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


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DOI: https://doi.org/10.1090/S0002-9939-1995-1277110-0
Article copyright: © Copyright 1995 American Mathematical Society