Bilal T. Bilalov | Searcharxiv

arXiv2025

Global solvability of the Laplace equation in weighted Sobolev spaces

We consider a non-local boundary value problem for the Laplace equation in unbounded studding the weak and strong solvability of that problem in the framework of the weighted Sobolev space $W^{1,p}_ν$, with a Muckenhoupt weight. We proved that if any weak solution belongs to the space $W_ν^{2,p}$, then it is also a strong solution and satisfies the corespding boundary conditions. It should be noted that such problems do not fit into the general theory of elliptic equations and require a special technique.

math.AP↗

arXiv2024

Higher order elliptic equations in weighted Banach spaces

We consider higher order linear, uniformly elliptic equations with non-smooth coefficients in Banach-Sobolev spaces generated by weighted general Banach Function Space (BFS). Supposing boundedness of the Hardy-Littlewood Maximal and Calderon-Zygmund singular operators in BFSs we obtain local solvability in the Sobolev-BFS and establish interior Schauder type a priori estimates for the. elliptic operator. These results will be used in order to obtain Fredholmness of the operator under consideration in weighted BFSs with suitable weight. In addition, we analyze some examples of weighted BFS that verify our assumptions and in which the corresponding Schauder type estimates and Fredholmness of the operator hold true.

math.AP↗