Oscillation theorems for the damped elliptic differential equation of second order

$\sum_{i, j=1}^{N}D_i[\,a_{ij}(x)D_jy\,]+\sum_{i=1}^{N}b_i(x)D_iy+c(x,y)=0$

are obtained. The results are extensions of averaging techniques due to Coles and Kamenev, and include earlier known results in literature.

In this paper, we obtain sharp norm estimates for the Bernardi integral transform of functions belonging to the class ${\mathcal K}(A,B)$, $-1\le B<A\le 1$, which is a subclass of the well-known class of convex univalent functions. As a consequence, a number of open questions arise naturally, concerning ${\mathcal S}^*(A,B)$--a subclass of the well-known class of starlike univalent functions, and many other classes.

A new proof, which is much simpler and which works in more generality, of a structure theorem on maximally differential graded ideals in a Noetherian graded ring containing a field of characteristic zero is given.

In this paper, we give a new definition of the flux of a superharmonic function defined outside a compact set in a Brelot space without positive potentials. We also give a new notion of potential in a BS space (that is, a harmonic space without positive potentials containing the constants) which leads to a Riesz decomposition theorem for the class of superharmonic functions that have a harmonic minorant outside a compact set. Furthermore, we give a characterization of the local axiom of proportionality in terms of a global condition on the space.

We consider quasilinear elliptic equations with lower order term and general measure data. We define renormalized solutions of Dirichlet problems and show the existence of such solutions. We also give uniqueness in some special cases.

Large time behavior of solutions to the compressible Navier-Stokes equation around a given constant state is considered in an infinite layer ${\bf R}^{n-1}\times (0,a)$, $n\geq2$, under the no slip boundary condition for the velocity. The $L^p$ decay estimates of the solution are established for all $1\leq p\leq \infty$. It is also shown that the time-asymptotic leading part of the solution is given by a function satisfying the $n-1$ dimensional heat equation. The proof is given by combining a weighted energy method with time-weight functions and the decay estimates for the associated linearized semigroup

Let $C$ be a smooth plane quartic curve defined over a field $k$ and $k(C)$ the rational function field of $C$. Let $\pi_P$ be the projection from $C$ to a line $\ell$ with a center $P\in C$. Then $\pi_P$ induces an extension of fields; $k(C)/k(\ell)$. Let $\widetilde C$ be a nonsingular model of the Galois closure of the extension, which we call the Galois closure curve of $k(C)/k(\ell)$. We give an answer to the problem for the genus of the Galois closure curve of quartic curve.

We give new representations of solutions for the periodic linear difference equation of the type $x(n+1)=B(n)x(n)+b(n)$, where complex nonsingular matrices $B(n)$ and vectors $b(n)$ are $\rho$-periodic. These are based on the Floquet multipliers and the Floquet exponents, respectively. By using these representations, asymptotic behavior of solutions is characterized by initial values. In particular, we can characterize necessary and sufficient conditions that the equation has a bounded solution(or a $\rho$-periodic solution), and the Massera type theorem by initial values.

We show that given a positive and finite Radon measure $\mu$, there is a $\Apx$ -superharmonic function $u$ which satisfies

$-\dive\A(x,Du)=\mu$

in the sense of distributions. Here $\A$ is an elliptic operator with $p(x)$-type nonstandard growth.