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.

We consider a Sturm-Liouville boundary value problem in a bounded domain D of R^n. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on bD. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact self-adjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types.

Peetre characterized local operators defined on the smooth section space over an open subset of an Euclidean space as ``linear differential operators'. We look for an extension to such maps of smooth vector sections of smooth Banach bundles. Since local operators are special disjointness preserving operators, it leads to the study of the disjointness structure of smooth Banach manifolds. In this thesis, we take an abstract approach to define the``smooth functions', via the so-called S-category. Especially, it covers the standard classes C^{n} and local Lipschitz functions, where 0≤n≤¡Û. We will study the structure of disjointness preserving linear maps between S-smooth functions defined on separable Banach manifolds. In particular, we will give an extension of Peetre's theorem to characterize disjointness preserving linear mappings between C^n or local Lipschitz functions defined on locally compact metric spaces.

A class of parametric optimal control problems for semilinear parabolic equations is considered. Using recent regularity results for solutions of such equations, sufficient conditions are derived under which the solutions to optimal control problems are locally Lipschitz continuous functions of the parameter in the L1-norm. It is shown that these conditions are also necessary, provided that the dependence of data on the parameter is sufficiently strong.

Technology trends toward lower power, higher speed and higher density devices have pushed the package performance to its limits. The high frequency effects e.g., crosstalk and signal distortion, may cause high bit error rates or malfunctioning of the circuit. Therefore, the successful release of a new product requires constant attention to the high frequency effects through the whole design process. Full-wave electromagnetic tools must be used for this purpose. Unfortunately, currently available full-wave tools require excessive computational resources to simulate large-scale interconnect structures.A prototype version of the Full-Wave Layered-Interconnect Simulator (UA-FWLIS), which employs the Method of Moments (MoM) technique, was developed in response to this design need. Instead of using standard numerical integration techniques, the MoM reaction elements were analytically evaluated, thereby greatly improving the computational efficiency of the simulator. However, the expansion and testing functions that are employed in the prototype simulator involve filamentary functions across the wire. Thus, many problems cannot be handled correctly. Therefore, a fundamental extension is made in this dissertation to incorporate rectangular-based, finite-width expansion and testing functions into the simulator. The critical mathematical issues and theoretical issues that were met during the extension are straightened out. The breakthroughs that were accomplished in this dissertation lay the foundation for future extensions. A new bend-cell expansion function is also introduced, thus allowing the simulator to handle bends on the interconnects with fewer unknowns. In addition, the Incomplete Lipschitz-Hankel integrals, which are used in the analytical solution, are studied. Two new series expansions were also developed to improve the computational efficiency and accuracy.

We present a novel approach for deriving analytical solutions to transport equations expressedin similarity variables. We apply a fixed-point iteration procedure to these transformedequations by formally solving for the highest derivative term and then integrating to obtainan expression for the solution in terms of a previous estimate. We are able to analyticallyobtain the Lipschitz condition for this iteration procedure and, from this (via requirements forconvergence given by the contraction mapping principle), deduce a range of values for the outerlimit of the solution domain, for which the fixed-point iteration is guaranteed to converge.

Globaliojo optimizavimo metodai, pagrįsti Lipšico rėžių apskaičiavimu, yra plačiai taikomi įvairių optimizavimo uždavinių sprendimui. Tačiau Lipšico metodai dažniausiai remiasi prielaida, kad Lipšico konstanta žinoma iš anksto, o tai retas atvejis sprendžiant praktinius uždavinius. Todėl Simonos Talačkaitės magistro darbe yra toliau nagrinėjama aktuali ir svarbi problematika iškylanti realizuojant Lipšico metodus nesiremiančius jokiomis išankstinėmis prielaidomis apie Lipšico konstantą. Praktinio tiriamojo pobūdžio magistro darbe iškeliamas toks pagrindinis tikslas: ištirti daugiamačių simpleksinių globaliojo optimizavimo algoritmų su nežinoma Lipšico konstanta efektyvumą priklausomai nuo naudojamo simplekso centro. Šiam tikslui pasiekti buvo iškelti šie uždaviniai: apžvelgti naujausią literatūrą skirta Lipšico metodams su nežinoma Lipšico konstanta; matematiškai išnagrinėti įvairių daugiamačių simplekso centrų apskaičiavimus bendru atveju bei juos realizuoti Matlab aplinkoje; papildyti simpleksinį globaliojo optimizavimo DISIMPL algoritmą šių simpleksų centrų apskaičiavimo paprogramėmis; eksperimentiškai ištirti pasiūlytų rezultatų praktiškumą sprendžiant testinius optimizavimo uždavinius. / This work analyzes Global optimization objectives, the most important it will be algorithms with simplicial Lipšico constant. Also, this work analyzes multidi- mensional DIRECT algorithm. We will provide dividing in higher dimennsions DIRECT algorithm. Then analyzes two simplex and apply the solutions. The hand simplex to smallerpartitions. Perceive multidimensional DIRECT algorithm division rules. In this work wrote a lot about simplicial center about dividing of hyoer-cube. Finally, the experiment it will be about the best way, how we can …nd circle center ir diferent way. Simplex centers using 8 test funkcions , changing the number of iterations and mistakes number. Create tables and to analyzes them. The purpose of this paper work is to introduce the simplex algorithm for global optimization with unknown Lipšicas constant depending on the e¢ ciency of the division of the rules used in the simplex.

Ergodic optimization is the study of which ergodic measures maximize the integral of a particular function. For sufficiently regular functions, e.g. Lipschitz/Holder continuous functions, it is conjectured that the set of functions optimized by measures supported on a periodic orbit is dense. Yuan and Hunt made great progress towards showing this for Lipschitz functions. This thesis presents clear proofs of Yuan and Hunt’s theorems in the case of the Shift as well as introducing a subset of Lipschitz functions, the super-continuous functions, where the set of functions optimized by measures supported on a periodic orbit is open and dense.

Thesis (Ph.D.)--Flinders University, School of Informatics and Engineering, Dept. of Mathematics. / Typescript bound. Includes bibliographical references: (leaves 135-140) and index. Also available online.

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (leaves [86]-88).