Decay estimates for Rivière’s equation, with applications to regularity and compactness

Sharp, Ben; Topping, Peter

doi:10.1090/S0002-9947-2012-05671-6

Decay estimates for Rivière’s equation, with applications to regularity and compactness
HTML articles powered by AMS MathViewer

by Ben Sharp and Peter Topping PDF

Trans. Amer. Math. Soc. 365 (2013), 2317-2339

Abstract:

We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in $\mathbb {R}^2$ introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmonic and almost-harmonic maps from surfaces.

References

David R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), no. 4, 765–778. MR 458158
R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), no. 3, 247–286 (English, with English and French summaries). MR 1225511
Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1034481, DOI 10.1090/cbms/074
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
Yuxin Ge, Estimations of the best constant involving the $L^2$ norm in Wente’s inequality and compact $H$-surfaces in Euclidean space, ESAIM Control Optim. Calc. Var. 3 (1998), 263–300. MR 1634837, DOI 10.1051/cocv:1998110
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
Frédéric Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 8, 591–596 (French, with English summary). MR 1101039
Frédéric Hélein, Harmonic maps, conservation laws and moving frames, 2nd ed., Cambridge Tracts in Mathematics, vol. 150, Cambridge University Press, Cambridge, 2002. Translated from the 1996 French original; With a foreword by James Eells. MR 1913803, DOI 10.1017/CBO9780511543036
Richard A. Hunt, On $L(p,\,q)$ spaces, Enseign. Math. (2) 12 (1966), 249–276. MR 223874
J. Li and X. Zhu. Small energy compactness for approximate harmonic mappings. Preprint, 2009.
Tristan Rivière, Conservation laws for conformally invariant variational problems, Invent. Math. 168 (2007), no. 1, 1–22. MR 2285745, DOI 10.1007/s00222-006-0023-0
Tristan Rivière. Integrability by compensation in the analysis of conformally invariant problems. Minicourse Notes, 2009.
Melanie Rupflin, An improved uniqueness result for the harmonic map flow in two dimensions, Calc. Var. Partial Differential Equations 33 (2008), no. 3, 329–341. MR 2429534, DOI 10.1007/s00526-008-0164-7
Stephen Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations 19 (1994), no. 1-2, 277–319. MR 1257006, DOI 10.1080/03605309408821017
Leon Simon, Theorems on regularity and singularity of energy minimizing maps, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996. Based on lecture notes by Norbert Hungerbühler. MR 1399562, DOI 10.1007/978-3-0348-9193-6
E. M. Stein, Note on the class $L$ $\textrm {log}$ $L$, Studia Math. 32 (1969), 305–310. MR 247534, DOI 10.4064/sm-32-3-305-310
William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3