Oblique Derivative Problems for Elliptic Equations in Conical Domains

The aim of our book is the investigation of the behavior of strong and weak solutions to the regular oblique derivative problems for second order elliptic equations, linear and quasi-linear, in the neighborhood of the boundary singularities. The main goal is to establish the precise exponent of the solution decrease rate and under the best possible conditions. The question on the behavior of solutions of elliptic boundary value problems near boundary singularities is of great importance for its many applications, e.g., in hydrodynamics, aerodynamics, fracture mechanics, in the geodesy etc. Only few works are devoted to the regular oblique derivative problems for second order elliptic equations in non-smooth domains. All results are given with complete proofs. The monograph will be of interest to graduate students and specialists in elliptic boundary value problems and their applications.

Provides new results in the oblique derivative probles for elliptic equations in nonsmooth domains Addresses problems previously studied only in smooth domains Investigates regular oblique derivative problems in nonsmooth domains

Auteur
Professor 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 at 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 at the Central Aeröydrodynamic Institute of Professor N. E. Zhukovskiy. Currently he is professor emeritus at the University of Warmia and Mazury in Olsztyn (Poland), here he worked for more than 20 years. He has published 3 monographs, 2 textbooks for students and about 80 scientific articles.

Résumé

"The book under review presents a comprehensive and meticulously structured exploration of strong and weak solutions concerning regular oblique derivative problems for second-order elliptic equations. Its systematic approach and detailed examination of boundary singularities make it a valuable resource for researchers and practitioners in various fields ... . Its comprehensive treatment, structured approach, and detailed analysis make it an indispensable resource for postgraduates and young researchers seeking to deepen their understanding of elliptic equations within conical domains." (Giuseppe Di Fazio, zbMATH 1532.35001, 2024)

Contenu

• 1. Introduction. - 2. Preliminaries. - 3. Eigenvalue Problems. - 4. Integral Inequalities. - 5. The Linear Oblique Derivative Problem for Elliptic Second Order Equation in a Domain with Conical Boundary Point. - 6. The Oblique Derivative Problem for Elliptic Second Order Semi-linear Equations in a Domain with a Conical Boundary Point. - 7. Behavior of Weak Solutions to the Conormal Problem for Elliptic Weak Quasi-Linear Equations in a Neighborhood of a Conical Boundary Point. - 8. Behavior of Strong Solutions to the Degenerate Oblique Derivative Problem for Elliptic Quasi-linear Equations in a Neighborhood of a Boundary Conical Point. - 9. The Oblique Derivative Problem in a Plane Sector for Elliptic Second Order Equation with Perturbed p( x )-Laplacian. - 10. The Oblique Derivative Problem in a Bounded n -Dimensional Cone for Strong Quasi-Linear Elliptic Second Order Equation with Perturbed p( x )-Laplacian. - 11. Existence of Bounded Weak Solutions.