On solvability in the small and Schauder-type estimates for higher order elliptic equations in grand Sobolev spaces (nonseparable case)

Abstract

In this work, it is considered an elliptic operator L of mth order with nonsmooth coefficients in a non-standard grand Sobolev space $W_{q)}^m\left(\mathrm{\Omega }\right)$ on a bounded domain $\mathrm{\Omega }\subset R^n$ generated by the norm of the grand Lebesgue space $L_{q)}\left(\mathrm{\Omega }\right)$. Under weaker restrictions on the coefficients of the operator, we prove the solvability (in the strong sense) in the small in $W_{q)}^m\left(\mathrm{\Omega }\right)$ and also establish interior Schauder-type estimates for these spaces. These estimates play the main role in establishing the Fredholmness of the Dirichlet problem for the equation Lu = f. The considered spaces are not separable, infinitely differentiable functions are not dense in them, and therefore many classical methods concerning Sobolev spaces are not applicable in this case. Nevertheless, it is possible to obtain the corresponding results under the assumption that the coefficients of the principal terms of the operator L are continuous, and the rest are essentially bounded in Ω.

Keywords:

• Elliptic operator
• Schauder estimates
• grand Sobolev and grand Lebesgue spaces

AMS CLASSIFICATIONS:

• 35A01
• 35J05
• 35K05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the Scientific and Technological Research Council of Turkey (TUBITAK) with Azerbaijan National Academy of Sciences (ANAS), Project Number: 19042020; by the Science Development Foundation under the President of the Republic of Azerbaijan – Grant No. EIF-BGM-4-RFTF1/2017- 21/02/1-M-19 and by TÜBİTAK 2211-A Domestic General Doctorate Scholarship Program.