Oblique Derivative Problems for Elliptic Equations in Conical Domains

ISBN-13: 9783031283802 / Angielski

Mikhail Borsuk

Opis

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.

Spis treści

Introduction 5

Chapter 1. Preliminaries 11

1.1. List of symbols

1.2. Elementary inequalities

1.3. Domains with a conical point

1.4. The quasi-distance function rε and its properties

1.5. Function spaces

1.5.1. Lebesgue spaces

1.5.2. Space M(G)

1.5.3. Regularization and Approximation by Smooth Functions

1.6. Hölder and Sobolev spaces

1.6.1. Notations and denitions

1.6.2. Sobolev embedding theorems

1.7. Weighted Sobolev spaces

1.8. Spaces of Dini continuous functions

1.9. Variable exponent spaces

1.10. The Nemyckij operator and its properties

1.11. Some functional analysis

1.12. The Cauchy problem for dierential inequalities

1.13. The dependence of the eigenvalues on the coecients of the dierential equation

1.14. Some informations about the gamma and Gegenbauer functions

1.15. Additional auxiliary results

1.15.1. Mean Value Theorem

1.15.2. Stampacchia's Lemma

1.15.3. Other assertions

1.16. Notes

Chapter 2. Eigenvalue problems

2.1. Linear eigenvalue problem

2.1.1. The eigenvalue problem for n = 2

2.1.2. The eigenvalue problem for n ≥ 3

2.1.3. On properties of eigenvalues

2.2. Nonlinear eigenvalue problem

Chapter 3. Integral inequalities

3.1. The classical Hardy inequalities

3.2. The Friedrichs - Wirtinger type inequality

Chapter 4. Linear oblique derivative problem for elliptic second-order

equation in a domain with boundary conical point

4.1. Preliminaries

4.2. Setting of the problem

4.3. Global integral weighted estimate

4.4. Local integral weighted estimates

4.5. The power modulus of continuity

4.6. Examples

4.7. Notes

Chapter 5. Oblique derivative problem for elliptic second-order semilinear equations in a domain with a conical boundary point

5.1. Setting of the problem

5.2. Main result

5.3. Global integral weighted estimate

5.4. Local integral weighted estimates

5.5. Power modulus of continuity

Chapter 6. Behavior of weak solutions to the conormal problem for elliptic weak quasi-linear equations in a neighborhood of a conical boundary point

6.1. Setting of the problem

6.2. Maximum principle

6.3. Comparison principle

6.4. The barrier function. The preliminary estimate of the solution modulus

6.5. Local estimate at the boundary

6.6. Global integral estimate

6.7. Local integral weighted estimates

6.8. The power modulus of continuity at the conical point for weak solutions

6.9. Example

6.10. Notes

Chapter 7. Behavior of strong solutions to the degenerate oblique derivative problem for elliptic quasi-linear equations in a neighborhood of a boundary conical point

7.1. Setting of the problem

7.2. The barrier function. The preliminary estimate of the solution modulus

7.3. Integral weighted estimates

7.4. The power modulus of the continuity at the conical point

7.5. Notes

Chapter 8. Oblique derivative problem in a plane sector for elliptic second-order equation with perturbed p(x)-Laplacian

8.1. Setting of the problem

8.2. Preliminary

8.3. Maximum Principle

8.4. Comparison Principle

8.5. The barrier function. Estimation of the solution modulus

8.6. Proof of the main Theorem 9.3

Chapter 9. Oblique derivative problem in a bounded n−dimensional cone for strong quasi-linear elliptic second-order equation with perturbed p(x)-Laplacian

9.1. Setting of the problem

9.2. Preliminary

9.3. Maximum Principle

9.4. Comparison Principle

9.5. The barrier function

9.6. Estimation of the solution modulus. The proof of the main Theorem 9.3

Chapter 10. Existence of bounded weak solutions

10.1. Setting of the problem

10.2. Proof of the existence theorem

Bibliography

Index

Notation Index

O autorze

​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 Aeroрydrodynamic 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.