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This volume discusses an in-depth theory of function spaces in an Euclidean setting, including several new features, not previously covered in the literature. In particular, it develops a unified theory of anisotropic Besov and Bessel potential spaces on Euclidean corners, with infinite-dimensional Banach spaces as targets. It especially highlights the most important subclasses of Besov spaces, namely Slobodeckii and Hölder spaces. In this case, no restrictions are imposed on the target spaces, except for reflexivity assumptions in duality results. In this general setting, the author proves sharp embedding, interpolation, and trace theorems, point-wise multiplier results, as well as Gagliardo-Nirenberg estimates and generalizations of Aubin-Lions compactness theorems. The results presented pave the way for new applications in situations where infinite-dimensional target spaces are relevant – in the realm of stochastic differential equations, for example.

Differential equations, Parabolic. --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial --- Functional analysis. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations

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Integro-differential equations --- Differential equations, Parabolic. --- Differential equations, Nonlinear. --- Mathematical models. --- Numerical solutions. --- Models, Mathematical --- Simulation methods --- Nonlinear differential equations --- Nonlinear theories --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial --- Numerical analysis

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Differential equations, Elliptic --- Differential equations, Parabolic --- Differential equations, Elliptic. --- Differential equations, Parabolic. --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear

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This monograph offers the reader a treatment of the theory of evolution PDEs with nonstandard growth conditions. This class includes parabolic and hyperbolic equations with variable or anisotropic nonlinear structure. We develop methods for the study of such equations and present a detailed account of recent results. An overview of other approaches to the study of PDEs of this kind is provided. The presentation is focused on the issues of existence and uniqueness of solutions in appropriate function spaces, and on the study of the specific qualitative properties of solutions, such as localization in space and time, extinction in a finite time and blow-up, or nonexistence of global in time solutions. Special attention is paid to the study of the properties intrinsic to solutions of equations with nonstandard growth.

Mathematics. --- Partial Differential Equations. --- Functional Analysis. --- Functional analysis. --- Differential equations, partial. --- Mathématiques --- Analyse fonctionnelle --- Differential equations, Hyperbolic. --- Differential equations, Parabolic. --- Quasilinearization. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Parabolic differential equations --- Parabolic partial differential equations --- Hyperbolic differential equations --- Partial differential equations. --- Differential equations, Nonlinear --- Differential equations, Partial --- Numerical solutions --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations

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The heart of the book is the development of a short-time asymptotic expansion for the heat kernel. This is explained in detail and explicit examples of some advanced calculations are given. In addition some advanced methods and extensions, including path integrals, jump diffusion and others are presented. The book consists of four parts: Analysis, Geometry, Perturbations and Applications. The first part shortly reviews of some background material and gives an introduction to PDEs. The second part is devoted to a short introduction to various aspects of differential geometry that will be needed later. The third part and heart of the book presents a systematic development of effective methods for various approximation schemes for parabolic differential equations. The last part is devoted to applications in financial mathematics, in particular, stochastic differential equations. Although this book is intended for advanced undergraduate or beginning graduate students in, it should also provide a useful reference for professional physicists, applied mathematicians as well as quantitative analysts with an interest in PDEs. .

Calculus --- Mathematics --- Physical Sciences & Mathematics --- Differential equations, Partial. --- Differential equations, Parabolic. --- Heat equation. --- Diffusion equation --- Heat flow equation --- Parabolic differential equations --- Parabolic partial differential equations --- Partial differential equations --- Mathematics. --- Partial differential equations. --- Partial Differential Equations. --- Differential equations, Parabolic --- Differential equations, Partial --- Differential equations, partial.