Quasilinear degenerate and nonuniformly elliptic and parabolic equations of second order

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Contents......Page 4
Abstract......Page 10
Preface......Page 14
Basic Notation......Page 16
PART I. QUASILINEAR, NONUNIFORMLY ELLIPTIC AND PARABOLIC EQUATIONS OF NONDIVERGENCE TYPE......Page 20
1. The basic characteristics of a quasilinear elliptic equation......Page 26
2. A conditional existence theorem......Page 28
3. Some facts about the barrier technique......Page 29
4. Estimates of IVul on the boundary ail by means of global barriers......Page 31
5. Estimates of jVul on the boundary by means of local barriers.......Page 35
6. Estimates of maxnIVul for equations with structure described in terms of the majorant ?......Page 40
7. The estimate of maxn I VuI for equations with structure described in terms of the majorant E2......Page 44
8. The estimate of maxolVul for a special class of equations......Page 47
9. The existence theorem for a solution of the Dirichlet problem in the case of an arbitrary domain 11 with a sufficiently smooth boundary......Page 51
10. Existence theorem for a solution of the Dirichlet problem in the case of a strictly convex domain fl......Page 53
1. A conditional existence theorem......Page 56
2. Estimates of IVul on r......Page 59
3. Estimates of maxQIVul......Page 62
4. Existence theorems for a classical solution of the first boundary value problem......Page 67
5. Nonexistence theorems......Page 69
1. Estimates of IVu(xo)I in terms of maxK,(xo)IuI......Page 73
2. An estimate of jVu(xo)I in terms of maxK,(xo)u (minK,(so)u). Hlarnack's inequality......Page 80
3. Two-sided Liouville theorems......Page 84
4. One-sided Liouville theorems......Page 87
PART II. QUASILINEAR (A, b)-ELLIPTIC EQUATIONS......Page 90
1. Generalized A-derivatives......Page 98
2. Generalized limit values of a function on the boundary of a domain......Page 102
3. The regular and singular parts of the boundary 31......Page 108
4. Some imbedding theorems......Page 111
5. Some imbedding theorems for functions depending on time......Page 115
6. General operator equations in a Banach space......Page 119
7. A special space of functions of scalar argument with values in a Banach space......Page 125
1. The structure of the equations and the classical formulation of the general boundary value problem......Page 131
2. The basic function spaces and the operators connected with the general boundary value problem for an (A, b, m, m)-elliptic equation......Page 141
3. A generalized formulation of the general boundary value problem for (A, b, m, m)-elliptic equations......Page 150
4. Conditions for existence and uniqueness of a generalized solution of the general boundary value problem......Page 152
5. Linear (A, b)-elliptic equations......Page 159
1. Nondivergence (A,b)-elliptic equations......Page 162
2. Existence and uniqueness of regular generalized solutions of the first boundary value problem......Page 165
3. The existence of regular generalized solutions of the first boundary value problem which are bounded in 11 together with their partial derivatives of first order......Page 176
PART III. (A, 0)-ELLIPTIC AND (A, 0)-PARABOLIC EQUATIONS......Page 186
1. The general boundary value problem for (A, 0, m, m)-elliptic equations......Page 190
2. (A, 0)-elliptic equations with weak degeneracy......Page 192
3. Existence and uniqueness of A -regular generalized solutions of the first boundary value problem for (A, 0)-elliptic equations......Page 204
1. The basic function spaces connected with the general boundary value problem for (A, 0, m, m)-parabolic equations......Page 216
2. The general boundary value problem for (A, 0, m, m)-parabolic equations......Page 229
3. (A, 0)-parabolic equations with weak degeneracy......Page 235
4. Linear A-parabolic equations with weak degeneracy......Page 251
PART IV. ON REGULARITY OF GENERALIZED SOLUTIONS OF QUASILINEAR DEGENERATE PARABOLIC EQUATIONS.......Page 256
1. The structure of the equations and their generalized solutions.......Page 258
2. On regularity of generalized solutions in the variable t......Page 263
3. The energy inequality......Page 266
4. Functions of generalized solutions......Page 268
5. Local estimates in LA PO......Page 275
6. Global estimates in LP,P?......Page 281
7. Exponential summability of generalized solutions......Page 283
8. Local boundedness of generalized solutions......Page 285
9. Boundedness of generalized solutions of the boundary value problem......Page 288
10. The maximum principle......Page 290
Bibliography......Page 294