Preface ........................................................ xi
Notational conventions ......................................... xv
1. Preliminary general material ................................. 1
1.1. Functional analysis ..................................... 1
1.1.1. Normed spaces, Banach spaces, locally convex
spaces ........................................... 1
1.1.2. Functions and mappings on Banach spaces, dual
spaces ........................................... 3
1.1.3. Convex sets ...................................... 6
1.1.4. Compactness ...................................... 7
1.1.5. Fixed-point theorems ............................. 8
1.2. Function spaces ......................................... 8
1.2.1. Continuous and smooth functions .................. 9
1.2.2. Lebesgue integrable functions ................... 10
1.2.3. Sobolev spaces .................................. 14
1.3. Nemytskii mappings ..................................... 19
1.4. Green formula and some inequalities .................... 20
1.5. Bochner spaces ......................................... 22
1.6. Some ordinary differential equations ................... 25
I. STEADY-STATE PROBLEMS ....................................... 27
2. Pseudomonotone or weakly continuous mappings ................ 29
2.1. Abstract theory, basic definitions, Galerkin method .... 29
2.2. Some facts about pseudomonotone mappings ............... 33
2.3. Equations with monotone mappings ....................... 35
2.4. Quasilinear elliptic equations ......................... 40
2.4.1. Boundary-value problems for 2nd-order
equations ....................................... 41
2.4.2. Weak formulation ................................ 42
2.4.3. Pseudomonotonicity, coercivity, existence of
solutions ....................................... 46
2.4.4. Higher-order equations .......................... 53
2.5. Weakly continuous mappings, semilinear equations ....... 56
2.6. Examples and exercises ................................. 58
2.6.1. General tools ................................... 59
2.6.2. Semilinear heat equation of type —div
(A(x, u) u) = g ................................. 62
2.6.3. Quasilinear equations of type — div(|u|p-2uu)
+c(u, u)=g ..................................... 69
2.7. Excursion to regularity for semilinear equations ....... 79
2.8. Bibliographical remarks ................................ 86
3. Accretive mappings .......................................... 89
3.1. Abstract theory ........................................ 89
3.2. Applications to boundary-value problems ................ 93
3.2.1. Duality mappings in Lebesgue and Sobolev
spaces .......................................... 93
3.2.2. Accretivity of monotone quasilinear mappings .... 95
3.2.3. Accretivity of heat equation .................... 99
3.2.4. Accretivity of some other boundary-value
problems ....................................... 102
3.2.5. Excursion to equations with measures in
right-hand sides ............................... 103
3.3. Exercises ............................................. 106
3.4. Bibliographical remarks ............................... 107
4. Potential problems: smooth case ............................ 109
4.1. Abstract theory ....................................... 109
4.2. Application to boundary-value problems ................ 114
4.3. Examples and exercises ................................ 120
4.4. Bibliographical remarks ............................... 124
5. Nonsmooth problems; variational inequalities ............... 125
5.1. Abstract inclusions with a potential .................. 125
5.2. Application to elliptic variational inequalities ...... 129
5.3. Some abstract nonpotential inclusions ................. 135
5.4. Excursion to quasivariational inequalities ............ 144
5.5. Exercises ............................................. 147
5.6. Some applications to free-boundary problems ........... 152
5.6.1. Porous media flow: a potential variational
inequality ..................................... 152
5.6.2. Continuous casting: a nonpotential
variational inequality ......................... 156
5.7. Bibliographical remarks ............................... 159
6. Systems of equations: particular examples .................. 161
6.1. Minimization-type variational method: polyconvex
functionals ........................................... 161
6.2. Buoyancy-driven viscous flow .......................... 168
6.3. Reaction-diffusion system ............................. 172
6.4. Thermistor ............................................ 175
6.5. Semiconductors ........................................ 178
II EVOLUTION PROBLEMS ......................................... 185
7. Special auxiliary tools .................................... 187
7.1. Sobolev-Bochner space W1,p,g(I,V1,V2) ................... 187
7.2. Gelfand triple, embedding W1,p,p'(I,V1,V*) ⊂ C(I;H) ..... 190
7.3. Aubin-Lions lemma ..................................... 193
8. Evolution by pseudomonotone or weakly continuous
mappings ................................................... 199
8.1. Abstract initial-value problems ....................... 199
8.2. Rothe method .......................................... 201
8.3. Further estimates ..................................... 215
8.4. Galerkin method ....................................... 221
8.5. Uniqueness and continuous dependence on data .......... 228
8.6. Application to quasilinear parabolic equations ........ 232
8.7. Application to semilinear parabolic equations ......... 239
8.8. Examples and exercises ................................ 242
8.8.1. General tools .................................. 242
8.8.2. Parabolic equation of type  u—div(|u|p-2
u)+c(u)=g ..................................... 244
8.8.3. Semilinear heat equation C(u)u -
div(k(u)u) = g ................................ 252
8.8.4. Navier-Stokes equation u+(u · )u - u +
π = g ......................................... 255
8.8.5. Some more exercises ............................ 257
8.9. Global monotonicity approach, periodic problems ....... 262
8.10.Problems with a convex potential: direct method ....... 267
8.11.Bibliographical remarks ............................... 272
9. Evolution governed by accretive mappings ................... 275
9.1. Strong solutions ...................................... 275
9.2. Integral solutions .................................... 280
9.3. Excursion to nonlinear semigroups ..................... 286
9.4. Applications to initial-boundary-value problems ....... 291
9.5. Applications to some systems .......................... 295
9.6. Bibliographical remarks ............................... 302
10.Evolution governed by certain set-valued mappings .......... 305
10.1.Abstract problems: strong solutions ................... 305
10.2.Abstract problems: weak solutions ..................... 309
10.3.Examples of unilateral parabolic problems ............. 313
10.4.Bibliographical remarks ............................... 318
11.Doubly-nonlinear problems .................................. 321
11.1 Inclusions of the type ∂Ψ(u) + ∂Φ(u) ∈  ............. 321
11.1.1.Potential Ψ valued in  ∪ +∞ ................... 321
11.1.2.Potential Φ valued in ∪ +∞ ................... 327
11.1.3.Uniqueness and continuous dependence on data ... 332
11.2.Inclusions of the type E(u) + (u) ∪ ............. 334
11.2.1. The case E:∂Φ ................................. 335
11.2.2. The case E nonpotential ....................... 339
11.2.3. Uniqueness .................................... 341
11.3. 2nd-order equations .................................. 342
11.4. Exercises ............................................ 351
11.5. Bibliographical remarks .............................. 355
12 Systems of equations: particular examples .................. 357
12.1. Thermo-visco-elasticity .............................. 357
12.2. Buoyancy-driven viscous flow ......................... 361
12.3. Predator-prey system ................................. 365
12.4. Semiconductors ....................................... 368
12.5. Phase-field model .................................... 372
12.6. Navier-Stokes-Nernst-Planck-Poisson-type system ...... 376
References .................................................... 383
Index ......................................................... 399
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