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Consider a general linear Hamiltonian system

This monograph is the first to provide a comprehensive, self-contained and rigorous presentation of some of the most powerful preconditioning methods for solving finite element equations in a common block-matrix factorization framework.

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let In another article to appear, we will prove that when

This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

Let On the other hand, the sharp blow-up analysis also allows us to show that if the mean curvature at

This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28-July 2, 2021, and hosted by University of Alcala, Alcala de Henares, Spain. The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, $q$-difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions. The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.

This monograph is devoted to the study of the weighted Bergman space $A^p_\\omega$ of the unit disc $\\mathbb{D}$ that is induced by a radial continuous weight $\\omega$ satisfying $\\lim_{r\\to 1^-}\\frac{\\int_r^1\\omega(s)\\,ds}{\\omega(r)(1-r)}=\\infty.$ Every such $A^p_\\omega$ lies between the Hardy space $H^p$ and every classical weighted Bergman space $A^p_\\alpha$. Even if it is well known that $H^p$ is the limit of $A^p_\\alpha$, as $\\alpha\\to-1$, in many respects, it is shown that $A^p_\\omega$ lies ``closer'' to $H^p$ than any $A^p_\\alpha$, and that several finer function-theoretic properties of $A^p_\\alpha$ do not carry over to $A^p_\\omega$.

In this paper, the author considers semilinear elliptic equations of the form $-\\Delta u- \\frac{\\lambda}{|x|^2}u +b(x)\\,h(u)=0$ in $\\Omega\\setminus\\{0\\}$, where $\\lambda$ is a parameter with $-\\infty0$. The author completely classifies the behaviour near zero of all positive solutions of equation (0.1) when $h$ is regularly varying at $\\infty$ with index $q$ greater than $1$ (that is, $\\lim_{t\\to \\infty} h(\\xi t)/h(t)=\\xi^q$ for every $\\xi>0$). In particular, the author's results apply to equation (0.1) with $h(t)=t^q (\\log t)^{\\alpha_1}$ as $t\\to \\infty$ and $b(x)=|x|^\\theta (-\\log |x|)^{\\alpha_2}$ as $|x|\\to 0$, where $\\alpha_1$ and $\\alpha_2$ are any real numbers.

The aim of this paper is to analyze some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. With this motivation we prove some new results in both directions, ranging from oscillation and nonoscillation conditions for ODE’s that improve on classical criteria, to estimates in the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications. To keep our investigation basically self-contained we also collect some, more or less known, material which often appears in the literature in various forms and for which we give, in some instances, new proofs according to our specific point of view.

This textbook provides a brief and lucid introduction to the theory of linear partial differential equations. It clearly explains the transition from classical to generalized solutions and the natural way in which Sobolev spaces appear as completions of spaces of continuously differentiable functions. The solution operators associated to non-homogeneous equations are used to make transition to the theory of nonlinear PDEs. Organized on three parts, this material is suitable for three one-semester courses, a beginning one in the frame of classical analysis, a more advanced course in modern theory and a master course in semi-linear equations.