Timeline for How to define Laplacian on $L_2$

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Jan 29, 2013 at 14:18	comment	added	Delio Mugnolo		of course. well, clearly i was trying to avoid to use explicitly sobolev space theory, which i was assuming not to be known by the OP. in fact, another possibility would be to study everything in $W^{2,1}$, if one already knows what an absolutely continuous function is, and in which sense one can think of it as differentiable.
Jan 29, 2013 at 8:19	comment	added	Daniel Spector		Part 1) is equivalent to the assumption $u \in H^2$ for which we know we can give meaning to the Laplacian in precisely the sense you describe. Part 2) is the assumption $u \in W^{2,\infty}$, which is a stronger hypothesis even than $H^2$. $H^1$ is enough to consider weak solutions to Laplace's equation, though defining the Laplacian on $H^1$ is another matter.
Jan 28, 2013 at 22:35	history	answered	Delio Mugnolo	CC BY-SA 3.0