Parabolic Systems with Polynomial Growth and Regularity
Parabolic Systems with Polynomial Growth and Regularity

Author: Frank Duzaar

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 135

ISBN-13: 0821849670

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The authors establish a series of optimal regularity results for solutions to general non-linear parabolic systems $ u_t- \mathrm{div} \ a(x,t,u,Du)+H=0,$ under the main assumption of polynomial growth at rate $p$ i.e. $ a(x,t,u,Du) \leq L(1+ Du ^{p-1}), p \geq 2.$ They give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the first estimates on the (parabolic) Hausdorff dimension of the related singular set, and the first Calderon-Zygmund estimates for non-homogeneous problems are achieved here.

The Regularity of General Parabolic Systems with Degenerate Diffusion
The Regularity of General Parabolic Systems with Degenerate Diffusion

Author: Verena Bögelein

Publisher: American Mathematical Soc.

Published: 2013-01-28

Total Pages: 155

ISBN-13: 0821889753

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The aim of the paper is twofold. On one hand the authors want to present a new technique called $p$-caloric approximation, which is a proper generalization of the classical compactness methods first developed by DeGiorgi with his Harmonic Approximation Lemma. This last result, initially introduced in the setting of Geometric Measure Theory to prove the regularity of minimal surfaces, is nowadays a classical tool to prove linearization and regularity results for vectorial problems. Here the authors develop a very far reaching version of this general principle devised to linearize general degenerate parabolic systems. The use of this result in turn allows the authors to achieve the subsequent and main aim of the paper, that is, the implementation of a partial regularity theory for parabolic systems with degenerate diffusion of the type $\partial_t u - \mathrm{div} a(Du)=0$, without necessarily assuming a quasi-diagonal structure, i.e. a structure prescribing that the gradient non-linearities depend only on the the explicit scalar quantity.

Linear and Quasilinear Parabolic Systems: Sobolev Space Theory
Linear and Quasilinear Parabolic Systems: Sobolev Space Theory

Author: David Hoff

Publisher: American Mathematical Soc.

Published: 2020-11-18

Total Pages: 226

ISBN-13: 1470461617

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This monograph presents a systematic theory of weak solutions in Hilbert-Sobolev spaces of initial-boundary value problems for parabolic systems of partial differential equations with general essential and natural boundary conditions and minimal hypotheses on coefficients. Applications to quasilinear systems are given, including local existence for large data, global existence near an attractor, the Leray and Hopf theorems for the Navier-Stokes equations and results concerning invariant regions. Supplementary material is provided, including a self-contained treatment of the calculus of Sobolev functions on the boundaries of Lipschitz domains and a thorough discussion of measurability considerations for elements of Bochner-Sobolev spaces. This book will be particularly useful both for researchers requiring accessible and broadly applicable formulations of standard results as well as for students preparing for research in applied analysis. Readers should be familiar with the basic facts of measure theory and functional analysis, including weak derivatives and Sobolev spaces. Prior work in partial differential equations is helpful but not required.

Jumping Numbers of a Simple Complete Ideal in a Two-Dimensional Regular Local Ring
Jumping Numbers of a Simple Complete Ideal in a Two-Dimensional Regular Local Ring

Author: Tarmo Järvilehto

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 93

ISBN-13: 0821848119

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The multiplier ideals of an ideal in a regular local ring form a family of ideals parameterized by non-negative rational numbers. As the rational number increases the corresponding multiplier ideal remains unchanged until at some point it gets strictly smaller. A rational number where this kind of diminishing occurs is called a jumping number of the ideal. In this manuscript the author gives an explicit formula for the jumping numbers of a simple complete ideal in a two-dimensional regular local ring. In particular, he obtains a formula for the jumping numbers of an analytically irreducible plane curve. He then shows that the jumping numbers determine the equisingularity class of the curve.

Elliptic Integrable Systems
Elliptic Integrable Systems

Author: Idrisse Khemar

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 234

ISBN-13: 0821869256

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In this paper, the author studies all the elliptic integrable systems, in the sense of C, that is to say, the family of all the $m$-th elliptic integrable systems associated to a $k^\prime$-symmetric space $N=G/G_0$. The author describes the geometry behind this family of integrable systems for which we know how to construct (at least locally) all the solutions. The introduction gives an overview of all the main results, as well as some related subjects and works, and some additional motivations.

Contemporary Research in Elliptic PDEs and Related Topics
Contemporary Research in Elliptic PDEs and Related Topics

Author: Serena Dipierro

Publisher: Springer

Published: 2019-07-12

Total Pages: 502

ISBN-13: 303018921X

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This volume collects contributions from the speakers at an INdAM Intensive period held at the University of Bari in 2017. The contributions cover several aspects of partial differential equations whose development in recent years has experienced major breakthroughs in terms of both theory and applications. The topics covered include nonlocal equations, elliptic equations and systems, fully nonlinear equations, nonlinear parabolic equations, overdetermined boundary value problems, maximum principles, geometric analysis, control theory, mean field games, and bio-mathematics. The authors are trailblazers in these topics and present their work in a way that is exhaustive and clearly accessible to PhD students and early career researcher. As such, the book offers an excellent introduction to a variety of fundamental topics of contemporary investigation and inspires novel and high-quality research.

On $L$-Packets for Inner Forms of $SL_n$
On $L$-Packets for Inner Forms of $SL_n$

Author: Kaoru Hiraga

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 110

ISBN-13: 0821853643

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The theory of $L$-indistinguishability for inner forms of $SL_2$ has been established in the well-known paper of Labesse and Langlands (L-indistinguishability forSL$(2)$. Canad. J. Math. 31 (1979), no. 4, 726-785). In this memoir, the authors study $L$-indistinguishability for inner forms of $SL_n$ for general $n$. Following the idea of Vogan in (The local Langlands conjecture. Representation theory of groups and algebras, 305-379, Contemp. Math. 145 (1993)), they modify the $S$-group and show that such an $S$-group fits well in the theory of endoscopy for inner forms of $SL_n$.

Second Order Analysis on $(\mathscr {P}_2(M),W_2)$
Second Order Analysis on $(\mathscr {P}_2(M),W_2)$

Author: Nicola Gigli

Publisher: American Mathematical Soc.

Published: 2012-02-22

Total Pages: 173

ISBN-13: 0821853090

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The author develops a rigorous second order analysis on the space of probability measures on a Riemannian manifold endowed with the quadratic optimal transport distance $W_2$. The discussion includes: definition of covariant derivative, discussion of the problem of existence of parallel transport, calculus of the Riemannian curvature tensor, differentiability of the exponential map and existence of Jacobi fields. This approach does not require any smoothness assumption on the measures considered.

Towards a Modulo $p$ Langlands Correspondence for GL$_2$
Towards a Modulo $p$ Langlands Correspondence for GL$_2$

Author: Christophe Breuil

Publisher: American Mathematical Soc.

Published: 2012-02-22

Total Pages: 127

ISBN-13: 0821852272

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The authors construct new families of smooth admissible $\overline{\mathbb{F}}_p$-representations of $\mathrm{GL}_2(F)$, where $F$ is a finite extension of $\mathbb{Q}_p$. When $F$ is unramified, these representations have the $\mathrm{GL}_2({\mathcal O}_F)$-socle predicted by the recent generalizations of Serre's modularity conjecture. The authors' motivation is a hypothetical mod $p$ Langlands correspondence.