Open AccessEstimates of Solutions of Elliptic Differential-Difference Equations with Degeneration
Author(s) -
В. А. Попов
Publication year - 2018
Publication title -
sovremennaâ matematika. fundamentalʹnye napravleniâ
Language(s) - English
Resource type - Journals
eISSN - 2949-0618
pISSN - 2413-3639
DOI - 10.22363/2413-3639-2018-64-1-131-147
Subject(s) - operator (biology) , bounded function , smoothness , elliptic operator , mathematics , domain (mathematical analysis) , mathematical analysis , pure mathematics , spectrum (functional analysis) , sobolev space , boundary (topology) , physics , biochemistry , chemistry , repressor , quantum mechanics , transcription factor , gene
We consider a second-order dierential-dierence equation in a bounded domain Q ⊂ Rn. We assume that the dierential-dierence operator contains some dierence operators with degeneration corresponding to dierentiation operators. Moreover, the dierential-dierence operator under consideration cannot be expressed as a composition of a dierence operator and a strongly elliptic dierential operator. Degenerated dierence operators do not allow us to obtain the G˚arding inequality. We prove a priori estimates from which it follows that the dierential-dierence operator under consideration is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well. It is well known that elliptic dierential-dierence equations may have solutions that do not belong even to the Sobolev space W 1(Q). However, using the obtained estimates, we can prove some smoothness of solutions, though not in the whole domain Q, but inside some subdomains Qr generated by the shifts of the boundary, where U Qr = Q.