Conservation Laws for Fourth Order Systems in Four Dimensions

By T. Lamm, T. Rivière. 2006

99 citations recorded by the source corpus.

Following an approach of the second author (Rivière, 2007) for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuity of weak solutions of this system. Moreover we use the conservation law to derive the existence of a unique global weak solution of the extrinsic biharmonic map flow in the energy space.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/82ff15a34300edf1d4ce67c2992e236a76057ba7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

Citations in the corpus

Regularity of n/2-harmonic maps into spheres (2010, result): “ts extend to other dimensions: In the biharmonic case, n= 4, regularity was proven in [CWY99] in the case of a sphere as a target manifold, and for more general targets in [Str03], [Wan04], [Sch08] , [LR08]. For even n≥ 6 similar regularity results hold, and we refer to [GS09], [GSZG09]. Regarding o”
Regularity of polyharmonic maps in the critical dimension (2009, background): “Lamm and Rivière [12] have given a different proof of continuity of weakly biharmonic maps in four dimensions using a completely different point of view based on conservation laws.”
Partial regularity for biharmonic maps, revisited (2008, methodology): “Acknowledgement It is a pleasure to thank Tristan Rivière for interesting discussions in an early stage of this work.”