2011 # 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.

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“…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.…”

confidence: 87%

“…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.…”

confidence: 87%

“…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].…”

confidence: 99%

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

confidence: 99%

“…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 .…”

confidence: 99%