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Regularity of the One-phase Free Boundaries

2023-02-24 Mathematics
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Release : 2023-02-24
Genre : Mathematics
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Book Rating : 386/5 ( reviews)

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Book Synopsis Regularity of the One-phase Free Boundaries by : Bozhidar Velichkov

Download or read book Regularity of the One-phase Free Boundaries written by Bozhidar Velichkov. This book was released on 2023-02-24. Available in PDF, EPUB and Kindle. Book excerpt: This open access book is an introduction to the regularity theory for free boundary problems. The focus is on the one-phase Bernoulli problem, which is of particular interest as it deeply influenced the development of the modern free boundary regularity theory and is still an object of intensive research. The exposition is organized around four main theorems, which are dedicated to the one-phase functional in its simplest form. Many of the methods and the techniques presented here are very recent and were developed in the context of different free boundary problems. We also give the detailed proofs of several classical results, which are based on some universal ideas and are recurrent in the free boundary, PDE and the geometric regularity theories. This book is aimed at graduate students and researches and is accessible to anyone with a moderate level of knowledge of elliptical PDEs.

Free Boundary Problems

2018-09-20 Mathematics
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Release : 2018-09-20
Genre : Mathematics
Kind : eBook
Book Rating : 798/5 ( reviews)

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Book Synopsis Free Boundary Problems by : Darya Apushkinskaya

Download or read book Free Boundary Problems written by Darya Apushkinskaya. This book was released on 2018-09-20. Available in PDF, EPUB and Kindle. Book excerpt: This book is concerned with several elliptic and parabolic obstacle-type problems with a focus on the cases where the free and fixed boundaries meet. The results presented complement those found in existing books in the subject, which mainly treat regularity properties away from the fixed boundary. The topics include optimal regularity, analysis of global solutions, tangential touch of the free and fixed boundaries, as well as Lipschitz- and $C^1$-regularity of the free boundary. Special attention is given to local versions of various monotonicity formulas. The intended audience includes research mathematicians and advanced graduate students interested in problems with free boundaries.

Regularity of Free Boundaries in Obstacle-Type Problems

2012 Mathematics
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Release : 2012
Genre : Mathematics
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Book Rating : 947/5 ( reviews)

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Book Synopsis Regularity of Free Boundaries in Obstacle-Type Problems by : Arshak Petrosyan

Download or read book Regularity of Free Boundaries in Obstacle-Type Problems written by Arshak Petrosyan. This book was released on 2012. Available in PDF, EPUB and Kindle. Book excerpt: The regularity theory of free boundaries flourished during the late 1970s and early 1980s and had a major impact in several areas of mathematics, mathematical physics, and industrial mathematics, as well as in applications. Since then the theory continued to evolve. Numerous new ideas, techniques, and methods have been developed, and challenging new problems in applications have arisen. The main intention of the authors of this book is to give a coherent introduction to the study of the regularity properties of free boundaries for a particular type of problems, known as obstacle-type problems. The emphasis is on the methods developed in the past two decades. The topics include optimal regularity, nondegeneracy, rescalings and blowups, classification of global solutions, several types of monotonicity formulas, Lipschitz, $C^1$, as well as higher regularity of the free boundary, structure of the singular set, touch of the free and fixed boundaries, and more. The book is based on lecture notes for the courses and mini-courses given by the authors at various locations and should be accessible to advanced graduate students and researchers in analysis and partial differential equations.

Geometric Methods in PDE’s

2015-10-31 Mathematics
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Release : 2015-10-31
Genre : Mathematics
Kind : eBook
Book Rating : 666/5 ( reviews)

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Book Synopsis Geometric Methods in PDE’s by : Giovanna Citti

Download or read book Geometric Methods in PDE’s written by Giovanna Citti. This book was released on 2015-10-31. Available in PDF, EPUB and Kindle. Book excerpt: The analysis of PDEs is a prominent discipline in mathematics research, both in terms of its theoretical aspects and its relevance in applications. In recent years, the geometric properties of linear and nonlinear second order PDEs of elliptic and parabolic type have been extensively studied by many outstanding researchers. This book collects contributions from a selected group of leading experts who took part in the INdAM meeting "Geometric methods in PDEs", on the occasion of the 70th birthday of Ermanno Lanconelli. They describe a number of new achievements and/or the state of the art in their discipline of research, providing readers an overview of recent progress and future research trends in PDEs. In particular, the volume collects significant results for sub-elliptic equations, potential theory and diffusion equations, with an emphasis on comparing different methodologies and on their implications for theory and applications.

The obstacle problem

1999-10-01 Mathematics
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Release : 1999-10-01
Genre : Mathematics
Kind : eBook
Book Rating : 492/5 ( reviews)

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Book Synopsis The obstacle problem by : Luis Angel Caffarelli

Download or read book The obstacle problem written by Luis Angel Caffarelli. This book was released on 1999-10-01. Available in PDF, EPUB and Kindle. Book excerpt: The material presented here corresponds to Fermi lectures that I was invited to deliver at the Scuola Normale di Pisa in the spring of 1998. The obstacle problem consists in studying the properties of minimizers of the Dirichlet integral in a domain D of Rn, among all those configurations u with prescribed boundary values and costrained to remain in D above a prescribed obstacle F. In the Hilbert space H1(D) of all those functions with square integrable gradient, we consider the closed convex set K of functions u with fixed boundary value and which are greater than F in D. There is a unique point in K minimizing the Dirichlet integral. That is called the solution to the obstacle problem.

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