NONUNIQUENESS IN VECTOR-VALUED CALCULUS OF VARIATIONS IN L-infinity AND SOME LINEAR ELLIPTIC SYSTEMS

作者：Katzourakis Nikos^*

来源：Communications on Pure and Applied Analysis, 2015, 14(1): 313-327.

DOI：10.3934/cpaa.2015.14.313

• Nonuniqueness
• vector-valued calculus of variations in L-infinity
• infinity-Laplacian
• Aronsson PDE
• quasiconformal maps

For a Hamiltonian H is an element of C-2(R-Nxn) and a map u: Omega subset of R-n -> R-N, we consider the supremal functional
E-infinity(u, Omega) := parallel to H(D-u)parallel to(L)infinity(Omega). (1)
The "Euler-Lagrange" PDE associated to (1) is the quasilinear system
A(infinity)u := (H-P circle times H-P + H[H-P]H-perpendicular to(PP))(Du) : D(2)u = 0. (2)
(1) and (2) are the fundamental objects of vector-valued Calculus of Variations in L-infinity and first arose in recent work of the author [28]. Herein we show that the Dirichlet problem for (2) admits for all n = N >= 2 infinitely-many smooth solutions on the punctured ball, in the case of H(P) = vertical bar P vertical bar(2) for the infinity-Laplacian and of H(P) = vertical bar P vertical bar(2)det(P+P)(-1/n) for optimised Quasiconformal maps. Nonuniqueness for the linear degenerate elliptic system A(x) : D(2)u = 0 follows as a corollary. Hence, the celebrated L-infinity scalar uniqueness theory of Jensen [24] has no counterpart when N >= 2. The key idea in the proofs is to recast (2) as a first order differential inclusion Du(x) is an element of K subset of R-nxn, x is an element of Omega.

• 出版日期2015-1