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Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation

Author: Benoît Mselati

Publisher: American Mathematical Society(RI)

Published: 2014-09-11

Total Pages: 121

ISBN-13: 9781470403966

An analytic approach to the equation $\Delta u = u^2$ A probabilistic approach to the equation $\Delta u = u^2$ Lower bounds for solutions Upper bounds for solutions The classification and representation of the solutions of $\Delta u = u^2$ in a domain Appendix A. Technical results Appendix. Bibliography Notation index Subject index.

Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation

Author: Benoît Mselati

Publisher: American Mathematical Soc.

Published: 2004

Total Pages: 146

ISBN-13: 0821835092

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Concerned with the nonnegative solutions of $\Delta u = u^2$ in a bounded and smooth domain in $\mathbb{R}^d$, this title intends to prove that they are uniquely determined by their fine trace on the boundary as defined in [DK98a], answering a major open question of [Dy02].

Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations

Author: Evgeniĭ Borisovich Dynkin

Publisher: American Mathematical Soc.

Published: 2004

Total Pages: 130

ISBN-13: 082183682X

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This book is devoted to the applications of probability theory to the theory of nonlinear partial differential equations. More precisely, it is shown that all positive solutions for a class of nonlinear elliptic equations in a domain are described in terms of their traces on the boundary of the domain. The main probabilistic tool is the theory of superdiffusions, which describes a random evolution of a cloud of particles. A substantial enhancement of this theory is presented that will be of interest to anyone who works on applications of probabilistic methods to mathematical analysis. The book is suitable for graduate students and research mathematicians interested in probability theory and its applications to differential equations. Also of interest by this author is Diffusions, Superdiffusions and Partial Differential Equations in the AMS series, Colloquium Publications.

Perspectives in Nonlinear Partial Differential Equations

Author: Henri Berestycki

Publisher: American Mathematical Soc.

Published: 2007

Total Pages: 522

ISBN-13: 0821841904

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In celebration of Haim Brezis's 60th birthday, a conference was held at the Ecole Polytechnique in Paris, with a program testifying to Brezis's wide-ranging influence on nonlinear analysis and partial differential equations. The articles in this volume are primarily from that conference. They present a rare view of the state of the art of many aspects of nonlinear PDEs, as well as describe new directions that are being opened up in this field. The articles, written by mathematicians at the center of current developments, provide somewhat more personal views of the important developments and challenges.

Holder Continuity of Weak Solutions to Subelliptic Equations with Rough Coefficients

Author: Eric T. Sawyer

Publisher: American Mathematical Soc.

Published: 2006

Total Pages: 176

ISBN-13: 0821838261

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This mathematical monograph is a study of interior regularity of weak solutions of second order linear divergence form equations with degenerate ellipticity and rough coefficients. The authors show that solutions of large classes of subelliptic equations with bounded measurable coefficients are H lder continuous. They present two types of results f

Nonlinear Second Order Elliptic Equations Involving Measures

Author: Moshe Marcus

Publisher: Walter de Gruyter

Published: 2013-11-27

Total Pages: 264

ISBN-13: 3110305313

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In the last 40 years semi-linear elliptic equations became a central subject of study in the theory of nonlinear partial differential equations. On the one hand, the interest in this area is of a theoretical nature, due to its deep relations to other branches of mathematics, especially linear and nonlinear harmonic analysis, dynamical systems, differential geometry and probability. On the other hand, this study is of interest because of its applications. Equations of this type come up in various areas such as problems of physics and astrophysics, curvature problems in Riemannian geometry, logistic problems related for instance to population models and, most importantly, the study of branching processes and superdiffusions in the theory of probability. The aim of this book is to present a comprehensive study of boundary value problems for linear and semi-linear second order elliptic equations with measure data. We are particularly interested in semi-linear equations with absorption. The interactions between the diffusion operator and the absorption term give rise to a large class of nonlinear phenomena in the study of which singularities and boundary trace play a central role. This book is accessible to graduate students and researchers with a background in real analysis and partial differential equations.

Shock-Wave Solutions of the Einstein Equations with Perfect Fluid Sources: Existence and Consistency by a Locally Inertial Glimm Scheme

Author: Jeff Groah

Publisher: American Mathematical Soc.

Published: 2004

Total Pages: 98

ISBN-13: 082183553X

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Demonstrates the consistency of the Einstein equations at the level of shock-waves by proving the existence of shock wave solutions of the spherically symmetric Einstein equations for a perfect fluid, starting from initial density and velocity profiles that are only locally of bounded total variation.

Equivalences of Classifying Spaces Completed at the Prime Two

Author: Robert Oliver

Publisher: American Mathematical Soc.

Published: 2006

Total Pages: 116

ISBN-13: 0821838288

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We prove here the Martino-Priddy conjecture at the prime $2$: the $2$-completions of the classifying spaces of two finite groups $G$ and $G'$ are homotopy equivalent if and only if there is an isomorphism between their Sylow $2$-subgroups which preserves fusion. This is a consequence of a technical algebraic result, which says that for a finite group $G$, the second higher derived functor of the inverse limit vanishes for a certain functor $\mathcal{Z}_G$ on the $2$-subgroup orbit category of $G$. The proof of this result uses the classification theorem for finite simple groups.

Twisted Tensor Products Related to the Cohomology of the Classifying Spaces of Loop Groups

Author: Katsuhiko Kuribayashi

Publisher: American Mathematical Soc.

Published: 2006

Total Pages: 98

ISBN-13: 0821838563

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Let $G$ be a compact, simply connected, simple Lie group. By applying the notion of a twisted tensor product in the senses of Brown as well as of Hess, we construct an economical injective resolution to compute, as an algebra, the cotorsion product which is the $E_2$-term of the cobar type Eilenberg-Moore spectral sequence converging to the cohomology of classifying space of the loop group $LG$. As an application, the cohomology $H^*(BLSpin(10); \mathbb{Z}/2)$ is explicitly determined as an $H^*(BSpin(10); \mathbb{Z}/2)$-module by using effectively the cobar type spectral sequence and the Hochschild spectral sequence, and further, by analyzing the TV-model for $BSpin(10)$.

Representation Type of Commutative Noetherian Rings III: Global Wildness and Tameness

Author: Lee Klingler

Publisher: American Mathematical Soc.

Published: 2005

Total Pages: 187

ISBN-13: 0821837389

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This memoir completes the series of papers beginning with [KL1,KL2], showing that, for a commutative noetherian ring $\Lambda$, either the category of $\Lambda$-modules of finite length has wild representation type or else we can describe the category of finitely generated $\Lambda$-modules, including their direct-sum relations and local-global relations. (There is a possible exception to our results, involving characteristic 2.)