Nonlinear Elliptic and Parabolic Equations of the Second Order

Approach your problems from the It isn't that they can't see the right end and begin with the solution. It is that they can't see answers. Then one day, perhaps the problem. you will find the final question. G.K. Chesterton. The Scandal of 'The Hermit Clad in Crane Father Brown 'The Point of a Pin'. Feathers' in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of mono­ graphs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theor.etical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

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.