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The goal of this book is to investigate the behavior of weak solutions to the elliptic interface problem in a neighborhood of boundary singularities: angular and conic points or edges. This problem is considered both for linear and quasi-linear equations, which are among the less studied varieties. As a second edition of Transmission Problems for Elliptic Second-Order Equations for Non-Smooth Domains (Birkhäuser, 2010), this volume includes two entirely new chapters: one about the oblique derivative problems for the perturbed p(x)-Laplacian equation in a bounded n-dimensional cone, and another about the existence of bounded weak solutions.
Researchers and advanced graduate students will appreciate this compact compilation of new material in the field.
Mikhail Borsuk is a well-known specialist in nonlinear boundary value problems for elliptic equations in non-smooth domains. He is a student-follower of eminent mathematicians Y. B. Lopatinskiy and V. A. Kondratiev. He graduated from the Steklov Mathematical Institute of the Russian Academy of Sciences (Moscow) for his postgraduate studies and then worked at the Moscow Institute of Physics and Technology and the Central Aerohydrodynamic Institute of Professor N. E. Zhukovskiy. He is a professor emeritus at the University of Warmia and Mazury in Olsztyn (Poland), where he worked for more than 20 years. He has published 4 monographs, 2 textbooks for students, and 81 scientific articles.
- 1. Preliminaries.- 2. Eigenvalue Problem and Integro-Differential Inequalities.- 3. Best Possible Estimates of Solutions to the Interface Problem for Linear Elliptic Divergence Second Order Equations in a Conical Domain.- 4. Interface Problem for the Laplace Operator with N Different Media.- 5. Interface Problem for Weak Quasi-Linear Elliptic Equations in a Conical Domain.- 6. Interface Problem for Strong Quasi-Linear Elliptic Equations in a Conical Domain.- 7. Best Possible Estimates of Solutions to the Interface Problem for a Quasi-Linear Elliptic Divergence Second Order Equation in a Domain with a Boundary Edge.- 8. Interface Oblique Derivative Problem for Perturbed p(x)-Laplacian Equation in a Bounded n- Dimensional Cone.- 9. Existence of Bounded Weak Solutions.
