**MSC2010** 35J93

### About an a priori estimate for the second order elliptic operator degenerate alog coordinate axis orthogonal to semi-plane boundary

#### G. A. Smolkin^{1}

Annotation | In the paper the methodology is demonstrated to derive an inequality of special type. The left-hand side of this inequality is a norm of the second-order derivative of a function along the normal to a half-plane boundary. The right-hand side of the inequality is a linear combination of two terms. The first is a norm of a function image generated by degenerate elliptic operator, and the second is a trace of function on the half-plane boundary. Paper deals with norms in Sobolev spaces and in Slobodetzky spaces. In the inequality proof two function continuations from half-plane to the entire plane are used. Using the first continuation which has derivatives up to the third order the inequality is reduced to estimation of mixed derivatives and derivatives with respect to boundary's tangents. This derivatives are obtained using the second continuation that is twice differentiable. |
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Keywords | Fouries transform, Sobolev spaces, a priori estimates, degenerate elliptic operator, function continuation. |

^{1}**Georgy A. Smolkin**, Associate Professor, Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68 Bolshevistskaya Str., Saransk 430005, Republic of Mordovia, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0001-5964-9814, smolkinga@yandex.ru

**Citation**: G. A. Smolkin, "[About an a priori estimate for the second order elliptic operator degenerate alog coordinate axis orthogonal to semi-plane boundary]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:3 (2017) 64–72 (In Russian)

**DOI** 10.15507/2079-6900.19.201703.64-72