In particular Clifford analysis has been used to solve, in certain Sobolev spaces, the full water wave problem in 3D . This method works in all dimensions greater than 2.
42.
The general theory also often requires that the functions belong to some given function space : often, the space of square-integrable functions is studied, and Sobolev spaces appear often.
43.
Starting from 1993, several conferences have been held to honor him : the first one, held in that year at the University of Kyoto, was a conference on Sobolev spaces.
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For example, we say that a function " u " belonging to the Sobolev space W ^ { 1, p } ( \ Omega ) is a weak solution of
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Can be solved locally in such a way that the radial limits of " G " and " F " tend locally to the same function in a higher Sobolev space.
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The theory of Bost is based on the use of Green functions which, up to logarithmic singularities, belong to the Sobolev space L " 2 " " 1 ".
47.
The Sobolev space for is the annihilator in the Sobolev space for of ( ? " c " ) } } and that for is the annihilator of ( ? ) } }.
48.
The Sobolev space for is the annihilator in the Sobolev space for of ( ? " c " ) } } and that for is the annihilator of ( ? ) } }.
49.
Usually, this is not a problem, since in the theory of " L " " p " spaces and Sobolev spaces, functions that are equal almost everywhere are identified.
50.
Partitions of unity therefore allow for certain other kinds of function spaces to be considered : for instance L " p " spaces, Sobolev spaces, and other kinds of spaces that require integration.
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