ON THE DIRICHLET PROBLEM TO ELLIPTIC EQUATION, THE ORDER OF WHICH DEGENERATES AT THE AXIS OF A CYLINDER

Stasys Rutkauskas

doi:10.3846/13926292.2017.1362053

Abstract

In this article, an elliptic equation, which type degenerates (either weakly or strongly) at the axis of 3-dimensional cylinder, is considered. The statement of a Dirichlet type problem in the class of smooth functions is given and, subject to the type of degeneracy, the classical solutions are composed. The uniqueness of the solutions is proved and the continuity of the solutions on the line of degeneracy is discussed.

Highlights

Introduction and statement of the problemS. Rutkauskas from the class C2(Q0) ∩ C(Q), and, in the case of the general boundary value conditions, the well-posedness of Dirichlet problem and the continuity of the solution on the line x = y = 0 is related with the behavior of the boundary functions in the vicinity of the points P0(0, 0, 0) and Ph(0, 0, h), in which this line crosses the bases of cylinder Q
Introduction and statement of the problemIn the cylinder Q = {(x, y, z): x2 + y2 < ρ2, 0 < z < h}, we consider the equation r2α(uxx + uyy + uzz) − cu = 0, α > 0, (1.1)where r = x2 + y2, c > 0 is a real constant
In [9, 10, 11], the Dirichlet problem is considered for equation uzz + r2α(uxx + uyy) − cu = 0, α > 0, (1.2)

Summary

Introduction and statement of the problem

S. Rutkauskas from the class C2(Q0) ∩ C(Q), and, in the case of the general boundary value conditions, the well-posedness of Dirichlet problem and the continuity of the solution on the line x = y = 0 is related with the behavior of the boundary functions in the vicinity of the points P0(0, 0, 0) and Ph(0, 0, h), in which this line crosses the bases of cylinder Q. There is shown that, in particular case α = 1, the solution of this problem is non-continuous on the liene x = y = 0, if the boundary value conditions are non-zero on the bases of cylinder Q. We shall show up here that, under the zero boundary value conditions on both bases of cylinder Q, this problem has the unique solution for all α > 1 from the same class C2(Q0)∩C(Q) as in the case of Eq (1.2). Such partial case of Problem D we call as Problem D1

Solutions of Problem D0

Conclusions