Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



$ n$-Laplacian in $ \mathcal{H}\sp 1\sb \mathrm{loc}$ does not lead to regularity

Author: Nikan B. Firoozye
Journal: Proc. Amer. Math. Soc. 123 (1995), 3357-3360
MSC: Primary 35J05; Secondary 42B30
MathSciNet review: 1277110
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that in two space dimensions, if a solution to Poisson's equation has right-hand side in $ \mathcal{H}_{{\text{loc}}}^1$, then this solution is actually continuous. The corresponding result for n-Laplacians is shown to be false for $ n \geq 3$; we construct two examples with right-hand sides in $ \mathcal{H}_{{\text{loc}}}^1({\Re ^n})$ such that the corresponding solutions to the n-Laplacian are unbounded in the first case, and bounded but discontinuous in the second.

References [Enhancements On Off] (What's this?)

  • [1] F. Bethuel, On the singular set of stationary harmonic maps, Manuscripta Math. 78 (1993), 417-443. MR 1208652 (94a:58047)
  • [2] E. DiBenedetto and J. J. Manfredi, Remarks on the regularity of solutions of certain degenerate elliptic systems, Univ. of Bonn, preprint no. 142, 1990.
  • [3] L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal. 116 (1991), 101-113. MR 1143435 (93m:58026)
  • [4] L. C. Evans and S. Müller, Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity, Univ. of Bonn preprint no. 262, 1992.
  • [5] F. Hélein, Régularité des applications faiblement harmoniques entre une surface et varieté Riemannienne, C. R. Acad. Sci. Paris. Sér. I Math. 312 (1991), 591-596. MR 1101039 (92e:58055)
  • [6] -, Regularity of weakly harmonic maps from a surface to a manifold with symmetries, Manuscripta Math. 70 (1991), 203-218. MR 1085633 (92a:58035)
  • [7] J. Malý and T. Kilpeläinen, The Wiener test and potential estimates for quasilinear elliptic equations, preprint, 1993.
  • [8] Libin Mou and P. Yang, Regularity for n-harmonic maps, J. Geom. Anal. (to appear). MR 1402388 (97h:58049)
  • [9] S. Semmes, A primer on Hardy space and some remarks on a theorem of Evans and Müller, preprint, 1993. MR 1257006 (94j:46038)
  • [10] E. M. Stein, Singular integrals and differentiability properties of functions (§5.2, p. 23), Princeton Univ. Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
  • [11] P. Tolksdorff, Everywhere regularity for some quasi-linear systems with lack of ellipticity, Ann. Mat. Pura Appl. 134 (1983), 241-266. MR 736742 (85h:35104)
  • [12] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), 210-240. MR 0474389 (57:14031)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35J05, 42B30

Retrieve articles in all journals with MSC: 35J05, 42B30

Additional Information

Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society