Conservation laws for conformally invariant variational problems

By T. Rivière. 2006

276 citations recorded by the source corpus.

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations,..., etc.) in divergence form. These divergence-free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE (see [Hel]). It enables us also to establish new results. In particular we solve a conjecture by E. Heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by S. Hildebrandt [Hil1] claiming that critical points of continuously differentiable elliptic conformally invariant Lagrangian in two dimensions are continuous.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/650c3232b7f569877b034d14aeae3a9863d641ae

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Citations in the corpus

Regularity of n/2-harmonic maps into spheres (2010, methodology): “Rivière [Riv07] was able to prove regularity for critical points of general conformally invariant functionals, thus solving a conjecture by S.”
Conservation Laws for Fourth Order Systems in Four Dimensions (2006, methodology): “In Rivière (2007) the second author found a procedure to rewrite second order elliptic systems of the form − u = · u (1.1) in divergence form.”
The blow up analysis of solutions of the elliptic sinh-Gordon equation (2007, result): “Good examples are Wente’s inequality [39] and Helein’s proof [12] of the regularity of weak harmonic maps. See also the recent work [ 30 ].”
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs (2005, methodology): “44 ([88]) Let u : D → N ⊂ R be a weakly harmonic map.”
Harmonic maps from degenerating Riemann surfaces (2008, background): “Away from these points, the convergence is strong.”
Three-term commutator estimates and the regularity of $\half$-harmonic maps into spheres (2011, background): “In general, W 1;2 solutions to (7) are not smooth in dimensions greater than 2; for a counterexample, see [Rivière 2007].”
Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps (2010, methodology): “1 In [10] the choice of P for systems of the form (3) was given by the geometrically relevant Coulomb Gauge satisfying”
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.”
Decay estimates for Rivière’s equation, with applications to regularity and compactness (2011, methodology): “MR2285745 (2008d:58010) [13] Tristan Rivière.”
Boundary regularity via Uhlenbeck–Rivière decomposition (2008, methodology): “We intend to apply the following nonlinear decomposition, which is due to Rivière and adapts Uhlenbeck’s technique in [Uhl82]: Lemma 2.1 (Rivière, 2007) (c.f. [Riv07], lemma A.3) There are εm ∈ (0, 1) and Cm > 0 such that for every a ∈ R2 and r > 0 and for every Ω ∈ L2(Br(a), som ⊗ R2)…”
Fractional div-curl quantities and applications to nonlocal geometric equations (2017, methodology): “Here, in the spirit of the celebrated work of Rivière [29], we develop the regularity theory of nonlocal antisymmetric systems of the form (4.1).”
Sharp commutator estimates via harmonic extensions (2016, background): “Just to name a few examples: the harmonic map equation [24, 33], the prescribed mean curvature equation [47, 23, 1, 33], or the conformally parametrized surface equation [31].”
Analysis aspects of Willmore surfaces (2008, background): “The results developed in [ Ri1 ] hinted that they could be extended to other conformally invariant equations such as the harmonic map equations into Lorentzian manifolds.”