The first step to cast the boundary value problem as in the language of Sobolev spaces is to rephrase it in its weak form.
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The idea here is that a Sobolev space provides the natural setting for both the idea of square-integrability and the idea of differentiation.
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Then take the Hilbert space closure with respect to this inner product, this is the Sobolev space H ^ 1 ( \ Omega ).
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These operators lie in every Schatten class and leave invariant the Sobolev spaces and the space of entire vectors for " W ".
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A quick calculation shows that " z & # x302; " is not just continuous, but also lies in a Sobolev space:
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Among all functions in the Sobolev space W ^ { 1, p } ( \ Omega ) satisfying the boundary conditions in the trace sense.
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Something similar to metric in Sobolev spaces W ^ k _ p ? ( talk ) 04 : 13, 7 November 2008 ( UTC ))
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Another approach to define fractional order Sobolev spaces arises from the idea to generalize the H�lder condition to the " L p "-setting.
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This is typically achieved by taking the Sobolev space H ^ 2 of functions on the edges of the graph and specifying matching conditions at the vertices.
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If however, u is the solution to some partial differential equation, it is in general a weak solution, so it belongs to some Sobolev space.
