Sobolev Spaces
The analysis of elliptic boundary value problems requires some fairly sophisticated tools of functional analysis. We require the space, the Sobolev space of "once-differentiable" functions on, such that both the function and its partial derivatives, are all square integrable. There is a subtlety here in that the partial derivatives must be defined "in the weak sense" (see the article on Sobolev spaces for details.) The space is a Hilbert space, which accounts for much of the ease with which these problems are analyzed.
The discussion in details of Sobolev spaces is beyond the scope of this article, but we will quote required results as they arise.
Unless otherwise noted, all derivatives in this article are to be interpreted in the weak, Sobolev sense. We use the term "strong derivative" to refer to the classical derivative of calculus. We also specify that the spaces, consist of functions that are times strongly differentiable, and that the th derivative is continuous.
Read more about this topic: Elliptic Boundary Value Problem
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“Le silence éternel de ces espaces infinis m’effraie. The eternal silence of these infinite spaces frightens me.”
—Blaise Pascal (1623–1662)