A relaxation of the intrinsic biharmonic energy

Abstract

The tension field τ (u) of a map u from a domain ⊂ R m into a manifold N is the negative L 2 -gradient of the Dirichlet energy. In this paper we study the intrinsic biharmonic energy functional T (u) = |τ (u)| 2 . In order to overcome the lack of coercivity of T , we extend it to a larger space. We construct minimizers of the extended functional via the direct method and we study the relation between these minimizers and critical points of T . Our results are restricted to dimensions m ≤ 4.

Existence of Equivariant Biharmonic Maps

“…The following is a consequence of the results of the first author [8]. Although the proofs in that paper are carried out for the special case of a flat domain only, it is not difficult to see that they can be generalised.…”

“…In order to overcome this problem, we use tools from geometric measure theory, encoding the concentrated energy in a measure on M . We follow an approach going back to Lin [11] and to Ambrosio and Soner [1] and developed further for the problem of biharmonic maps by the authors [14,8].…”

“…in terms of E 2 with a Pohozaev type argument (used in the work of the first author [8] and extended by the second author [17]). In the absence of such a condition, the main ideas from this paper will still work for functionals such as E 2 + aE 1 for a > 0, but we leave it to the reader to work out the details.…”

A construction of biharmonic maps into homogeneous spaces

“…An approach to minimizing the functional through a relaxation has been explored by Hornung [3]. He proved existence of minimizers of a relaxed functional.…”

“…While a direct minimization of the functional E 2 appears to be difficult in general, it was shown by Hornung [3] that after a modification of the functional, minimizers can be found with the direct method if m = 3 or 4. The functional studied by Hornung can be regarded as a relaxation of E 2 .…”