By Siegfried Carl, Vy K. Le, Dumitru Motreanu
This monograph focuses totally on nonsmooth variational difficulties that come up from boundary worth issues of nonsmooth information and/or nonsmooth constraints, equivalent to multivalued elliptic difficulties, variational inequalities, hemivariational inequalities, and their corresponding evolution difficulties. It offers a scientific and unified exposition of comparability ideas in keeping with a definitely prolonged sub-supersolution method.
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Additional resources for Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications
Example text
According to Sect. 4, 1/p u V = Ω |∇u|p dx defines an equivalent norm in V . From the inequalities for the function ξ → |ξ|p−2 ξ, we see that the operator −Δp : W01,p (Ω) → (W01,p (Ω))∗ has the mapping properties given in the following lemma. 111. Let V be a closed subspace of W 1,p (Ω) such that W01,p (Ω) ⊂ V ⊂ W 1,p (Ω). Then one has: (i) −Δp : V → V ∗ is continuous, bounded, pseudomonotone, and has the (S+ )-property. (ii) −Δp : W01,p (Ω) → (W01,p (Ω))∗ is (a) strictly monotone if 1 < p < ∞. (b) strongly monotone if p = 2 (Laplacian).
Let X and Y be Banach spaces, and let f : U ⊂ X → Y be a map whose domain D(f ) = U is an open subset of X. The directional derivative of f at u ∈ U in the direction h ∈ X is given by f (u + th) − f (u) δf (u; h) = lim t→0 t provided this limit exists. If δf (u; h) exists for every h ∈ X, and if the mapping DG f (u) : X → Y defined by DG f (u)h = δf (u; h) is linear and continuous, then we say that f is Gˆ ateaux-differentiable at u, and we call DG f (u) the Gˆ ateaux derivative of f at u. 44 (Fr´ echet Derivative).
The following sum rule for the subdifferential is due to Moreau and Rockafellar. 57 (Sum Rule). Let X be a Banach space, and let φ1 , φ2 : X → R ∪ {+∞} be convex functionals. If there is a point u0 ∈ dom(φ1 ) ∩ dom(φ2 ) at which φ1 is continuous, then the following holds: ∂(φ1 + φ2 )(u) = ∂φ1 (u) + ∂φ2 (u) for all u ∈ X. 58. Let f : R → R be a nondecreasing function with its one-sided limits f and f¯. Define φ : R → R by x φ(x) = x f (s) ds = x0 f¯(s) ds. , dom(φ) = R, and thus φ is even locally Lipschitz.