Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and ApplicationsThis monograph focuses primarily on nonsmooth variational problems that arise from boundary value problems with nonsmooth data and/or nonsmooth constraints, such as multivalued elliptic problems, variational inequalities, hemivariational inequalities, and their corresponding evolution problems. It provides a systematic and unified exposition of comparison principles based on a suitably extended sub-supersolution method. |
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Results 1-5 of 64
Page vii
... 2.5.1 Clarke's Generalized Gradient 2.5.2 Some Calculus 2.5.3 Critical Point Theory 55 59 62 63 63 68 73 3 4 Variational Equations . 3.1 Semilinear Elliptic Equations 81. 2.5.4 Linking Theorem 77 น 0 on ΘΩ and P in Vo ,
... 2.5.1 Clarke's Generalized Gradient 2.5.2 Some Calculus 2.5.3 Critical Point Theory 55 59 62 63 63 68 73 3 4 Variational Equations . 3.1 Semilinear Elliptic Equations 81. 2.5.4 Linking Theorem 77 น 0 on ΘΩ and P in Vo ,
Page 1
... ΘΩ , ( 1.1 ) CRN is a bounded domain with smooth boundary ΘΩ , f : ∩ → R is some given smooth function , and assume the existence of a classic subsolution u and supersolution ū of ( 1.1 ) , i.e. , u , ū∈ C2 ( Ω ) ∩C ( Ω ) ...
... ΘΩ , ( 1.1 ) CRN is a bounded domain with smooth boundary ΘΩ , f : ∩ → R is some given smooth function , and assume the existence of a classic subsolution u and supersolution ū of ( 1.1 ) , i.e. , u , ū∈ C2 ( Ω ) ∩C ( Ω ) ...
Page 2
... ΘΩ . ( 1.5 ) However , if u and ū are appropriate ( weak ) sub- and supersolutions of ( 1.5 ) satisfying , in addition , u < ū , then ( weak ) solutions of ( 1.5 ) ( not necessarily unique ) exist within the interval [ u , ū ] formed by ...
... ΘΩ . ( 1.5 ) However , if u and ū are appropriate ( weak ) sub- and supersolutions of ( 1.5 ) satisfying , in addition , u < ū , then ( weak ) solutions of ( 1.5 ) ( not necessarily unique ) exist within the interval [ u , ū ] formed by ...
Page 3
... ΘΩ and p - 2 Vu √ √ up - 2 Ju√ pdx + S Ω 4 Ω g ( u ) dx ≤ ( f , 4 ) for all y ∈ Von L ( Ω ) . ( 1.12 ) Comparison principles for solutions of nonlinear elliptic and parabolic varia- tional equations including the special case ( 1.7 ) ...
... ΘΩ and p - 2 Vu √ √ up - 2 Ju√ pdx + S Ω 4 Ω g ( u ) dx ≤ ( f , 4 ) for all y ∈ Von L ( Ω ) . ( 1.12 ) Comparison principles for solutions of nonlinear elliptic and parabolic varia- tional equations including the special case ( 1.7 ) ...
Page 4
... ΘΩ such that F ( u ) ≤ 0 and F ( ū ) ≥ 0 . Then the existence and comparison result as well as the characterization of the solution set for ( 1.6 ) given above hold . Note that in view of ( i * ) , the operator F : Vo → Vo * is ...
... ΘΩ such that F ( u ) ≤ 0 and F ( ū ) ≥ 0 . Then the existence and comparison result as well as the characterization of the solution set for ( 1.6 ) given above hold . Note that in view of ( i * ) , the operator F : Vo → Vo * is ...
Contents
| 1 | |
| 11 | |
Variational Equations | 81 |
Multivalued Variational Equations | 143 |
Variational Inequalities | 211 |
Hemivariational Inequalities | 279 |
VariationalHemivariational Inequalities | 319 |
Nonsmooth Variational Problems and | 378 |
Index | 392 |
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Common terms and phrases
assume boundary value problems c₁ Carathéodory Carathéodory function Chap Clarke's generalized gradient coercive compact embedding comparison principle completes the proof convergence convex convex function convex set Corollary critical point defined Definition denote differential dj(x dual space dx dt dxdt equations equivalent existence and comparison extremal solutions Fatou's lemma function f given greatest solution growth condition hemivariational inequalities hypotheses IBVP implies Lemma linear Lipschitz boundary Lipschitz continuous Lipschitz function Lº(Q locally Lipschitz function lower semicontinuous LP(Q multivalued Nemytskij operator nonlinear nonsmooth norm obtain ordered interval parabolic prove pseudomonotone quasilinear satisfies Sect sequence smallest solution solution set sub-supersolution subdifferential subset subsolution supersolution test function u₁ Un+1 variational inequalities variational-hemivariational inequality weak convergence γυ θε ΘΩ Ω Ω ик
Popular passages
Page 78 - Let 5 be a nonempty subset of the Banach space X and let Q be a compact topological submanifold of X with nonempty boundary dQ (in the sense of manifolds with boundary). We say that S and Q link if the next properties hold S n dQ = 0 and whenever / 6 F, where The theorem below is our main result of this Section.
Page 13 - If X is a normed linear space and Y is a Banach space, then B(X, Y) is a Banach space.
Page 182 - The main result of this section is given by the following theorem. Theorem 4.3.
Page 13 - Theorem 3.B (The uniform boundedness theorem). Let F be a nonempty set of continuous maps F: X -» Y, where X is a Banach space over К and Y is a normed space over K.
Page 17 - Let X be a Banach space. Then X is reflexive if and only if X* is reflexive.
Page 15 - The set of all linear continuous functionals on X is called the adjoint or conjugate space of X and is denoted X*.
Page 12 - Then the following two conditions are equivalent: (i) A is continuous, (ii) G = 0, and g is bounded in D.
Page 18 - Let X be a normed linear space. Then, the closed unit ball in X is compact if and only if X is finite-dimensional.
Page 14 - Theorem). Let X and Y be Banach spaces and A : X — » Y be a linear continuous operator.
Page 14 - X -> Y, where X is a Banach space and Y is a normed linear space.