Wang Jing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 55-59). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Introduction --- p.3 / Chapter 1 --- Formulation of the Problem --- p.10 / Chapter 1.1 --- Reformulated Navier-Stokes Equations --- p.10 / Chapter 1.2 --- Linearized Problems --- p.15 / Chapter 2 --- Construction of the Approximate Solution --- p.19 / Chapter 2.1 --- Two-scale Asymptotic Expansions --- p.19 / Chapter 2.2 --- Determination of Each Inner and Boundary Terms --- p.22 / Chapter 2.3 --- Truncation Terms --- p.31 / Chapter 3 --- Estimates of the Error Term of the Approximate Solution and Main Results --- p.33 / Chapter 3.1 --- Error Equations --- p.33 / Chapter 3.2 --- Energy Estimates --- p.36 / Chapter 3.2.1 --- BasicL2 Estimates --- p.36 / Chapter 3.2.2 --- Tangential Derivatives Estimates --- p.38 / Chapter 3.2.3 --- Normal Derivatives Estimates --- p.49 / Chapter 3.3 --- Pointwise Estimates --- p.52 / Bibliography --- p.55

This thesis is motivated by some of the recent results of the solvability of elliptic PDE in Lipschitz domains and the relationships between the solvability of different boundary value problems. The parabolic setting has received less attention, in part due to the time irreversibility of the equation and difficulties in defining the appropriate analogous time-varying domain. Here we study the solvability of boundary value problems for second order linear parabolic PDE in time-varying domains, prove two main results and clarify the literature on time-varying domains. The first result shows a relationship between the regularity and Dirichlet boundary value problems for parabolic equations of the form Lu = div(A∇u)−ut = 0 in Lip(1, 1/2) time-varying cylinders, where the coefficient matrix A = [aij(X, t)] is uniformly elliptic and bounded. We show that if the Regularity problem (R)p for the equation Lu = 0 is solvable for some 1 < p < then the Dirichlet problem (D*) 1 p, for the adjoint equation L*v = 0 is also solvable, where p' = p/(p − 1). This result is analogous to the one established in the elliptic case. In the second result we prove the solvability of the parabolic Lp Dirichlet boundary value problem for 1 < p ≤ ∞ for a PDE of the form ut = div(A∇u)+B ·∇u on time-varying domains where the coefficients A = [aij(X, t)] and B = [bi(X, t)] satisfy a small Carleson condition. This result brings the state of affairs in the parabolic setting up to the current elliptic standard. Furthermore, we establish that if the coefficients of the operator A and B satisfy a vanishing Carleson condition, and the time-varying domain is of VMO-type then the parabolic Lp Dirichlet boundary value problem is solvable for all 1 < p ≤ ∞. This is related to elliptic results where the normal of the boundary of the domain is in VMO or near VMO implies the invertibility of certain boundary operators in Lp for all 1 < p < ∞. This then (using the method of layer potentials) implies solvability of the Lp boundary value problem in the same range for certain elliptic PDE. We do not use the method of layer potentials, since the coefficients we consider are too rough to use this technique but remarkably we recover Lp solvability in the full range of p's as the elliptic case. Moreover, to achieve this result we give new equivalent and localisable definitions of the appropriate time-varying domains.

The investigation is initially a continuation of Neuberger's work on linear boundary value problems. A very general iterative procedure for solution of these problems is described. The alternating-projection theorem of von Neumann is the mathematical starting point for this study. Later theorems demonstrate the validity of numerical approximation for Neuberger's method under certain conditions. A sampling of differential equations within the scope of our iterative method is given. The numerical evidence is that the procedure works well on neutral-state equations, for which no software is written now.

This thesis studies the existence, multiplicity, bifurcation and the stability of the solutions to semilinear elliptic boundary value problems. These problems are motivated both by the mathematical structure and the numerous applications in fluid mechanics chemical reactions, nuclear reactors, Riemannian geometry and elasticity theory. This study considers the problem for different classes of nonlinearities and obtain the existence and multiplicity of positive solutions.

In this paper results concerning a semilinear elliptic boundary value problem are proven. This problem has five solutions when the range of the derivative of the nonlinearity ƒ includes the first two eigenvalues. The existence and multiplicity or radially symmetric solutions under suitable conditions on the nonlinearity when Ω is a ball in R^N.

In this research report we explore some applications of symmetry methods for boundary value problems in the pricing of barrier options. Various nancial instruments satisfy the Black-Scholes partial di erential equation (pde) but with di erent domain, maturity date and boundary conditions. We nd Lie symmetries that leave the Black-Scholes (pde) invariant and will guarantee that the relevant solutions satisfy the boundary conditions. Using these sym- metries, we can thus generate group-invariant solutions to the boundary value problem.

by Yu Wai-kuen. / Thesis (M. Phil.)--Chinese University of Hong Kong, 1981. / Bibliography: l. 52-55.

Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaves 38-39.

by Lei Ka Keung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 70-71). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Dirichlet Problem at infinity --- p.9 / Chapter 1.1 --- The Geometric Boundary --- p.9 / Chapter 1.2 --- Dirichlet Problem --- p.15 / Chapter 2 --- The Martin Boundary --- p.29 / Chapter 2.1 --- The Martin Metric --- p.30 / Chapter 2.2 --- The Representation Formula --- p.31 / Chapter 2.3 --- Uniqueness of Representation --- p.36 / Chapter 3 --- The Geometric boundary and the Martin boundary --- p.42 / Chapter 3.1 --- Estimates for harmonic functions in cones --- p.42 / Chapter 3.2 --- A Harnack Inequality at Infinity --- p.49 / Chapter 3.3 --- The kernel function --- p.54 / Chapter 3.4 --- The Main Theorem --- p.55 / Chapter 4 --- Positive Harmonic Functions on Product of Manifolds --- p.61 / Chapter 4.1 --- Splitting Theorem --- p.61 / Chapter 4.2 --- Riemannian Halfspace and the parabolic Martin boundary --- p.62 / Chapter 4.3 --- Splitting of parabolic Martin kernels --- p.63 / Chapter 4.4 --- Proof of theorem 4.1 --- p.66 / Bibliography

by Ho Yum Bun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 93-94). / Text in English; abstracts in English and Chinese. / by Ho Yum Bun. / List of Figures --- p.3 / Abstract --- p.9 / Chinese Abstract --- p.10 / Acknowledgement --- p.11 / Chapter 1 --- Introduction --- p.12 / Chapter 1.1 --- Motivation --- p.12 / Chapter 1.2 --- What is Sonoluminescence? --- p.13 / Chapter 1.3 --- The Main Task of this Project --- p.13 / Chapter 1.4 --- Organization of this Thesis --- p.13 / Chapter 2 --- Reviews on One-dimensional Dynamical Cavity Problem --- p.15 / Chapter 2.1 --- Introduction --- p.15 / Chapter 2.2 --- Formulation --- p.15 / Chapter 2.3 --- Moore's R Function Method --- p.18 / Chapter 2.4 --- Mode Expansion Method --- p.19 / Chapter 2.5 --- Transformation method --- p.20 / Chapter 2-6 --- Summary --- p.21 / Chapter 3 --- Numerical Results For One-dimensional Dynamical Cavity Prob- lem --- p.22 / Chapter 3.1 --- Introduction --- p.22 / Chapter 3.2 --- Evolution of a Cavity System --- p.23 / Chapter 3.3 --- Motion of the Moving Mirror --- p.23 / Chapter 3.4 --- R(z) Function --- p.24 / Chapter 3.4.1 --- Construction of R(z) Function --- p.24 / Chapter 3.4.2 --- Numerical R(z) Function --- p.27 / Chapter 3.5 --- Results --- p.27 / Chapter 3.5.1 --- Results with Moore's R(z) Function Method --- p.27 / Chapter 3.5.2 --- Results with the Mode Expansion Method --- p.29 / Chapter 3.5.3 --- Results with the Transformation Method --- p.36 / Chapter 3.6 --- Summary --- p.36 / Chapter 4 --- Spherical Dynamical Cavity Problem --- p.37 / Chapter 4.1 --- Introduction --- p.37 / Chapter 4.2 --- Formulation --- p.37 / Chapter 4.3 --- Motion of a Moving Spherical Mirror --- p.39 / Chapter 4.4 --- Summary --- p.40 / Chapter 5 --- The G(z) Function Method --- p.41 / Chapter 5.1 --- Introduction --- p.41 / Chapter 5.2 --- G(z) Function --- p.42 / Chapter 5.2.1 --- Ideas of Deriving the G(z) Function --- p.42 / Chapter 5.2.2 --- Formalism --- p.42 / Chapter 5.2.3 --- Initial G(z) Function --- p.45 / Chapter 5.3 --- Construction of the G(z) Function --- p.46 / Chapter 5.3.1 --- Case I : l=0 --- p.46 / Chapter 5.3.2 --- Case II : l > 0 --- p.49 / Chapter 5.4 --- Asymptotic Series Solution of G(z) --- p.50 / Chapter 5.5 --- Application to Resonant Mirror Motion --- p.52 / Chapter 5.6 --- Regularization of G(z) --- p.58 / Chapter 5.7 --- Behaviors of the Fields --- p.58 / Chapter 5.7.1 --- z vs tf Graph --- p.61 / Chapter 5.7.2 --- Case 1: l= 0 --- p.61 / Chapter 5.7.3 --- "Case2: l= 1,2" --- p.62 / Chapter 5.7.4 --- Case 3: l= 3 --- p.73 / Chapter 5.7.5 --- Section Summary --- p.73 / Chapter 5.8 --- Summary --- p.73 / Chapter 6 --- Three-dimensional Mode Expansion Method and Transforma- tion Method --- p.75 / Chapter 6.1 --- Introduction --- p.75 / Chapter 6.2 --- Mode Expansion Method --- p.75 / Chapter 6.2.1 --- Formalism --- p.75 / Chapter 6.2.2 --- Application of Floquet's Theory --- p.78 / Chapter 6.2.3 --- Results --- p.80 / Chapter 6.3 --- The Transformation Method --- p.80 / Chapter 6.3.1 --- The Method --- p.80 / Chapter 6.3.2 --- Numerical Schemes --- p.86 / Chapter 6.3.3 --- Results --- p.89 / Chapter 6.4 --- Summary --- p.89 / Chapter 7 --- Conclusion --- p.90 / Chapter 7.1 --- The One-dimensional Dynamical Cavity Problem --- p.90 / Chapter 7.2 --- The Dynamical Spherical Cavity Problem --- p.91 / Chapter 7.3 --- Numerical Methods --- p.91 / Chapter 7.4 --- Further Investigation --- p.92 / Bibliography --- p.93