đť‘›-Laplacian in â„‹Âą_{}fi] does not lead to regularity
Nick B. Firoozye · Proceedings of the American Mathematical Society · 1995
It is well known that in two space dimensions, if a solution to Poisson’s equation has right-hand side in H loc 1 \mathcal {H}_{{\text {loc}}}^1 , then this solution is actually continuous. The corresponding result for n -Laplacians is shown to be false for n ≥ 3 n \geq 3 ; we construct two examples with right-hand sides in H loc 1 ( ℜ n ) \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.