Cartesian Currents in the Calculus of Variations II: by Mariano Giaquinta

By Mariano Giaquinta

Non-scalar variational difficulties look in numerous fields. In geometry, for in­ stance, we stumble upon the fundamental difficulties of harmonic maps among Riemannian manifolds and of minimum immersions; comparable questions look in physics, for instance within the classical concept of a-models. Non linear elasticity is one other instance in continuum mechanics, whereas Oseen-Frank conception of liquid crystals and Ginzburg-Landau conception of superconductivity require to regard variational difficulties with the intention to version relatively complex phenomena. ordinarily one is drawn to discovering strength minimizing representatives in homology or homotopy sessions of maps, minimizers with prescribed topological singularities, topological fees, reliable deformations i. e. minimizers in sessions of diffeomorphisms or extremal fields. within the final or 3 many years there was becoming curiosity, wisdom, and knowing of the overall thought for this sort of difficulties, also known as geometric variational difficulties. as a result of loss of a regularity conception within the non scalar case, unlike the scalar one - or in different phrases to the prevalence of singularities in vector valued minimizers, frequently comparable with focus phenomena for the strength density - and thanks to the actual relevance of these singularities for the matter being thought of the query of singling out a vulnerable formula, or thoroughly figuring out the importance of assorted vulnerable formulations becames non trivial.

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Extra info for Cartesian Currents in the Calculus of Variations II: Variational Integrals

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Fix E: > O. We may also assume that :F(Uk, il) ~ A+E:. J ;=1 and the conclusion follows as E: -+ O. c. with respect to the strong convergence in Ll and convex. Equivalently its epigraph is strongly closed, and therefore by Theorem 3, weakly closed. c. of :F with respect to the weak convergence in Ll. 0 Remark that we in fact have proved the following abstract theorem to which Theorem 1 may be subsumed 14 1. Regular Variational Integrals Theorem 4. Let V be a Banach space and let :F : V --+ JR+ U {+oo} be convex.

C. in all variables and convex in ~ IM(G)I if and only if F(x, u, ~ II II· e) (ii) f(x, u, G) e e for any x, u. 3 The Parametric Extension of Regular Integrals Let f(x, u, G) : n x IRN x M(N, n) -+ R+ be a non-negative Lagrangian. c. envelop in the sense of Sec. 2. Suppose that f(x, u, G) is a regular integrand. In this case maps with equibounded energies (1) :F(u, n) := J f(x, u, Du) dx [} have graphs of equibounded masses. We now associate to :F a variational integral defined on currents with finite mass.

C. of :F, we infer f is trivial. 0) liminf f(Pk) . k ..... L(B) measurable set E C B such that 0 < JL(E) < JL(B). > 0 we can find another 12 1. Regular Variational Integrals XEk ->. L L XE k ->. L L t as measures3 . Let Q be a cube with Q cc il and let t E [0,1]. We can then find a sequence of measurable sets Ek C Q such that XEk ->. L). For p,q E aN we now set U(X) := tp + (1 - t)q if x E Q, and Uk(X) = u(x) = 0 if x fI. Q. L). c. L k-+oo n ->. L. L(Q) [tf(p) + (1 - t)f(q) - f(tp + (1 - t)q)] > O.

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