Nonlinear Elliptic and Parabolic Equations of the Second Order

1 / Auxiliary Notions.- 1.1. Domains and Functional Spaces of Smooth Functions.- 1.2. The Sobolev Spaces, Measures, and Lattices.- 1.3. Conditional Existence Theorems for the Solution of a Nonlinear Equation.- 1.4. Comments.- 2 / Nonlinear Equations with Constant 'Coefficients' in the Whole Space.- 2.1. Constructing the Resolvent.- 2.2. The Existence Theorem for the Solution and the Estimates of its Derivatives.- 2.3. Comments.- 3 / A Priori Estimates in Lp for Solutions of Nonlinear Elliptic and Parabolic Equations.- 3.1. Some Properties of ?-Concave Functions.- 3.2. Estimation of Solutions of Special Nonlinear Equations.- 3.3. The Aleksandrov Estimate.- 3.4. The Maximum Principle and the Uniqueness of a Solution in the Sobolev Class.- 3.5. Passage to the Limit for Nonlinear Operators in the Sobolev Class.- 3.6. Passage to the Limit for Nonlinear Operators in Classes of Convex Functions.- 3.7. The Uniqueness of a Solution and the Comparison Theorems for Nonlinear Operators in the Class of Convex Functions.- 3.8. Comments.- 4 / A Priori Estimates in C? for Solutions of Linear and Nonlinear Equations.- 4.1. Research into Properties of Two Special Functions.- 4.2. The Harnack Inequality and the Hölder Condition for Solutions of Linear Equations with Measurable Coefficients.- 4.3. The Hölder Condition for Solutions of Simultaneous Linear Inequalities.- 4.4. The Holder Condition for Solutions of Elliptic and Parabolic Equations near the Boundary.- 4.5. The Hölder Condition for Solutions of Simultaneous Linear Inequalities near the Boundary and Estimation of their Normal Derivatives.- 4.6. The Hölder Condition for Solutions of Some Degenerate Linear Equations on the Boundary.- 4.7 Comments.- 5 / A Priori Estimates in C2+? for Solutions of Nonlinear Equations.- 5.1. The Boundedness of and the Hölder Condition for the Derivatives of Solutions on the Boundary.- 5.2. The Estimates of the First Derivative with Respect to X.- 5.3. The Estimates of the Derivative with Respect to t.- 5.4. Estimation of the Solution in the Norms of C2 and $$ W_2^{1,2} $$.- 5.5. The Estimates of u in the Norm of C2+?.- 5.6. Discussion of the First-Order Matching Conditions.- 5.7. Comments.- 6 / Existence Theorems for Solutions of Nondegenerate Equations.- 6.1. The Class F?, the Uniqueness of a Solution, and the Estimate of |u|.- 6.2. The Existence of a Solution for F ? F. The First Boundary Value Problem.- 6.3. The Existence of a Solution for F ? F in a Nonsmooth Domain and Cauchy's Problem.- 6.4. The Existence of a Solution for F?. Examples.- 6.5. The Existence of a Solution for F ? F?0.- 6.6. Comments.- 7 / Degenerate Nonlinear Equations in the Whole Space.- 7.1. Permutations of a Differential Operator with an Elliptic Operator.- 7.2. A Priori Estimates of the First and the Second Derivatives.- 7.3. The Existence of a Solution in the Class of Concave Functions.- 7.4. The Existence of a Solution in the Class of Concave Functions for the Normalized Bellman Equation.- 7.5. An Example of a One-Dimensional Degenerate Equation.- 7.6. Comments.- 8 / Degenerate Nonlinear Equations in a Domain.- 8.1. Equations with Constant 'Coefficients' in a Ball and a Circular Cylinder.- 8.2. Examples of Equations with Monge-Ampère Operators, and Other Examples.- 8.3. Interrelation Between the Equation in a Domain of a Euclidean Space and the Equation on a Manifold.- 8.4. Permutation of a Differential Operator with an Elliptic Operator on a Manifold.- 8.5. The Estimates of the Derivatives of a Solution to a Nonlinear Equation on a Manifold.- 8.6. The Estimates of the Second Mixed Derivatives on the Boundary of a Domain.- 8.7. The Solvability of Equations Weakly Nondegenerate Along the Normal.- 8.8. The Solvability of Degenerate Equations in a Domain.- 8.9. Comments.- Appendix 1 / Proof of Lemma 4.1.6.- Appendix 2 / Aleksandrov-Busemann-Feller Theorem.