Contents
Preface
General Notation
Contents of Volume 1
10 Auxiliary Results In Partial Differential
Equations
1. Schauder's estimates for elliptic and parabolic
equations
2. Sobolev's inequality
3. Lp estimates for elliptic equations
4. Lp estimates for parabolic equations
Problems
11 Nonattalnablllty
1. Basic definitions; a lemma
2. A fundamental lemma
3. The case d(x)>3
4. Thecased(x)>2
5. M consists of one point and d=l
6. The case d(i) = 0
7. Mixed case
Problems

12 Stability and Splrallng of Solutions
1. Criterion for stability 270
2. Stable obstacles 278
3. Stability of point obstacles 283
4. The method of descent 286
5. Spiraling of solutions about a point obstacle 290
6. Spiraling of solutions about any obstacle 300
7. Spiraling for linear systems 303
Problems ^ 306
13 The Dlrlchlet Problem for Degenerate
Elliptic Equations
1. A general existence theorem 308
2. Convergence of paths to boundary points 315
3. Application to the Dirichlet problem 318
Problems 322
14 Small Random Perturbations of Dynamical
Systems
1. The functional Jr(<J>) 326
2. The first Ventcel-Freidlin estimate 332
3. The second Ventcel-Freidlin estimate 334
4. Application to the first initial-boundary
value problem 346
5. Behavior of the fundamental solution as
t-+0 348
6. Behavior of Green's function as e—>0 354
7. The problem of exit 359
8. The problem of exit (continued) 367
9. Application to the Dirichlet problem 371
10. The principal eigenvalue 373
11. Asymptotic behavior of the principal eigenvalue 376
Problems 383
I5  Fundamental Solutions for Degenerate
Parabolic Equations
1. Construction of a candidate for a funda-
fundamental solution 388
2. Interior estimates 396
3. Boundary estimates 399
4. Estimates near infinity 406
5. Relation between К and a diffusion process 409
6. The behavior of |(f) near S 414
7. Existence of a generalized solution in the case of
a two-sided obstacle 420
8. Existence of a fundamental solution in the case of
a strictly one-sided obstacle 423
9. Lower bounds on the fundamental solution 426
10. The Cauchy problem 428
Problems 432
16 Stopping Time Problems and Stochastic Games
Part I. The Stationary Case
1. Statement of the problem 433
2. Characterization of saddle points 436
3. Elliptic variational inequalities in bounded domains 440
4. Existence of saddle points in bounded domains 444
5. Elliptic estimates for increasing domains 447
6. Elliptic variational inequalities 457
7. Existence of saddle points in unbounded domains 462
8. The stopping time problem 463
Part II. The Nonstationary Case
9. Characterization of saddle points 464
10. Parabolic variational inequalities 466
11. Parabolic variational inequalities (con-
(continued) 478
12. Existence of a saddle point 486
13. The stopping time problem 488
Problems 490

17 Stochastic Differential Games

1. Auxiliary results 494
2. N-person stochastic differential games with
perfect observation 498
3. Stochastic differential games with stopping time 502
4. Stochastic differential games with partial observation 507
Problems 518
Bibliographical Remarks 520
References 523
Index 527