-Laplacian in 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

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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 , then this solution is actually continuous. The corresponding result for *n*-Laplacians is shown to be false for ; we construct two examples with right-hand sides in such that the corresponding solutions to the *n*-Laplacian are unbounded in the first case, and bounded but discontinuous in the second.

**[1]**Fabrice Bethuel,*On the singular set of stationary harmonic maps*, Manuscripta Math.**78**(1993), no. 4, 417–443. MR**1208652**, 10.1007/BF02599324**[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]**Lawrence C. Evans,*Partial regularity for stationary harmonic maps into spheres*, Arch. Rational Mech. Anal.**116**(1991), no. 2, 101–113. MR**1143435**, 10.1007/BF00375587**[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]**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****[6]**Frédéric Hélein,*Regularity of weakly harmonic maps from a surface into a manifold with symmetries*, Manuscripta Math.**70**(1991), no. 2, 203–218. MR**1085633**, 10.1007/BF02568371**[7]**J. Malý and T. Kilpeläinen,*The Wiener test and potential estimates for quasilinear elliptic equations*, preprint, 1993.**[8]**Libin Mou and Paul Yang,*Regularity for 𝑛-harmonic maps*, J. Geom. Anal.**6**(1996), no. 1, 91–112. MR**1402388**, 10.1007/BF02921568**[9]**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**, 10.1080/03605309408821017**[10]**Elias M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095****[11]**Peter Tolksdorf,*Everywhere-regularity for some quasilinear systems with a lack of ellipticity*, Ann. Mat. Pura Appl. (4)**134**(1983), 241–266. MR**736742**, 10.1007/BF01773507**[12]**K. Uhlenbeck,*Regularity for a class of non-linear elliptic systems*, Acta Math.**138**(1977), no. 3-4, 219–240. MR**0474389**

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DOI:
https://doi.org/10.1090/S0002-9939-1995-1277110-0

Article copyright:
© Copyright 1995
American Mathematical Society