Singular Solutions of Nonlinear Elliptic and Parabolic Equations

✍ Scribed by Alexander A. Kovalevsky; Igor I. Skrypnik; Andrey E. Shishkov

• Publisher: De Gruyter
• Year: 2016
• Tongue: English
• Leaves: 448
• Series: De Gruyter Series in Nonlinear Analysis and Applications; 24
• Category: Library

✦ Synopsis

This monograph looks at several trends in the investigation of singular solutions of nonlinear elliptic and parabolic equations. It discusses results on the existence and properties of weak and entropy solutions for elliptic second-order equations and some classes of fourth-order equations with L1-data and questions on the removability of singularities of solutions to elliptic and parabolic second-order equations in divergence form. It looks at localized and nonlocalized singularly peaking boundary regimes for different classes of quasilinear parabolic second- and high-order equations in divergence form.
The book will be useful for researchers and post-graduate students that specialize in the field of the theory of partial differential equations and nonlinear analysis.

Contents:
Foreword
Part I: Nonlinear elliptic equations with L^1-data
Nonlinear elliptic equations of the second order with L^1-data
Nonlinear equations of the fourth order with strengthened coercivity and L^1-data
Part II: Removability of singularities of the solutions of quasilinear elliptic and parabolic equations of the second order
Removability of singularities of the solutions of quasilinear elliptic equations
Removability of singularities of the solutions of quasilinear parabolic equations
Quasilinear elliptic equations with coefficients from the Kato class
Part III: Boundary regimes with peaking for quasilinear parabolic equations
Energy methods for the investigation of localized regimes with peaking for parabolic second-order equations
Method of functional inequalities in peaking regimes for parabolic equations of higher orders
Nonlocalized regimes with singular peaking
Appendix: Formulations and proofs of the auxiliary results
Bibliography

✦ Table of Contents

Contents
Foreword
I. Nonlinear elliptic equations with L1-data
1. Nonlinear elliptic equations of the second order with L1-data
1.1 Introduction
1.2 General conditions for the limiting summability of solutions
1.3 The cases where the right-hand side of the equation belongs to logarithmic classes
1.4 On the integrally logarithmic conditions for the limiting summability of solutions
1.5 The case where the right-hand side of equation belongs to Lebesgue spaces close to L1(O)
1.6 On the convergence of functions from satisfying special integral inequalities
1.7 On the existence of entropy solutions for the equations with degenerate coercivity and L1-data
1.8 A priori properties of the entropy solutions of equations with degenerate coercivity and L1-data
2. Nonlinear equations of the fourth order with strengthened coercivity and L1-data
2.1 Introduction
2.2 Set of functions
2.3 Definition and some properties of entropy solutions
2.4 One a priori estimate for the entropy solutions
2.5 Notion of H-solution
2.6 On uniqueness of the entropy solution
2.7 Theorems on existence
2.8 Entropy solutions as elements of the Sobolev spaces and the existence of W-solutions
2.9 On the summability of functions from satisfying certain integral inequalities
2.10 Improvement of the properties of summability for the solutions of problem (2.1.6), (2.1.7)
2.11 Some characteristics of the set of functions
2.12 Set of functions
2.13 Definition and a priori estimates of the proper entropy solutions
2.14 Existence of the proper entropy solutions
2.15 Relationship with the entropy solutions and the theorem on uniqueness
2.16 Relationship with H-solutions and W-solutions
2.17 Properties of summability of the proper entropy solutions
2.18 Relationship with generalized solutions
2.19 Examples of coefficients and the right-hand sides of Eq. (2.1.6)
II. Removability of singularities of the solutions of quasilinear elliptic and parabolic equations of the second order
3. Removability of singularities of the solutions of quasilinear elliptic equations
3.1 Introduction
3.2 Removability of isolated singularities
3.2.1 Formulation of assumptions and principal results
3.2.2 Integral estimates of the solutions for 1 < p < n
3.2.3 Pointwise estimates of the solutions for 1 < p < n
3.3 Removability of singularities of the solutions of elliptic equations on manifolds
3.3.1 Formulation of assumptions and main results
3.3.2 Integral estimates for the gradient of solution in the case 1 < p < n - s
3.3.3 Pointwise integral estimates for the solution in the case 1 < p < n - s
3.4 Removability of isolated singularities of the solutions of elliptic equations with absorption
3.4.1 Formulation of the assumptions and main results
3.4.2 Proof of Theorem 3:4:2
3.4.3 Integral estimates for the gradient of the solution
3.4.4 Proof of Theorem 3:4:1
4. Removability of singularities of the solutions of quasilinear parabolic equations
4.1 Introduction
4.2 Removability of isolated singularities
4.2.1 Formulation of assumptions and main results
4.2.2 Integral estimates for the solution
4.2.3 Pointwise estimates of the solution
4.3 Removability of isolated singularities for the solutions of quasilinear parabolic equations with absorption
4.3.1 Formulation of the assumptions and main results
4.3.2 Proof of Theorem 4:3:2
4.3.3 Integral estimates for the solution
4.3.4 Proof of Theorem 4:3:1
5. Quasilinear elliptic equations with coefficients from the Kato class
5.1 Introduction
5.2 Harnack’s inequality
5.2.1 Formulation of assumptions and main results
5.2.2 Proof of Theorem 5:2:1
5.2.3 Proof of Theorem 5:2:2
5.3 Removability of isolated singularities
5.3.1 Statement of propositions and main results
5.3.2 Proof of Theorem 5:3:2
5.3.3 Proof of Theorem 5:3:1
5.4 Removability of isolated singularities for the solutions of quasilinear elliptic equations with absorption
5.4.1 Formulation of assumptions and main results
5.4.2 Proof of Theorem 5:4:2
5.4.3 Integral and pointwise estimates for the gradient of the solution
III. Boundary regimes with peaking for quasilinear parabolic equations
6. Energy methods for the investigation of localized regimes with peaking for parabolic second-order equations
6.1 Introduction: localized and nonlocalized singular boundary regimes
6.2 Sufficient conditions for the localization of boundary regimes with peaking
6.3 Sharp conditions for the effective localization of boundary regimes: the case of slow diffusion p > q
6.4 Effective localization of singular boundary regimes for quasihomogeneous parabolic equations
6.5 Effective localization of singular boundary regimes for the equations of nonstationary fast-diffusion type
7. Method of functional inequalities in peaking regimes for parabolic equations of higher orders
7.1 Boundary peaking regimes for quasilinear parabolic equations of higher orders
7.2 Energy functions of the solutions and the main system of functional inequalities
7.3 Localized singular boundary regimes: the case of slow diffusion p > q
7.4 Localized boundary regimes: the case p = q
8. Nonlocalized regimes with singular peaking
8.1 Propagation of blow-up waves
8.2 Estimates for the blow-up wave in the equation of slow-diffusion type
8.3 Blow-up waves in quasihomogeneous parabolic equations
9. Appendix: Formulations and proofs of the auxiliary results
9.1 Interpolation inequalities
9.2 Systems of differential inequalities
9.3 Functional inequalities
Bibliography