Keywords: Boundary value problems Differential-operator equations Banach-valued Besov spaces Operator-valued multipliers Interpolation of Banach spaces a b s t r a c t The boundary value problems for linear and nonlinear degenerate differential-operator equations in Banach-valued Besov spaces are studied. Several conditions for the separability of linear elliptic problems are given. Moreover, t...

Multigrid algorithms are known to be highly efficient in solving systems of elliptic equations. However, standard multigrid algorithms fail to achieve optimal grid-independent convergence rates in solving non-elliptic problems. In many practical cases, the non-elliptic part of a problem is represented by the convection operator. Downstream marching, when it is viable, is the simplest and most e...

We consider the problem of the random fluctuations in the solutions to elliptic PDEs with highly oscillatory random coefficients. In our setting, as the correlation length of the fluctuations tends to zero, the heterogeneous solution converges to a deterministic solution obtained by averaging. When the Green’s function to the unperturbed operator is sufficiently singular (i.e., not square integ...

We introduce two integral representations of monodromy on Lamé equation. By applying them, we obtain results on hyperelliptic-to-elliptic reduction integral formulae, finite-gap potential and eigenvalues of Lamé operator.

1. deRham Cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. The Gauss-Manin Connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Frobenius and the Hodge Filtration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4. Applications to Elliptic Curves. . . . . . . . . . . . . . . . . . . . . . . . . . ....

Let (M,F) be a compact Riemannian foliated manifold, equipped with a bundlelike metric gM . We will also use the following notation: N ∗F is the conormal bundle to F , G is the holonomy groupoid of F . In Introduction, we will formulate our main results for the geometric case of the transverse signature operator, referring the reader to the main body of the paper for the formulations in the cas...

We study the spectral structure of the complex linearized operator for a class of nonlinear Schrödinger systems, obtaining as byproduct some interesting properties of non-degenerate ground state of the associated elliptic system, such as being isolated and orbitally stable.

ly, we may think of a differential operator ~ on M as a mapping on the smooth functions on M. That is, let C~(M) denote the space of smooth real-valued functions on M. Then 3' : C~(M) ~ C~(M). Bifurcations on Hemispheres 215 In our extension theory we consider only those operators that respect the symmetries on M and N. We define these operators as follows. There is a natural action of ISO(M) o...

A differential operator D commuting with an S-action is said to be rigid if the non-constant Fourier coefficients of kerD and coker D are the same. Somewhat surprisingly, the study of rigid differential operators turns out to be closely related to the problem of defining Chern numbers on singular varieties. This relationship comes into play when we make use of the rigidity properties of the com...

The bounds of eigenvalues which are independent of both degrees of high-order elements and mesh sizes are shown for the preconditioned system by bilinear elements for the highorder finite elements discretizations applied to a model uniformly elliptic operator.