Elliptic problems and Hörmander spaces

Details Elliptic problems and Hörmander spaces Elliptic problems and Hörmander spaces

2009-04-02

arxiv.org/abs/0904.0372v1

Operator Theory: Advances and Applications, vol. 191, Birkh\"auser-Verlag, Basel, 2009, pp. 447--470.

Mathematics

Analysis of PDEs

20 papges

Scientific paper

The paper gives a survey of the modern results on elliptic problems on the H\"ormander function spaces. More precisely, elliptic problems are studied on a Hilbert scale of the isotropic H\"ormander spaces parametrized by a real number and a function slowly varying at $+\infty$ in the Karamata sense. This refined scale is finer than the Sobolev scale and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this scale. A local refined smoothness of the elliptic problem solution is studied. An abstract construction of classes of function spaces in which the elliptic problem is a Fredholm one is found. In particular, some generalizations of the Lions-Magenes theorems are given.

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