Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications

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Springer Science & Business Media, Jun 7, 2007 - Mathematics - 398 pages

This 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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Variational Equations
Multivalued Variational Equations
Obstacle Problem
Hemivariational Inequalities
Nonsmooth Variational Problems and

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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.

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