Singular Solutions of Nonlinear Elliptic and Parabolic Equations.

Intro -- 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 &lt -- p &lt -- n -- 3.2.3 Pointwise estimates of the solutions for 1 &lt -- p &lt -- 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 &lt -- p &lt -- n - s -- 3.3.3 Pointwise integral estimates for the solution in the case 1 &lt -- p &lt -- 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 &gt -- 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 &gt -- 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.

The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome. Please submit book proposals toJürgen Appell.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.