445 research outputs found
Extension Operators and Finite Elements for Fractal Boundary Value Problems
The dissertation is organized into two main parts. The first part considers fractal extension operators. Although extension operators are available for general subsets of Euclidean domains or metric spaces, our extension operator is unique in that it utilizes both the iterative nature of the fractal and finite element approximations to construct the operator. The resulting operator is especially well suited for future numerical work on domains with prefractal boundaries. In the dissertation we prove the existence of a linear extension operator, Π from the space of Hölder continuous functions on a fractal set S to the space of Hölder continuous functions on a larger domain Ω. Moreover this same extension operator maps functions of finite energy on the fractal to H1 functions on the larger domain Ω.
In the second part, we consider boundary value problems in domains with fractal boundaries. First we consider the Sierpinski prefractal and how we might apply the technique of singular homogenization to thin layers constructed on the prefractal. We will also discuss numerical approximation in domains with fractal boundaries and introduce a finite element mesh developed for studying problems in domains with prefractal Koch boundaries. This mesh exploits the self-similarity of the Koch curve for arbitrary rational values of α and its construction is crucial for future numerical study of problems in domains with prefractal Koch curve boundaries. We also show a technique for mesh refinement so that singularities in the domain can be handled and present sample numerical results for the transmission problem
Acoustic scattering from corners, edges and circular cones
Consider the time-harmonic acoustic scattering from a bounded penetrable obstacle imbedded in an isotropic homogeneous medium. The obstacle is supposed to possess a circular conic point or an edge point on the boundary in three dimensions and a planar corner point in two dimensions. The opening angles of cones and edges are allowed to be non-convex. We prove that such an obstacle scatters any incoming wave non-trivially (i.e., the far field patterns cannot vanish identically), leading to the absence of real non-scattering wavenumbers. Local and global uniqueness results for the inverse problem of recovering the shape of a penetrable scatterers are also obtained using a single incoming wave. Our approach relies on the singularity analysis of the inhomogeneous Laplace equation in a cone
Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems
For a strongly elliptic second-order operator on a bounded domain
it has been known for many years how to interpret
the general closed -realizations of as representing boundary
conditions (generally nonlocal), when the domain and coefficients are smooth.
The purpose of the present paper is to extend this representation to nonsmooth
domains and coefficients, including the case of H\"older
-smoothness, in such a way that pseudodifferential
methods are still available for resolvent constructions and ellipticity
considerations. We show how it can be done for domains with
-smoothness and operators with -coefficients, for
suitable and . In particular, Kre\u\i{}n-type resolvent
formulas are established in such nonsmooth cases. Some unbounded domains are
allowed.Comment: 62 page
Singular integrals and boundary value problems for elliptic systems
Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas, Fecha de lectura: 15-11-201
The Dirichlet Problem for L\'evy-stable operators with -data
We prove Sobolev regularity for distributional solutions to the Dirichlet
problem for generators of -stable processes and exterior data,
inhomogeneity in weighted -spaces. This class of operators includes the
fractional Laplacian. For these rough exterior data the theory of weak
variational solutions is not applicable. Our regularity estimate is robust in
the limit which allows us to recover the local theory.Comment: 21 pages, 1 figur
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