come up with a non smooth solution of $L(phi)=0$ but there is a nonzero chance that I am screwing up the computation and really the equation is $ L(phi)=mu$ where $mu$ some distribution supported at the origin.
thanks
Craig

Elliptic Fourier Descriptors – how to derive the coefficients?

In Kuhl and Giardina’s 1982 paper, it derives the coefficients, $a_n$ and $b_n$, by equating the two definitions of $dot{x}(t)$:

$$
begin{align*}
dot{x}(t) &= sum_{n=1}^infty alpha_n cosfrac{2n pi t}{T} + beta_n sinfrac{2n pi t}{T} \
dot{x}(t) &= sum_{n=1}^infty -frac{2npi}{T} a_n sinfrac{2n pi t}{T} + frac{2npi}{T} b_n cosfrac{2n pi t}{T}
end{align*}
$$

Where
$$
alpha_n = frac{2}{T} sum_{p=1}^K frac{Delta x_p}{Delta t_p} left( sinfrac{2n pi t_p}{T} – sinfrac{2n pi t_{p-1}}{T} right) \
beta_n = -frac{2}{T} sum_{p=1}^K frac{Delta x_p}{Delta t_p} left( cosfrac{2n pi t_p}{T} – cosfrac{2n pi t_{p-1}}{T} right)
$$

Their result is
$$
a_n = frac{T}{2n^2 pi^2} sum_{p=1}^K frac{Delta x_p}{Delta t_p} left( cosfrac{2n pi t_p}{T} – cosfrac{2n pi t_{p-1}}{T} right)\
b_n = frac{T}{2n^2 pi^2} sum_{p=1}^K frac{Delta x_p}{Delta t_p} left( sinfrac{2n pi t_p}{T} – sinfrac{2n pi t_{p-1}}{T} right)
$$

However, when I tried to derive them, I get:
$$
frac{2npi}{T} b_n = alpha_n = frac{2}{T} sum_{p=1}^K frac{Delta x_p}{Delta t_p} left( sinfrac{2n pi t_p}{T} – sinfrac{2n pi t_{p-1}}{T} right)\
b_n = frac{1}{npi} sum_{p=1}^K frac{Delta x_p}{Delta t_p} left( sinfrac{2n pi t_p}{T} – sinfrac{2n pi t_{p-1}}{T} right)
$$

and similar for $a_n$. The results are off exactly by $frac{2npi}{T}$, the coefficient before $b_n$ in the second definition. So maybe I missed something in my derivation.

Showing the Monge-Ampere equation is elliptic

I have a question about my books definition of ellipticy on how it relates to the Monge-Ampere equation in $mathbb R^2$.

The Monge-Ampere equation is as follows. Let $Omega$ be an open subset of $ mathbb R^2={x=(x_1,x_2)}$ and let $uin C^2(Omega)$ satisfy $u_{x_1x_1}(x)u_{x_2x_2}(x)-u^2_{x_1x_2}(x)-f(x)=0$ where $f>0$ in $Omega$.

The definition of ellipticy given in my book is as follows:

Consider a general differential equation $F[u]=F(x,u(x),Du(x),D^2u(x))=0$ where $F:S:=Omega times mathbb R times mathbb R^d times S(d,mathbb R) to mathbb R$ where $S(d,mathbb R)$ is the space of real symmetric $dtimes d$ matricies. Elements of $ S$ are written as $(x,z,p,r)$ where $p=(p_1,…,p_d)$ and $r=(r_{ij})_{1le i,j le d}$ The differential equation $F[u]$ is said to be elliptic at $u$ if $left( dfrac{partial F}{partial r_{i,j}}(x,u(x),Du(x),D^2u(x))right)_{1 le i,j le d} $ is positive definite.

So using this definition how do we compute $ dfrac{partial F}{partial r_{i,j}}(x,u(x),Du(x),D^2u(x)) $ for each pair of $(i,j)$? We can see that $r_{i,j}=u_{x_ixj}$ so for example does $$ dfrac{partial F}{partial r_{1,1}}(x,u(x),Du(x),D^2u(x)) = dfrac{partial}{u_{x_1x_1}}left(u_{x_1x_1}(x)u_{x_2x_2}(x)-u^2_{x_1x_2}(x)-f(x)right)=u_{x_2x_2}(x) $$
Then by this logic $left( dfrac{partial F}{partial r_{i,j}}(x,u(x),Du(x),D^2u(x))right)_{1 le i,j le d}= begin{bmatrix}
u_{x_2x_2} & -2u_{x_1x_2} \
-2u_{x_1x_2} & u_{x_1x_1} \
end{bmatrix}$
but if this is correct why is this matrix positive definite? Also where does the condition of $f>0$ come into play.

Any help is appreciated!

Quotient of quarter periods K’ and K of Jacobi elliptic functions

There are several ways to express the quarter period $K$,
$$
K(m)=int_0^{pi/2}frac{mathrm{d}theta}{sqrt{1-msin^2theta}},
$$

as a power series (and thus for $K’=K(1-m)$ there are, too) and also efficient ways of calculation (like agm).
But are there any known formulas for the quotient
$$
K’/K
$$

? I’m new to the field of elliptic functions.

Any ideas, general results etc.?

Elliptic curve with same number of point over two different fields

Following a discussion with a number theory professor, we arrived at the following question :
Can we find an elliptic curve (short form : $y^2=x^3+ax+b$) with an identical number of points on two different finite fields: $GF(q)$ and $GF(q^r)$ with $q$ prime.

Looking at some brute force results, we found only one result :
the elliptic curve defined by $y^2=x^3+2x+1$ with fields of order 3 and 9.

We only looked at first 100’s primes $q$, with power up to 10 so far, but we cannot understand what’s specific about this choice, and why is this the only solution we can find so far…

any idea? Any clue as how we could attack this problem?

Thanks

Edit : Using Weil conjecture, we can start with, $forall q$, $exists alpha in mathbb{C}$ such that $forall r$ :
$$ #E(mathbb{F}_{q^r})= (q^r+1) – (alpha^r+bar{alpha}^r) $$
with $alpha$ and $bar{alpha}$ conjugate, such that $midalphamid=sqrt{q}$