|
Showing 1 - 5 of
5 matches in All Departments
The purpose of this CIME summer school was to present current areas
of research arising both in the theoretical and applied setting
that involve fully nonlinear partial different equations. The
equations presented in the school stem from the fields of Conformal
Mapping Theory, Differential Geometry, Optics, and Geometric Theory
of Several Complex Variables. The school consisted of four courses:
Extremal problems for quasiconformal mappings in space by Luca
Capogna, Fully nonlinear equations in geometry by Pengfei Guan,
Monge-Ampere type equations and geometric optics by Cristian E.
Gutierrez, and On the Levi Monge Ampere equation by Annamaria
Montanari.
The Monge-Ampere equation has attracted considerable interest in
recent years because of its important role in several areas of
applied mathematics. Monge-Ampere type equations have applications
in the areas of differential geometry, the calculus of variations,
and several optimization problems, such as the Monge-Kantorovitch
mass transfer problem. This book stresses the geometric aspects of
this beautiful theory, using techniques from harmonic analysis -
covering lemmas and set decompositions.
This book concerns the theory of optimal transport (OT) and its
applications to solving problems in geometric optics. It is a
self-contained presentation including a detailed analysis of the
Monge problem, the Monge-Kantorovich problem, the transshipment
problem, and the network flow problem. A chapter on Monge-Ampère
measures is included containing also exercises. A detailed analysis
of the Wasserstein metric is also carried out. For the applications
to optics, the book describes the necessary background concerning
light refraction, solving both far-field and near-field refraction
problems, and indicates lines of current research in this area.
Researchers in the fields of mathematical analysis, optimal
transport, partial differential equations (PDEs), optimization, and
optics will find this book valuable. It is also suitable for
graduate students studying mathematics, physics, and engineering.
The prerequisites for this book include a solid understanding of
measure theory and integration, as well as basic knowledge of
functional analysis.
Now in its second edition, this monograph explores the Monge-Ampere
equation and the latest advances in its study and applications. It
provides an essentially self-contained systematic exposition of the
theory of weak solutions, including regularity results by L. A.
Caffarelli. The geometric aspects of this theory are stressed using
techniques from harmonic analysis, such as covering lemmas and set
decompositions. An effort is made to present complete proofs of all
theorems, and examples and exercises are offered to further
illustrate important concepts. Some of the topics considered
include generalized solutions, non-divergence equations, cross
sections, and convex solutions. New to this edition is a chapter on
the linearized Monge-Ampere equation and a chapter on interior
Hoelder estimates for second derivatives. Bibliographic notes,
updated and expanded from the first edition, are included at the
end of every chapter for further reading on Monge-Ampere-type
equations and their diverse applications in the areas of
differential geometry, the calculus of variations, optimization
problems, optimal mass transport, and geometric optics. Both
researchers and graduate students working on nonlinear differential
equations and their applications will find this to be a useful and
concise resource.
Now in its second edition, this monograph explores the Monge-Ampere
equation and the latest advances in its study and applications. It
provides an essentially self-contained systematic exposition of the
theory of weak solutions, including regularity results by L. A.
Caffarelli. The geometric aspects of this theory are stressed using
techniques from harmonic analysis, such as covering lemmas and set
decompositions. An effort is made to present complete proofs of all
theorems, and examples and exercises are offered to further
illustrate important concepts. Some of the topics considered
include generalized solutions, non-divergence equations, cross
sections, and convex solutions. New to this edition is a chapter on
the linearized Monge-Ampere equation and a chapter on interior
Hoelder estimates for second derivatives. Bibliographic notes,
updated and expanded from the first edition, are included at the
end of every chapter for further reading on Monge-Ampere-type
equations and their diverse applications in the areas of
differential geometry, the calculus of variations, optimization
problems, optimal mass transport, and geometric optics. Both
researchers and graduate students working on nonlinear differential
equations and their applications will find this to be a useful and
concise resource.
|
You may like...
Catan
(16)
R1,150
R887
Discovery Miles 8 870
|