Abstract
We prove a new regularity result for systems of nonlinear elliptic equations with quadratic Jacobian type nonlinearity in dimension two. Our proof is based on an adaptation of John Lewis’ method which has not been used for such systems so far.
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Acerbi E., Fusco N.: An approximation lemma for W 1,p functions. In: Ball, J.M.(eds) Material Instabilities in Continuum Mechanics (Edinburgh, 1985–1986), pp. 1–5. Oxford Science Publishers, Oxford University Press, New York (1988)
Bethuel F.: Un résultat de régularité pour les solutions de l’équation de surfaces à courbure moyenne prescrite. C. R. Acad. Sci. Paris Sér. I Math. 314(13), 1003–1007 (1992)
Chanillo S.: Sobolev inequalities involving divergence free maps. Comm. Partial Differ. Equ. 16, 1969–1994 (1991)
Chanillo S., Li Y.-Y.: Continuity of solutions of uniformly elliptic equations in R 2. Manuscripta Math. 77, 415–433 (1992)
Coifman R., Lions P.L., Meyer Y., Semmes S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286 (1993)
Faraco D., Koskela P., Zhong X.: Mappings of finite distortion: the degree of regularity. Adv. Math. 190, 300–318 (2005)
Faraco D., Zhong X.: A short proof of the self-improving regularity of quasiregular mappings. Proc. Am. Math. Soc. 134, 187–192 (2006)
Frehse J.: A discontinuous solution of a mildly nonlinear elliptic system. Math. Z. 134, 229–230 (1973)
Hajłasz P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5, 403–415 (1996)
Hélein F.: Regularité des applications faiblement harmoniques entre une surface et une variété riemannienne. C. R. Acad. Sci. Paris 312, 591–596 (1991)
Hélein, F.: Phenomena of compensation and estimates for partial differential equations. In: Proceedings of the ICM’1998. Documenta Math. Extra vol. III, pp. 21–30 (1998)
Hélein F.: Harmonic maps, conservation laws and moving frames, 2nd edn. Cambridge University Press, Cambridge (2002)
Lewis J.L.: On very weak solutions of certain elliptic systems. Comm. Partial Differ. Equ. 18(9–10), 1515–1537 (1993)
Mikkonen, P.: On the Wolff potential and quasilinear elliptic equations involving measures. Ann. Acad. Sci. Fenn. Math. Diss. 104, p. 71 (1996)
Moscariello G.: On weak minima of certain integral functionals. Ann. Polon. Math. 69, 37–48 (1998)
Rivière T.: Conservation laws for conformally invariant variational problems. Invent. Math. 168, 1–22 (2007)
Strzelecki P.: A new proof of regularity of weak solutions of the H-surface equation. Calc. Var. PDE 16, 227–242 (2003)
Zatorska-Goldstein A.: Very weak solutions of nonlinear subelliptic equations. Ann. Acad. Sci. Fenn. Math. 30, 407–436 (2005)
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Parts of this work have been done while the second and the third author had been enjoying the hospitality of the Department of Mathematics of the University of Pittsburgh. P.H. was supported by NSF grant DMS-0500966. P.S. was partially supported by the MNiSzW grant no 1 PO 3A 005 29. X.Z. was supported by the Academy of Finland, project 207288.
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Hajłasz, P., Strzelecki, P. & Zhong, X. A new approach to interior regularity of elliptic systems with quadratic Jacobian structure in dimension two. manuscripta math. 127, 121–135 (2008). https://doi.org/10.1007/s00229-008-0199-2
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DOI: https://doi.org/10.1007/s00229-008-0199-2