we consider the semilinear elliptic boundary value problem  = ∈∂ω− δ = ∈ωu x xu x f u x x( ) 0;( ) λ ( ( ));where λ > 0 is a parameter, ω is a bounded region in rn with a smooth boundary, and f is asmooth function. we prove, under some additional conditions, the existence of a positive solution for λlarge. we prove that our solution u for λ large is such that = →∞∈ω|| u ||: sup| u(x) |xas λ ...

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for H initial data, s > −1/2, and for any s1 < min(3s+1, s+ 1), the difference of the nonlinear and linear evolutions is in H1 for all times, with at most polynomially growing H1 norm. The result also extends to KdV with a smooth, mean zero, time-dependent potential in the case s ≥ 0. Our resu...

Recent theoretical and simulation studies indicate that grain boundary motion is coupled to the translation and rotation of the adjacent grains. However, the geometry of the system can strongly modify this coupling. We simulate the evolution of grains in two geometries that are similar to experimental systems that have been studied. The first, a small circular grain embedded in a matrix consist...

In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compact set has nonnegative Ricci curvature and nonnegative mean curvature (of the bo...

Stability and Extremal Theorems for Plane Curves with Bounded Curvature Ralph Howard and Andrejs Treibergs In this note we dis uss some elementary theorems about the relation between area and length of losed embedded plane urves with bounded urvature. Our main result (see Theorem 4.1) solves the extremal problem of whi h domain has largest boundary length among embedded disks in the plane whose...