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Interpolation inequalities for partial regularity

  • Christoph Hamburger EMAIL logo


We propose two new direct methods for proving partial regularity of solutions of nonlinear elliptic or parabolic systems. The methods are based on two similar interpolation inequalities for solutions of linear systems with constant coefficient. The first results from an interpolation inequality of L p norms in combination with an L p estimate with low exponent p > 1 . For the second, we provide a functional-analytic proof, that also sheds light upon the A-harmonic approximation lemma of Duzaar and Steffen. Both methods use a Caccioppoli inequality and avoid higher integrability. We illustrate the methods in detail for the case of a quasilinear elliptic system.

Communicated by Giuseppe Mingione


[1] J.-P. Aubin, Un théorème de compacité, C. R. Acad. Sci., Paris 256 (1963), 5042–5044. Search in Google Scholar

[2] F. Duzaar and J. F. Grotowski, Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation, Manuscr. Math. 103 (2000), 267–298. 10.1007/s002290070007Search in Google Scholar

[3] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, 1983. 10.1515/9781400881628Search in Google Scholar

[4] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1998. Search in Google Scholar

[5] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003. 10.1142/5002Search in Google Scholar

[6] C. Hamburger, Partial regularity of solutions of nonlinear quasimonotone systems, Hokkaido Math. J. 32 (2003), 291–316. 10.14492/hokmj/1350657525Search in Google Scholar

[7] C. Hamburger, Partial regularity of solutions of nonlinear quasimonotone systems via a lower L p estimate, Q. J. Math., to appear. Search in Google Scholar

[8] C. Hamburger, The heat flow in nonlinear Hodge theory under general growth, J. Differ. Equations, to appear. Search in Google Scholar

[9] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Gauthier-Villars, Paris, 1969. Search in Google Scholar

[10] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1991. Search in Google Scholar

[11] J. Simon, Compact sets in the space L p ( 0 , T ; B ) , Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. 10.1007/BF01762360Search in Google Scholar

Received: 2021-05-10
Revised: 2021-08-08
Accepted: 2022-01-28
Published Online: 2022-08-30
Published in Print: 2023-07-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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