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\title{Continuous Dependence of Functional Differential Equations on the Scaling Parameter}
\speaker{L.\,E.~Rossovskii}{Peoples' Friendship University of
Russia, Moscow, Russia}
\author{A.\,A.~Tovsultanov}{Chechen State University, Grozny, Russia}
\ctitle{L.\,E.~Rossovskii, A.\,A.~Tovsultanov}{Continuous
dependence of functional differential equations on the scaling
parameter}


Let $\Omega$ be a bounded domain in $\mathbb R^n$, containing the
origin. Consider the Dirichl{\'e}t problem
$$
-\sum\limits_{i,j=1}^n\left(a_{ij}u_{x_i}(x)+b_{ij}u_{x_i}(x/p)+c_{ij}u_{x_i}(x/q)\right)_{x_j}=f(x)\quad(x\in\Omega\subset\mathbb R^n),\quad u|_{\partial\Omega}=0,
$$
with $a_{ij},b_{ij},c_{ij}\in\mathbb C$ $(i,j=1,\ldots,n)$ and
$f\in L_2(\Omega)$. Assuming that $q$ is a fixed number greater
than $1$ while $p$ ranges over a certain segment $[1,p_0]$, we are
interested in the unique solvability of the problem in the Sobolev
space $\mathring H^1(\Omega)$ (by a solution, we mean a
generalized solution understood in the standard way) for all the
considered values of $p$ and the behaviour of the family of
solutions $u=u_p$ as $p\to1$.

Exact solvability conditions in terms of the coefficients were
earlier found~\cite{ross1} in the case where different scaling
parameters were all integer powers, positive or negative, of one
parameter. Now, there is no tie-up between the two parameters $p$
and $q$. On the other hand, the dependence of a solution to a
functional differential equation on a parameter defining the
argument transformation, is studied for the first time.

Introduce the symbols (the summation is over $i,j=1,\ldots,n$)
$$a(\xi)=\sum a_{ij}\xi_i\xi_j,\quad b(\xi)=\sum
b_{ij}\xi_i\xi_j,\quad c(\xi)=\sum
c_{ij}\xi_i\xi_j\quad(\xi\in\mathbb R^n).$$

\begin{theorem}\label{ross-thm}
If
\begin{equation}\label{ross-cond}
|b(\xi)|+q^{n/2}|c(\xi)|<\mathrm{Re}\,a(\xi)\quad(\xi\neq0),
\end{equation}
then there exists a number $p_0>1$ such that the Dirichl{\'e}t
problem has a unique solution $u=u_p\in\mathring H^1(\Omega)$ for
all values of $p\in[1,p_0]$ with $u_p\to u_1$ in $\mathring
H^1(\Omega)$ as $p\to1$.
\end{theorem}

The proof is based on the use of the G\aa{}rding-type inequality
$$
\mathrm{Re}\sum\limits_{i,j=1}^n\int\limits_{\Omega}\!\left
(a_{ij}u_{x_i}(x)+b_{ij}u_{x_i}(x/p)+c_{ij}u_{x_i}(x/q)\right)\bar
u_{x_j}\,dx\geqslant\gamma\|u\|^2_{H^1(\Omega)}\;\;(\forall
u\in\mathring H^1(\Omega)).
$$
Condition \eqref{ross-cond} in Theorem \ref{ross-thm} is actually
necessary and sufficient for this inequality to hold with a common
constant $\gamma>0$ for all $p\in[1,p_0]$ provided $p_0>1$ is
sufficiently close to $1$.

This work was supported by the State contract of the Russian
Ministry of Education and Science (contract \No~1.1974.2014/K).

\begin{thebibliography}{9}

\bibitem{ross1} Rossovskii~L.\,E. Boundary value problems for
elliptic functional differential equations with dilatations and
compressions of the arguments, {\it Trans. Moscow Math. Soc.},
{\bf 62}, 185--212 (2001).

\bibitem{ross2} Skubachevskii~A.\,L. {\it Elliptic Functional
Differential Equations and Applications}. ---
Basel--Boston--Berlin: Birkh\"auser, 1997.

\end{thebibliography}

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