| preface chapter i. introduction 1. outline of this book 2. further remarks 3. notation chapter ii. maximum principles 1. the weak maximum priciple 2. the strong maximum principle 3. a priori estimates notes exercises chapter iii. introduction to the theory of weak solutions 1. the theory of weak derivatives 2. the method of continuity 3. problems in small balls 4. global existence and the perron process notes exercises chapter iv. holder estimates .1. hslder continuity 2. campanato spaces 3. interior estimates 4. estimates near a fiat boundary 5. regularized distance 6. intermediate schauder estimates 7. curved boundaries and nonzero boundary data 8. a special mixed problem notes exercises chapter v. existence, uniqueness, and regularity of solutions 1. uniqueness of solutions 2. the cauchy-dirichlet problem with bounded coefficients 3. the cauchy-dirichlet problem with unbounded coefficients 4. the oblique derivative problem notes exercises chapter vi. further theory of weak solutions 1. notation and basic results 2. differentiability of weak derivatives 3. sobolev inequalities 4. poincare's inequality 5. global boundedness 6. local estimates 7. consequences of the local estimates 8. boundary estimates 9. more sobolev-type inequalities 10. conormal problems 11. solvability in holder spaces 12. the parabolic degiorgi classes notes exercises chapter vii. strong solutions 1. maximum principles 2. basic results from harmonic analysis 3. lp estimates for constant coefficient equations 4. interior lp estimates 5. boundary and global estimates 6. the oblique derivative problem 7. the local maximum principle 8. the weak harnack inequality 9. boundary estimates notes exercises chapter viii. fixed point theorems and their applications 1. the schauder fixed point theorem 2. applications of the schauder theorem 3. a theorem of caristi and its applications notes exercises chapter ix. comparison and maximum principles 1. comparison principles 2. maximum estimates 3. comparison principles for divergence form operators 4. maximum estimates for divergence form operators notes exercises chapter x. boundary gradient estimates 1. the boundary gradient estimate in general domains 2. convex-increasing domains 3. the spatial distance function 4. curvature conditions 5. nonexistence results 6. the case of one space dimension 7. continuity estimates notes exercises chapter xi. global and local gradient bounds 1. global estimates for general equations 2. examples 3. local gradient bounds 4. the sobolev theorem of michael and simon 5. estimates for equations in divergence form 6. the case of one space dimension 7. a gradient bound for an intermediate situation notes exercises chapter xii. hslder gradient estimates and existence theorems 1. interior estimates for equations in divergence form 2. equations in one space dimension 3. interior estimates for equations in general form 4. boundary estimates 5. improved results for nondivergence equations 6. selected existence results notes exercises chapter xiii. the oblique derivative problem for quasilinear parabolic equations 1. maximum estimates 2. gradient estimates for the conormal problem 3. gradient estimates for uniformly parabolic problems in general form 4. the hslder gradient estimate for the conormal problem 5. nonlinear boundary conditions with linear equations 6. the hslder gradient estimate for quasilinear equations 7. existence theorems notes exercises chapter xiv. fully nonlinear equations i. introduction 1. comparison and maximum principles 2. simple uniformly parabolic equations 3. higher regularity of solutions 4. the cauchy-dirichlet problem 5. boundary second derivative estimates 6. the oblique derivative problem 7. the case of one space dimension notes exercises chapter xv. fully nonlinear equations ii. hessian equations 1. general results for hessian equations 2. estimates on solutions 3. existence of solutions 4. properties of symmetric polynomials 5. the parabolic analog of the monge-ampere equation notes exercises references index |
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