Frobenius actions on the de Rham cohomology of Drinfeld modules

We study the action of endomorphisms of a Drinfeld A-module \phi on its de Rham cohomology H_{DR}(\phi,L) and related modules, in the case where \phi is defined over a field L of finite A-characteristic \mathfrak{p}. Among others, we find that the nilspace H_{0} of the total Frobenius Fr_{DR} on H_{DR}(\phi,L) \mathfrak{p} has dimension h = height of \phi. We define and study a pairing between the \mathfrak{p}-torsion _{\mathfrak{p}}\phi of \phi and H_{DR}(\phi,L), which becomes perfect after dividing out H_{0}

A formula for the probability of the exponents of finite p-groups

In this paper, I will introduce a link between the volume of a finite p-group in the Cohen-Lenstra measure and partitions of a certain type. These partitions will be classified by the output of an algorithm. As a corollary, I will give a formula for the probability of a p-group to have a specific exponent

On the torsion of optimal elliptic curves over function fields

For an optimal elliptic curve E over \mathbb{F}_{q}(t) of conductor \mathfrak{p}\cdot\infty, where \mathfrak{p} is prime, we show that E(F)_{tor} is generated by the image of the cuspidal divisor group

Estimates of the second-order derivatives for solutions to the two-phase parabolic problem

The L_{\infty}-estimates of the second derivatives for solutions of the parabolic free boundary problem with two phases \Delta u-\partial_{t}u=\lambda^{+}\chi_{\left\{ u>0\right\} }-\lambda^{-}\chi_{\left\{ u<0\right\} }\textrm{in }B_{1}^{+}\times]-1,0],\textrm{ }\lambda^{\pm}\geq0,\lambda^{+}+\lambda^{-}>0, satisfying the non-zero Dirichlet condition on \Pi_{1}:=\left\{ (x,t):\left|x\right|\leq1,x_{1}=0,-1<t\leq0\right\}, are proved

Partial regularity for local minimizers of splitting-type variational integrals

We consider local minimizers u:\mathbb{R}^{n}\supset\Omega\rightarrow\mathbb{R}^{N} of anisotropic variational integrals of (p,q)-growth with exponents 2\leq p\leq q\leq\mbox{min}\left\{ 2+p,p\frac{n}{n-2}\right\}. If the integrand is of splitting-type, then partial C^{1}-regularity of u is established

A 2D-variant of a theorem of Uraltseva and Urdaletova for higher order variational problems

If \Omega is a domain in \mathbb{R}^{2} and if u:\Omega\rightarrow\mathbb{R} locally minimizes the energy \int_{\Omega}\left[h_{1}(\left|(\nabla^{2}u)_{I}\right|)+h_{2}(\left|(\nabla^{2}u)_{II}\right|)\right]dx, where (\nabla^{2}u)_{I}, (\nabla^{2}u)_{II} denotes a decomposition of the Hessian matrix \nabla^{2}u, then we prove the higher integrability and even the continuity of \nabla^{2}u under rather general assumptions imposed on the N-functions h_{1}, h_{2}

Existence of global solutions for a parabolic system related to the nonlinear Stokes problem

In this note we consider an initial-boundary value problem describing a nonlinear variant of the nonstationary Stokes equation. We prove the existence of a (unique) global solution with Galerkin-type arguments

A regularity theory for scalar local minimizers of splitting-type variational integrals

Starting from Giaquinta\u27s counterexample [Gi] we introduce the class of splitting functionals being of (p,q)-growth with exponents pleq q<infty and show for the scalar case that locally bounded local minimizers are of class C^{1,mu}. Note that to our knowledge the only C^{1,mu}-results without imposing a relation between p and q concern the case of two independent variables as it is outlined in Marcellini\u27s paper [Ma1], Theorem A, and later on in the work of Fusco and Sbordone [FS], Theorem 4.2