By Mariano Giaquinta, Guiseppe Modica, Jiri Soucek

ISBN-10: 354064010X

ISBN-13: 9783540640103

This monograph (in volumes) bargains with non scalar variational difficulties coming up in geometry, as harmonic mappings among Riemannian manifolds and minimum graphs, and in physics, as reliable equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and available to non experts. themes are taken care of so far as attainable in an easy approach, illustrating effects with basic examples; in precept, chapters or even sections are readable independently of the overall context, in order that elements might be simply used for graduate classes. Open questions are usually pointed out and the ultimate component of each one bankruptcy discusses references to the literature and infrequently supplementary effects. ultimately, a close desk of Contents and an in depth Index are of aid to refer to this monograph

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Extra resources for Cartesian currents in the calculus of variations

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Thus Ln is the Legendre harmonic of degree n and dimension q. From Lemma 2, §7 we get where ~(q-l) is any point of Sq-2 and t E [-1,1]. Evaluation ofthe integral in standard coordinates now yields Laplace's first integral. 42 2. The Specific Theories Theorem 1 will be referred to as the Laplace representation of the Legendre polynomials. To demonstrate its usefulness we derive another recurrence formula for the Pn(q; t) as well as some estimates. 8) Next, we prove inequalities for Pn(q; t). 8) this estimate remains valid for q = 2.

5) is zero and we have a nontrivial set of constants Ct, C 2, . 4) vanishes in all §6 Homogeneous Harmonics 33 al, ... ,aN. Therefore a system is singular if there is an element of Yn(q) that is zero in all N(q, n) points. Conversely, we have Lemma 1: Suppose al, ... , aN is a regular system of degree n; then every Y n E Yn(q), with Yn(aj) = O,j = 1, ... , N, vanishes identically. Of course not every system of N points is regular, although singularity is somehow the exception. The functions Pn(q; t; ·aj),j = 1, ...

The Specific Theories because the sum over j yields N~~~~;I) Pm{q - 1; ~(q-l) addition theorem in (q - 1) dimensions. We set u : = ~(q-l) . 13) to Lemma 2: For t, s, u E [-1,1] and q to by the 3 we have ~ ~) N(q,n)p (. Isq-11 ~ . T'/(q-l») nq,st+uVl-s~vl-t- = ISql_21 N(q - 1, m)A~(q, t)A~(q, s)Pn(q - 1; u) We multiply by Pdq -1; u)(l- U2)~ on both sides and integrate over [-1,1]. 17) We have now gained a rather closed and explicit theory of complete systems of functions on spheres, and it is an interesting question if this knowledge can be used in other situations.

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