On Some Transformations Associated to a Certain Cone

TL;DR

This paper investigates elliptic pseudo-differential equations in 4-faced cones, providing explicit solutions and analyzing boundary value problems with integral conditions within Sobolev-Slobodetskii spaces.

Contribution

It introduces explicit formulas for solutions of elliptic equations in conical domains and establishes unique solvability for boundary value problems with integral conditions.

Findings

Explicit formula for the solution in the cone

Unique solvability of boundary value problems with integral conditions

Evaluation of the Bochner kernel for the cone

Abstract

A model elliptic pseudo-differential equation in $4$-faced cone is studied in Sobolev--Slobodetskii space. The Bochner kernel for such a cone is evaluated and explicit formula for unique solution to the considered equation is presented under certain restrictions on the symbol. Boundary value problem with additional integral condition is considered and unique solvability to the boundary value problem is proved.