Browsing by Subject "Sobolev spaces"

Sobolev functions generalize the concept of differentiability for functions beyond classical settings. The spaces of Sobolev functions are fundamental in mathematics and physics, particularly in the study of partial differential equations and functional analysis. This thesis provides an overview of construction of an extension operator on the space of Sobolev functions on a locally uniform domain. The primary reference is Luke Rogers' work "A Degree-Independent Sobolev Extension Operator". Locally uniform domains satisfy certain geometric properties, for example there are not too thin cusps. However locally uniform domains can possess highly non-rectifiable boundaries. For instance, the interior of the Koch snowflake represents a locally uniform domain with a non-rectifiable boundary. First we will divide the interior points of the complement of our locally uniform domain into dyadic cubes and use a collection of the cubes having certain geometric properties. The collection is called Whitney decomposition of the locally uniform domain. To extend a Sobolev function to a small cube in the Whitney decomposition one approach is to use polynomial approximations to the function on an nearby piece of the domain. We will use a polynomial reproducing kernel in order to obtain a degree independent extension operator. This involves defining the polynomial reproducing kernel in sets of the domain that we call here twisting cones. These sets are not exactly cones, but have some similarity to cones. Although a significant part of Rogers' work deals extensively with proving the existence of the kernel with the desired properties, our focus will remain in the construction of the extension operator so we will discuss the polynomial reproducing kernel only briefly. The extension operator for small Whitney cubes will be defined as convolution of the function with the kernel. For large Whitney cubes it is enough to set the extension to be 0. Finally the extension operator will be the smooth sum of the operators defined for each cube. Ultimately, since the domain is locally uniform the boundary is of measure zero and no special definition for the extension is required there. However it is necessary to verify that the extension "matches" the function correctly at the boundary, essentially that their k-1-th derivatives are Lipschitz there. This concludes the construction of a degree independent extension operator for Sobolev functions on a locally uniform domain.

Maxwell’s equations are a set of equations which describe how electromagnetic fields behave in a medium or in a vacuum. This means that they can be studied from the perspective of partial differential equations as different kinds of initial value problems and boundary value problems. Because often in physically relevant situations the media are not regular or there can be irregular sources such as point sources, it’s not always meaningful to study Maxwell’s equations with the intention of finding a direct solution to the problem. Instead in these cases it’s useful to study them from the perspective of weak solutions, making the problem easier to study. This thesis studies Maxwell’s equations from the perspective of weak solutions. To help understand later chapters, the thesis first introduces theory related to Hilbert spaces, weak derivates and Sobolev spaces. Understanding curl, divergence, gradient and their properties is important for understanding the topic because the thesis utilises several different Sobolev spaces which satisfy different kinds of geometrical conditions. After going through the background theory, the thesis introduces Maxwell’s equations in section 2.3. Maxwell’s equations are described in both differential form and timeharmonic differential forms as both are used in the thesis. Static problems related to Maxwell’s equations are studied in Chapter 3. In static problems the charge and current densities are stationary in time. If the electric field and magnetic field are assumed to have finite energy, it follows that the studied problem has a unique solution. The thesis demonstrates conditions on what kind of form the electric and magnetic fields must have to satisfy the conditions of the problem. In particular it’s noted that the electromagnetic field decomposes into two parts, out of which only one arises from the electric and magnetic potential. Maxwell’s equations are also studied with the methods from spectral theory in Chapter 4. First the thesis introduces and defines a few concepts from spectral theory such as spectrums, resolvent sets and eigenvalues. After this, the thesis studies non-static problems related to Maxwell’s equations by utilising their time-harmonic forms. In time-harmonic forms the Maxwell’s equations do not depend on time but instead on frequencies, effectively simplifying the problem by eliminating the time dependency. It turns out that the natural frequencies which solve the spectral problem we study belong to the spectrum of Maxwell’s operator iA . Because the spectrum is proved to be discrete, the set of eigensolutions is also discrete. This gives the solution to the problem as the natural frequency solving the problem has a corresponding eigenvector with finite energy. However, this method does not give an efficient way of finding the explicit form of the solution.