Boundedness of the L-index in a direction of entire solutions of second order partial differential equation

Abstract

We construct a continuous function L : C 2 → (0, +∞) 2 such that every entire solution of a certain second order partial differential equation has bounded L-index in a direction b = (b1, b2) ∈ C 2 \ {0}. On the other hand, the entire bivariate function F (z1, z2) = cos √ z1z2 is a solution of this equation. The function F has unbounded index in each direction b ∈ C 2 \ {0}. The constructed function L is a full solution of a problem posed by A. A. Kondratyuk's in 2007 about the existence of the function L with specified properties. We also suggest a continuous function L1 : C 2 → (0, +∞) 2 such that the function F has bounded L1-index in every direction b ∈ C n \ {0}.

Composition, product and sum of analytic functions of bounded L-index in direction in the unit ball

“…. Indeed by Theorem 9 inequality 14is satisfied for the function G. Substituting z 0 α , t α and t 0 α instead z 0 , t and t 0 in (14) we obtain…”

“…These functions have regular behavior, uniform distribution of zeros in some sense and its growth estimates [15,24]. It is known many various conditions providing index boundedness for every analytic solutions of some ordinary and partial differential equations and its system [14,15,24,27].…”

Exhaustion by balls and entire functions of bounded $\mathbf{L}$-index in joint variables

“…It should be noted that the concepts of bounded -index in a direction and bounded L-index in joint variables have few advantages in the comparison with traditional approaches to study properties of entire solutions of differential equations. In particular, if an entire solution has a bounded index [7,8,19], this implies immediately estimates for its growth rate, an uniform distribution of its zeros, a certain regular behavior of the solution, etc. A full bibliography on applications in theory of ordinary and partial differential equations can be found in [1,23].…”

On some problem for entire functions of unbounded index in any direction

“…Mostly, these functions were investigated by A. Bandura and O. Skaskiv (see the bibliography in their monograph [7]). These functions have many applications in analytic theory of differential equations [6,7,13,14].…”