On an estimate in the Sobolev space generated by the second order degenerate elliptic operator defined in the half-plane

G. A. Smolkin

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 2, pp. 206-214

Résumé

The article considers an elliptic operator that is defined in the half-plane and degenerates along the normal to the boundary of the half-plane. The results obtained by the author earlier are made more precise. A partition of unity of a dual variable is constructed that allows to «freeze» the derivatives along the orthogonal direction to the degeneracy set and to carry out a smooth continuation of the function to the whole plane. It is shown that this and the «standard» continuations examined in detail by L.N. Slobodetsky, is sufficient for obtaining the necessary a priori estimate. Moreover, the inequalities are proved by the Fourier transform with respect to the part of variables and by the use of Schwartz inequality. It is established that the Sobolev norm of the function’s second order derivatives will be finite if its restriction to the boundary of the half-plane and function’s image both belong to the Sobolev spaces with indicators 3, 2, respectively. The results obtained can be spread to a wider class of operators; also they may be used in the research of boundary value problems for the degenerate elliptic and quasi-elliptic operators defined in half-spaces.

Détail

Keywords: degenerate elliptic operator, Sobolev space, a priori estimates.
Mots-clés : Fourier transform