Global gradient estimates for asymptotically regular problems of p(x)-Laplacian type

We study an asymptotically regular problem of [Formula: see text]-Laplacian type with discontinuous nonlinearity in a nonsmooth bounded domain. A global Calderón–Zygmund estimate is established for such a nonlinear elliptic problem with nonstandard growth under the assumption that the associated nonlinearity has a more general kind of the asymptotic behavior near the infinity with respect to the gradient variable. We also address an optimal regularity requirement on the nonlinearity as well as a minimal geometric assumption on the boundary of the domain for the nonlinear Calderón–Zygmund theory in the setting of variable exponent Sobolev spaces.

On a Newton-Type Method for Differential-Algebraic Equations

This paper deals with the approximation of systems of differential-algebraic equations based on a certain error functional naturally associated with the system. In seeking to minimize the error, by using standard descent schemes, the procedure can never get stuck in local minima but will always and steadily decrease the error until getting to the solution sought. Starting with an initial approximation to the solution, we improve it by adding the solution of some associated linear problems, in such a way that the error is significantly decreased. Some numerical examples are presented to illustrate the main theoretical conclusions. We should mention that we have already explored, in some previous papers (Amat et al., in press, Amat and Pedregal, 2009, and Pedregal, 2010), this point of view for regular problems. However, the main hypotheses in these papers ask for some requirements that essentially rule out the application to singular problems. We are also preparing a much more ambitious perspective for the theoretical analysis of nonlinear DAEs based on this same approach.

SAMPLING AND BIRKHOFF REGULAR PROBLEMS

We establish new sampling representations for linear integral transforms associated with arbitrary general Birkhoff regular boundary value problems. The new approach is developed in connection with the analytical properties of Green’s function, and does not require the root functions to be a basis or complete. Unlike most of the known sampling expansions associated with eigenvalue problems, the results obtained are, generally speaking, of Hermite interpolation type.

RELATIONSHIPS BETWEEN REGULAR AND IRREGULAR COLLECTIVE COMMUNICATION OPERATIONS ON CLUSTERED MULTIPROCESSORS

We characterize collective communication operations on (clustered) multiprocessor systems in terms of their communication volume, and arrive at useful relationships between regular and irregular operations over sets of processors and sets of cluster-nodes, respectively. We show that regular problems over sets of processors induce corresponding irregular problems over sets of nodes. We hereby identify a symmetric variant of the personalized all-to-all communication problem that might be worth studying in its own right, and discuss an algorithm for solving this problem. From a simple algorithm for the regular all-gather problem over sets of processors, we derive an algorithm for the irregular all-gather problem over both sets of processors and sets of nodes. For communication libraries like MPI, the relationships emphasize the need for efficient algorithms for the irregular collective communication operations.