We study the question of well-posedness of the Cauchy problem for Schrödinger maps from R×R to the sphere S or to H, the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schrödinger system of equations and then study this modified Schrödinger map system (MSM). We then prove local well posedness of the Cauchy ...

The Szego projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of Diederich, Fornaess, and Catlin. It preserves local smoothness near boundary points of finite type. The Bergman projection, being linked to the 3 problem, plays an important role in several complex variables, as demonstrated by recent applications to the boundary behavior of holomorp...

Consider the Cauchy problem for an ordinary diierential equation _ x = g(t; x); x(0) = x; t 2 0; T]: (1:1) When g is continuous, the local existence of solutions is provided by Peano's theorem. Several existence and uniqueness results are known also in the case of a discontinuous right hand side 7]. We recall here the classical theorem of Carath eodory 8]: Theorem A. Let g : 0; T] IR n 7 ! IR n...

The computability of the solution operator of the Cauchy problem for the Fifth-order CamassaHolm equation is studied in this paper. Firstly, a nonlinear map KR : H → C (R;H (R)) is defined from the initial value φ to the solution u. Then we used the relevant knowledge of type-2 theory of effectivity, functional analysis and Sobolev space to prove that when s > (6 √ 10− 17) / 4, the solution ope...

We study the periodic non-linear Schrodinger equation −iu t +u xx = ±|u| p−1 u for initial data which are assumed to be small in some negative order Sobolev space H s (T) (s < 0), but which may have large L 2 mass. In [6], [7] these equations were shown to be ill-posed in H s (T), in the sense that the solution map was not uniformly continuous from H s (T) to itself even for short times and sma...

We propose and investigate a method for the stable determination of a harmonic function from knowledge of its value and its normal derivative on a part of the boundary of the (bounded) solution domain (Cauchy problem). We reformulate the Cauchy problem as an operator equation on the boundary using the Dirichlet-to-Neumann map. To discretize the obtained operator, we modify and employ a method d...

We construct a gauge theoretic change of variables for the wave map from R × R into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation n ≥ 4 for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact L...

Abstract The presented work summarizes the relationships between stability results and separation theorems. We prove equivalence different types of theorems on by an additive map concerning Cauchy functional equation.

In [LLT] Lascoux, Leclerc and Thibon introduced symmetric functions Gλ which are spin and weight generating functions for ribbon tableaux. This article is aimed at studying these functions in analogy with Schur functions. In particular we will describe: • a Pieri and dual-Pieri formula for ribbon functions, • a ribbon Murnagham-Nakayama formula, • ribbon Cauchy and dual Cauchy identities, • and...