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This book is intended for graduate students and research mathematicians working in several complex variables and analytic spaces.
This memoir considers the Dirichlet problem for parabolic operators in a half space with singular drift terms. Chapter I begins the study of a parabolic PDE modelled on the pullback of the heat equation in certain time varying domains considered by Lewis-Murray and Hofmann-Lewis. Chapter II obtains mutual absolute continuity of parabolic measure and Lebesgue measure on the boundary of this halfspace and also that the $L DEGREESq(R DEGREESn)$ Dirichlet problem for these PDEs has a solution when $q$ is large enough. Chapter III proves an analogue of a theorem of Fefferman, Kenig, and Pipher for certain parabolic PDEs with singular drift terms. Each of the chapters that comprise this memoir has its own numbering system and list
This book is intended for graduate students and research mathematicians interested in mechanics of particle systems.
In the case where the norms are induced by metrics on the fibres of ${\mathcal L}$, we establish the functoriality of the sectional capacity under base change, pullbacks by finite surjective morphisms, and products. We study the continuity of $S Gamma(\overline{\mathcal L})$ under variation of the metric and line bundle, and we apply this to show that the notion of $v$-adic sets in $X(\mathbb C v)$ of capacity $0$ is well-defined. Finally, we show that sectional capacities for arbitrary norms can be well-approximated using objects of finite type.
This book is intended for graduate students and research mathematicians interested in algebraic topology.
There are several generalizations of the classical theory of Sobolev spaces as they are necessary for the applications to Carnot-Caratheodory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms. The aim of this paper is to present a unified approach to the theory of Sobolev spaces that covers applications to many of those areas. The variety of different areas of applications forces a very general setting. We are given a metric space $X$ equipped with a doubling measure $\mu$. A generalization of a Sobolev function and its gradient is a pair $u\in L^{1}_{\rm loc}(X)$, $0\leq g\in L^{p}(X)$ such that for every ball $B\subset X$ the Poincare-type inequality $ \intbar_{B} u-u_{B} \, d\mu \leq C r ( \intbar_{\sigma B} g^{p}\, d\mu)^{1/p}\,$ holds, where $r$ is the radius of $B$ and $\sigma\geq 1$, $C>0$ are fixed constants. Working in the above setting we show that basically all relevant results from the classical theory have their counterparts in our general setting. These include Sobolev-Poincare type embeddings, Rellich-Kondrachov compact embedding theorem, and even a version of the Sobolev embedding theorem on spheres. The second part of the paper is devoted to examples and applications in the above mentioned areas.
This book is intended for graduate students and research mathematicians interested in calculus of variations and optimal control; optimization.
Let $\mathcal S$ be a second order smoothness in the $\mathbb{R} DEGREESn$ setting. We can assume without loss of generality that the dimension $n$ has been adjusted as necessary so as to insure that $\mathcal S$ is also non-degenerate. This title describes how $\mathcal S$ must fit into one of three mutually exclusive cases, and in each of these cases the authors characterize, by a simple intrinsic condition, the second order smoothnesses $\mathcal S$ whose canonical Sobolev projection $P_{\mathcal{S}}$ is of weak type $(1,1)$ in the $\mathbb{R} DEGR
This title provides a comprehensive examination of non-uniform lattices on uniform trees. Topics include graphs of groups, tree actions and edge-indexed graphs; $Aut(x)$ and its discrete subgroups; existence of tree lattices; non-uniform coverings of indexed graphs with an arithmetic bridge; non-uniform coverings of indexed graphs with a separating edge; non-uniform coverings of indexed graphs with a ramified loop; eliminating multiple edges; existence of arithmetic bridges. This book is intended for graduate students and research mathematicians interested in group theory and generalizations.