Preamble

In 1967, M. I. Krutitskaya investigated the equation $Z_n Z_{n-1} \dots Z_1 u = 0$, where the differential operators $Z_i$ are defined as:
$$Z_i = A_i \frac{\partial^2}{\partial x^2} + 2B_i \frac{\partial^2}{\partial x \partial y} + C_i \frac{\partial^2}{\partial y^2} + D_i \frac{\partial}{\partial x} + E_i \frac{\partial}{\partial y} + F_i$$
The coefficients $A_i, B_i, C_i, D_i, E_i,$ and $F_i$ are functions of $(x, y)$. This work builds upon the foundational methods established in \cite{1} for the analysis of higher-order partial differential equations.

For a domain $D$, let $v_{im}(x, y)$ be a system of functions satisfying the conditions discussed in \cite{1} and \cite{2}. We consider the solution to the equation $Z_1 u = 0$ in $D_1$. The general solution can be represented using the integral form:
$$v_{im}(x, y) = -\iint_{D} \Gamma(x, y; \xi, \eta) v_{lm}(\xi, \eta) d\xi d\eta$$
where $\Gamma(x, y; \xi, \eta)$ is the Green's function for the operator $Z_1$ in the domain $D_1$, as defined in \cite{2}.

The functions $u_{mk}(x, y)$ are constructed to satisfy the boundary conditions on the domain $D_1$. Following the methodology in \cite{3} and \cite{4}, we establish that for the iterative operator $Z_n Z_{n-1} \dots Z_1 u = 0$, the solution $u(x, y)$ can be decomposed into a sum of functions $v_{im}(x, y)$ and $w_m(x, y)$, where each component corresponds to the kernel of the respective operator $Z_i$.

Properties of Solutions

Let $u(x, y)$ be a solution to the equation in a domain $D$. We assume that the boundary conditions are satisfied such that $u(x, y) = \phi_i(x, y)$ for $i = 0, 1, \dots, n-1$. As shown in \cite{2}, if the coefficients of the operators $Z_i$ are sufficiently smooth, the solution $u(x, y)$ is unique and depends continuously on the boundary data.

Consider the specific case of the operator $( \Delta + 1)(\Delta - 1) u = 0$, where $\Delta$ is the Laplace operator. If we define the domain as a square with $x \in [0, \pi]$ and $y \in [0, \pi]$, the solution can be expressed via a series of eigenfunctions, such as $\sin x \sin y$. This approach aligns with the results presented by P. Montel in \cite{5} and further developed in \cite{4} regarding the mean value properties of polyharmonic functions.

Mean Value Theorems and Convergence

A significant property of the solutions to $Z_1 u = 0$ is the mean value theorem. Let $u(x, y)$ be a solution in a disk of radius $R$ centered at $(0, 0)$. The value of the solution at the origin can be bounded by the integral of the solution over the boundary $x^2 + y^2 = R^2$. Specifically, if $u(x, y)$ is continuous in the closed disk, then:
$$|u(0, 0)| \leq Q(a, 0)$$
where $Q(a, 0)$ is a constant depending on the boundary values and the coefficients of the operator.

Furthermore, if we consider a sequence of solutions $u_m(x, y)$ that converges uniformly on the boundary $x^2 + y^2 = 1$, then the sequence converges to a solution $u(x, y)$ within the interior of the domain. This property is analogous to the behavior of harmonic functions. As noted by B. M. Gakhov in \cite{6}, the behavior of these solutions near singular points or boundaries is critical for the stability of the mathematical model.

In conclusion, the iterative application of the operators $Z_i$ allows for the modeling of complex physical processes. The analytical framework provided in \cite{7} ensures that the solutions $u(x, y)$ maintain regularity and satisfy the necessary physical constraints within the specified domains.