Competing Operators and Their Applications to Boundary Value Problems

Marek Galewski (u. a.)

ISBN 10: 3032154448 / ISBN 13: 9783032154446

Published by Springer, 2026

Competing Operators and Their Applications to Boundary Value Problems | Marek Galewski (u. a.) | Taschenbuch | SpringerBriefs in Mathematics | x | Englisch | 2026 | Springer | EAN 9783032154446 | Verantwortliche Person für die EU: Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg, juergen[dot]hartmann[at]springer[dot]com | Anbieter: preigu. Seller Inventory # 134595281

Synopsis:

This book addresses problems driven by differential operators that lack monotonicity. The authors’ methods rely on coercivity and continuity, allowing for the construction of an approximative scheme whose convergence is induced by coercivity.

This observation leads to a new type of solution, which is precisely a limit of finite-dimensional approximation schemes and leads to the weak solution, provided that the operator driving the equation is at least pseudomonotone. This new type of solution is called a generalized solution. To systematically treat its existence, the authors introduce an abstract existence tool that serves as a counterpart to the Browder-Minty Theorem in the non-variational case and the Weierstrass-Tonelli Theorem if the problem is potential. Thus, the authors utilize many already developed techniques, suitably modified due to the absence of the monotonicity assumption.

The authors obtain three abstract results, also in the non-smooth case, which they apply to nonlinear boundary value problems. In their applications, they also deal with problems depending on an unbounded weight, which forces them to implement a suitable truncation technique.

The book includes an extended chapter covering analysis on abstract tools from the theory of monotone operators and minimization techniques, supplied with proofs and comments that allow for a better understanding of the authors’ approach towards generalized solutions. It includes necessary background on Sobolev spaces, introduces the non-variational generalized solution, and investigates the existence of solutions for variational problems and inclusions.

From the Back Cover:

This book addresses problems driven by differential operators that lack monotonicity. The authors’ methods rely on coercivity and continuity, allowing for the construction of an approximative scheme whose convergence is induced by coercivity.

This observation leads to a new type of solution, which is precisely a limit of finite-dimensional approximation schemes and leads to the weak solution, provided that the operator driving the equation is at least pseudomonotone. This new type of solution is called a generalized solution. To systematically treat its existence, the authors introduce an abstract existence tool that serves as a counterpart to the Browder-Minty Theorem in the non-variational case and the Weierstrass-Tonelli Theorem if the problem is potential. Thus, the authors utilize many already developed techniques, suitably modified due to the absence of the monotonicity assumption.

The authors obtain three abstract results, also in the non-smooth case, which they apply to nonlinear boundary value problems. In their applications, they also deal with problems depending on an unbounded weight, which forces them to implement a suitable truncation technique.

The book includes an extended chapter covering analysis on abstract tools from the theory of monotone operators and minimization techniques, supplied with proofs and comments that allow for a better understanding of the authors’ approach towards generalized solutions. It includes necessary background on Sobolev spaces, introduces the non-variational generalized solution, and investigates the existence of solutions for variational problems and inclusions.

Bibliographic Details

Title: Competing Operators and Their Applications ...
Publisher: Springer
Publication Date: 2026
Binding: Taschenbuch
Condition: Neu