Sobolev mappings and Cantor type counterexamples Author: Martin Fiala Supervisor: doc. RNDr. Stanislav Hencl, Ph.D. Abstract: The aim of this work is to show one of the general con- structions of the mappings, which can be used to create different coun- terexamples in the theory of Sobolev mappings. The construction is described in detail and then it is used for a number of examples. The last chapter is devoted to a slight generalization of this construction. 1

Partial differential equations on domains presenting point singularities have always been of interest for applied mathematicians; this interest stems from the difficulty to prove regularity results for non-smooth domains, which has important consequences in the numerical solution of partial differential equations. In my thesis I address those consequences in the case of conforming and penalty finite element methods. The main results here contained concerns a priori error estimates for conforming and penalty finite element methods with respect to the energy norm, the $\mathcal{L}^2(\Omega)$ norm in both the standard and weighted setting.

A mapping f from R^{n} to R^{n} is said to satisfy the Luzin condition N if f maps sets of measure zero to sets of measure zero. It is known to be valid for mappings in the Sobolev space W^{1,p} for p > n and for p <= n there are counterexamples. The aim of this thesis is to summarize known results and study the validity of Luzin condition N for mappings in the Sobolev space W^{2,p}.

<p>The Laplace equation and the related p-Laplace equation are closely associated with Sobolev spaces. During the last 15 years people have been exploring the possibility of solving partial differential equations in general metric spaces by generalizing the concept of Sobolev spaces. One such generalization is the Newtonian space where one uses upper gradients to compensate for the lack of a derivative.</p><p>All papers on this topic are written for an audience of fellow researchers and people with graduate level mathematical skills. In this thesis we give an introduction to the Newtonian spaces accessible also for senior undergraduate students with only basic knowledge of functional analysis. We also give an introduction to the tools needed to deal with the Newtonian spaces. This includes measure theory and curves in general metric spaces.</p><p>Many of the properties of ordinary Sobolev spaces also apply in the generalized setting of the Newtonian spaces. This thesis includes proofs of the fact that the Newtonian spaces are Banach spaces and that under mild additional assumptions Lipschitz functions are dense there. To make them more accessible, the proofs have been extended with comments and details previously omitted. Examples are given to illustrate new concepts.</p><p>This thesis also includes my own result on the capacity associated with Newtonian spaces. This is the theorem that if a set has p-capacity zero, then the capacity of that set is zero for all smaller values of p.</p>

The Laplace equation and the related p-Laplace equation are closely associated with Sobolev spaces. During the last 15 years people have been exploring the possibility of solving partial differential equations in general metric spaces by generalizing the concept of Sobolev spaces. One such generalization is the Newtonian space where one uses upper gradients to compensate for the lack of a derivative. All papers on this topic are written for an audience of fellow researchers and people with graduate level mathematical skills. In this thesis we give an introduction to the Newtonian spaces accessible also for senior undergraduate students with only basic knowledge of functional analysis. We also give an introduction to the tools needed to deal with the Newtonian spaces. This includes measure theory and curves in general metric spaces. Many of the properties of ordinary Sobolev spaces also apply in the generalized setting of the Newtonian spaces. This thesis includes proofs of the fact that the Newtonian spaces are Banach spaces and that under mild additional assumptions Lipschitz functions are dense there. To make them more accessible, the proofs have been extended with comments and details previously omitted. Examples are given to illustrate new concepts. This thesis also includes my own result on the capacity associated with Newtonian spaces. This is the theorem that if a set has p-capacity zero, then the capacity of that set is zero for all smaller values of p.

The method of steepest descent is applied to a nonlinearly constrained optimization problem which arises in the study of liquid crystals. Let Ω denote the region bounded by two coaxial cylinders of height 1 with the outer cylinder having radius 1 and the inner having radius ρ. The problem is to find a mapping, u, from Ω into R^3 which agrees with a given function v on the surfaces of the cylinders and minimizes the energy function over the set of functions in the Sobolev space H^(1,2)(Ω; R^3) having norm 1 almost everywhere. In the variational formulation, the norm 1 condition is emulated by a constraint function B. The direction of descent studied here is given by a projected gradient, called a B-gradient, which involves the projection of a Sobolev gradient onto the tangent space for B. A numerical implementation of the algorithm, the results of which agree with the theoretical results and which is independent of any strong properties of the domain, is described. In chapter 2, the Sobolev space setting and a significant projection in the theory of Sobolev gradients are discussed. The variational formulation is introduced in Chapter 3, where the issues of differentiability and existence of gradients are explored. A theorem relating the B-gradient to the theory of Lagrange multipliers is stated as well. Basic theorems regarding the continuous steepest descent given by the Sobolev and B-gradients are stated in Chapter 4, and conditions for convergence in the application to the liquid crystal problem are given as well. Finally, in Chapter 5, the algorithm is described and numerical results are examined.

<p>This paper deals with Lebesgue points and studies properties of the set of Lebesgue points for various classes of functions. We consider continuous functions, L¹ functions and Sobolev functions. In the case of uniformly continuous functions and Hölder continuous functions we develop a characterization in terms of Lebesgue points. For Sobolev functions we study the dimension of the set of non-Lebesgue points.</p>

The bending of an orthotropic cusped plate in energetic and weighted Sobolev spaces has been considered. The existence and uniqueness of generalized and weak solutions of admissible boundary value problems (BVPs) have been investigated.

This paper deals with Lebesgue points and studies properties of the set of Lebesgue points for various classes of functions. We consider continuous functions, L1 functions and Sobolev functions. In the case of uniformly continuous functions and Hölder continuous functions we develop a characterization in terms of Lebesgue points. For Sobolev functions we study the dimension of the set of non-Lebesgue points.

We establish the upper bounds for the difference between the first two eigenvalues of the relative and absolute eigenvalue problems. Relative and absolute boundary conditions are generalization of Dirichlet and Neumann boundary conditions on functions to differential forms respectively. The domains are taken to be a family of symmetric regions in Rn consisting of two cavities joined by a straight thin tube. Our operators are Hodge Laplacian operators acting on k-forms given by the formula Δ(k) = dδ+δd, where d and δ are the exterior derivatives and the codifferentials respectively. A result has been established on Dirichlet case (0-forms) by Brown, Hislop, and Martinez [2]. We use the same techniques to generalize the results on exponential decay of eigenforms, singular perturbation on domains [1], and matrix representation of the Hodge Laplacian restricted to a suitable subspace [2]. From matrix representation, we are able to find exponentially small upper bounds for the difference between the first two eigenvalues.