## 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. |

### From inside the book

Results 1-5 of 57

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**subset**KC Vo . This leads to the following well - known variational inequality for the operator Au = -4pu + g ( u ) : u Є K : ( Auf , -u ) ≥0 , for all K. ( 1.17 ) Introducing the indicator function Ik of the set K , we see that ( 1.17 ) ... Page 12

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**subset**M of X into itself . Then A has a fixed point provided M is bounded , closed , and convex . In finite - dimensional normed linear spaces , Theorem 2.2 reduces to Brouwer's fixed - point theorem . Corollary 2.3 ( Brouwer's Fixed ... Page 13

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**subset**Gr(A) = {(u, Au) : u ∈ D(A)} of the product space X × Y. The operator A is called closed (or graph-closed) iff Gr(A) is closed in X×Y, which means that for each sequence (un ) in D(A), it follows from un → u in X and Aun → v ... Page 20

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**subset**K of X is convex iff € u , vK implies tu + ( 1 - t ) v K for all 0 ≤ t≤ 1 . Theorem 2.34 . Let H be a Hilbert space with inner product ( , ) , and let K be a nonempty , closed , and convex**subset**of H. Then to each u € H ... Page 21

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**subset**M of a normed linear space X is called weakly sequentially closed if the limit of every weakly convergent sequence ( un ) C M belongs to M ; i.e. , ( un ) CM and un → u implies uЄ M. Simple examples show that , in general ...### Contents

10 | |

Variational Equations | 81 |

Multivalued Variational Equations | 143 |

Obstacle Problem | 247 |

Hemivariational Inequalities | 279 |

7 | 317 |

Nonsmooth Variational Problems and | 378 |

Index | 392 |

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### Common terms and phrases

assume assumption 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 dx dt dxdt element equivalent existence and comparison extremal solutions Fatou's lemma given greatest solution growth condition hemivariational inequalities hypotheses IBVP implies L²(N Lª(N Lemma lim sup linear Lipschitz boundary Lipschitz continuous Lipschitz function Lº(N Lº(Q locally Lipschitz function lower semicontinuous Math multivalued Nemytskij operator nonlinear nonsmooth norm obtain ordered interval p-Laplacian parabolic positive constant prove quasilinear quasilinear elliptic respectively right-hand side satisfies Sect sequence smallest solution solution set sub-supersolution subdifferential subset subsolution supersolution test function u₁ Un+1 variational inequalities variational-hemivariational inequality verify weak convergence

### 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.