Download Unbounded Functionals in the Calculus of Variations: by Luciano Carbone, Riccardo De Arcangelis PDF

By Luciano Carbone, Riccardo De Arcangelis

During the last few many years, study in elastic-plastic torsion idea, electrostatic screening, and rubber-like nonlinear elastomers has pointed find out how to a few fascinating new periods of minimal difficulties for power functionals of the calculus of adaptations. This advanced-level monograph addresses those matters via constructing the framework of a normal thought of indispensable illustration, rest, and homogenization for unbounded functionals.

The first a part of the booklet builds the root for the overall concept with thoughts and instruments from convex research, degree idea, and the speculation of variational convergences. The authors then introduce a few functionality areas and discover a few decrease semicontinuity and minimization difficulties for power functionals. subsequent, they survey a few particular features the speculation of normal functionals.

The moment 1/2 the ebook conscientiously develops a conception of unbounded, translation invariant functionals that ends up in effects deeper than these already recognized, together with exact extension houses, illustration as integrals of the calculus of diversifications, rest idea, and homogenization methods. during this learn, a few new phenomena are mentioned. The authors' strategy is unified and stylish, the textual content good written, and the consequences fascinating and helpful, not only in a variety of fields of arithmetic, but additionally in a number of utilized arithmetic, physics, and fabric technology disciplines.

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Additional resources for Unbounded Functionals in the Calculus of Variations: Representation, Relaxation, and Homogenization

Example text

5) f ∗∗ (z) ≤ co(sc− f )(z) for every z ∈ Rn . 1) follow. 2. 3), the following examples prove that, in general, for a given function f , co(sc− f ) and f ∗∗ may be different. 2. Let n = 2, and let f be defined by z2 − z1 ez2 0 +∞ 2 f : (z1 , z2) ∈ R → if z2 ≥ 0 and 0 < z1 ≤ z2 e−z2 if z2 ≥ 0 and z2e−z2 < z1 otherwise. Then domf is convex, f is upper semicontinuous in R2 and locally Lipschitz in domf . Moreover, it is clear that f ∗∗ (z1 , z2 ) = 0 if z1 ≥ 0 and z2 ≥ 0 for every (z1 , z2 ) ∈ R2, +∞ otherwise whilst it is easy to see that co(sc− f)(z1 , z2 ) = z2 0 +∞ if z2 ≥ 0 and z1 = 0 if z2 ≥ 0 and z1 > 0 for every (z1 , z2 ) ∈ R2 .

E. the function sc−f : z ∈ Rn → sup{φ(z) : φ: Rn → [−∞, +∞] lower semicontinuous, φ ≤ f in Rn }. It is clear that sc− f turns out to be lower semicontinuous, and that sc− f(z) ≤ f (z) for every z ∈ Rn . 8) y→z from which it follows that domf ⊆ dom(sc− f ) ⊆ domf . 9) Given S ⊆ Rn , it results that sc− IS = IS . 7. Let f : Rn → [−∞, +∞]. Then sc−f (z) = inf{λ ∈ R : (z, λ) ∈ epif } for every z ∈ Rn . Proof. For the sake of simplicity, let us set j: z ∈ Rn → inf{λ ∈ R : (z, λ) ∈ epif }. 9, epif ⊆ epiφ.

For example this happens for the linearity property, as it can be easily checked by means of simple examples. To overcome such difficulties, the notion of measurability of a function is introduced. It provides a quite natural and general tool, that allows the development of a complete and flexible theory of integration provided it is concerned with measurable functions. Because of this, we will deal mainly with integrals of measurable functions, even if occasionally the integral of non-necessarily measurable ones might be taken into account.

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