# Math Technical Report Series

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- ItemInnovative Solution of a 2-D Elastic Transmission Problem(2006-07-12T14:01:23Z) Hsiao, George C.; Nigam, Nilima; Sändig, Anna-MargareteThis paper is concerned with a boundary-field equation approach to a class of boundary value problems exterior to a thin domain. A prototype of this kind of problems is the interaction problem with a thin elastic structure. We are interested in the asymptotic behavior of the solution when the thickness of the elastic structure approaches to zero. In particular, formal asymptotic expansions will be developed, and their rigorous justification will be considered. As will be seen, the construction of these formal expansions hinges on the solutions of a sequence of exterior Dirichlet problems, which can be treated by employing boundary element methods. On the other hand, the justification of the corresponding formal procedure requires an independence on the thickness of the thin domain for the constant in the Korn inequality. It is shown that in spite of the reduction of the dimensionality of the domain under consideration, this class of problems are in general not singular perturbation problems, because of appropriate interface conditions.
- ItemA Newton-Imbedding Procedure for Solutions of Semilinear Boundary Value Problems in Sobolev Spaces(2006-07-12T13:54:38Z) Hsiao, G.C.This paper is concerned with the application of the Newton-imbedding iteration procedure to nonlinear boundary value problems in Sobolev spaces. A simple model problem for the second-order semilinear elliptic equations is considered to illustrate the main idea. The essence of the method hinges on the a priori estimates of solutions of the associated linear problem in appropriate Sobolev spaces. It is to our surprise that H1(Ω)-solution is not smooth enough to guarantee the convergence of the sequence generated by the procedure. Existence and uniqueness of solution to the original nonlinear problem are established constructively. An application of this approach to the Lamé system with nonlinear body force and its generalization to contain a nonlinear surface traction in elasticity will also be discussed.
- ItemAn Initial-Boundary Value Problem for the Viscous Compressible Flow(2006-07-07T14:03:08Z) Hebeker, Friedrich-Karl; Hsiao, George CA constructive approach is presented to treat an initial boundary value problem for isothermal Navier Stokes equations. It is based on a characteristics (Lagrangean) approximation locally in time and a boundary intergral equation method via nonstationary potentials. As a basic problem, the later leads to a Volterra integral equation of the first kind which is proved to be uniquely solvable and even coercive in some anistropic Sobolev spaces. The solution depends continuously upon the data and may be constructed by a quasioptimal Galerkin procedure.
- ItemOn the Boundary Integral Equation Method for a Mixed Boundary Value Problem of the Biharmonic Equation(Department of Mathematical Sciences, 2005) Cakoni, F.; Hsiao, G.C.; Wendland, W.
- ItemFlow of a Model for Dilute Wormlike Micellar Solutions(Depatment of Mathematical Sciences, 2006) Rossi, L.F.; McKinley, G.; Cook, L.P