We have recently demonstrated' that such a Green's function exists if the polynomial Epl+ -..+pn <2h ap,. ..Pn(itl) PI. .. (itn)P" and both its highest and lowest homogeneous parts occurring, are all three different from zero for any real ti, ..., tn not all zero. THEOREM 1. In En, ifD is a bounded domain having a connected boundary B, and if U is a neighborhood of B, and if an analytic functio...

Motivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography that appears as an important application in Underwater Acoustics, we analyze a general Schrödinger-type equation posed on two-dimensional variable domains with mixed boundary conditions. The resulting initialand boundary-value problem is transformed into an equivalent one posed on a ...

A class of optimal control problems for quasilinear elliptic equations is considered, where the coefficients of the elliptic differential operator depend on the state function. Firstand second-order optimality conditions are discussed for an associated control-constrained optimal control problem. Main emphasis is laid on second-order sufficient optimality conditions. To this aim, the regularity...

The existence of a unique weak solution to the homogeneous closed Dirichlet problem is proven for an elliptic-hyperbolic equation. The equation arises in a model for electromagnetic wave propagation in cold plasma. A class of open boundary value problems for the equation is shown to possess strong solutions. MSC2000 : 35M10, 35D05, 82D10.

Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere S. Our method is based on the Hamiltonian-Jacobi approach, where the corresponding Hamitonian system is solved with mixed boundary condit...

In this work, the multiscale problem of modeling fluctuations in boundary layers in stochastic elliptic partial differential equations is solved by homogenization. A homogenized equation for the covariance of the solution of stochastic elliptic PDEs is derived. In addition to the homogenized equation, a rate for the covariance and variance as the cell size tends to zero is given. For the homoge...

A new multigrid algorithm is constructed for the solution of linear systems of equations which arise from the discretization of elliptic PDEs. It is defined in terms of the difference scheme on the fine grid only, and no rediscretization of the PDE is required. Numerical experiments show that this algorithm gives high convergence rates for several classes ofproblems: symmetric, nonsymmetdc and ...

This is the second paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial diierential system which describes the ow of two incom-pressible, immiscible uids in porous media. In this paper we consider a nite element approximation for this system. The elliptic equation for the pressure and velocity is approximated by ...

An abstract robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed. The term “robust” refers to the convergence rate of the method being independent of discretization parameters, i.e., the problem size, and problem parameters. Important instances of such problem parameters are in particular (highl...

We study solution techniques for a linear-quadratic optimal control problem involving fractional powers of elliptic operators. These fractional operators can be realized as the Dirichlet-toNeumann map for a nonuniformly elliptic problem. Thus, we consider an equivalent formulation with a nonuniformly elliptic operator as state equation. The rapid decay of the solution to this problem suggests a...