ISSN 0001-4346, Mathematical Notes, 2015, Vol. 97, No. 5, pp. 745–758. © Pleiades Publishing, Ltd., 2015.
Original Russian Text © L. E. Rossovskii, A. L. Tasevich, 2015, published in Matematicheskie Zametki, 2015, Vol. 97, No. 5, pp. 733–748.
The First Boundary-Value Problem
for Strongly Elliptic Functional-Differential
Equations with Orthotropic Contractions
L. E. Rossovskii
*
and A. L. Tasevich
**
Peoples’ Friendship University of Russia, Moscow, Russia
Received October 9, 2014
Abstract—We obtain a number of necessary and sufficient strong ellipticity conditions for a
functional-differential equation containing, in its leading part, orthotropic contractions of the
argument of the unknown function. We establish the unique solvability of the first boundary-value
problem and the discreteness, semiboundedness, and sectorial structure of its spectrum.
DOI: 10.1134/S0001434615050090
Keywords: strong elliptic functional-differential equation, first boundary-value problem, or-
thotropic contraction, G ˚ arding-type inequality, strong ellipticity condition, Plancherel’s
theorem, Fourier transform, Riesz theorem, difference operator.
1. INTRODUCTION
The paper deals with the boundary-value problem
A
R
u ≡−
2
i,j =1
(R
ij
u
x
i
)
x
j
= f (x), x ∈ Ω, u|
∂Ω
=0 (1.1)
in a bounded domain Ω ⊂ R
2
, where
R
ij
v(x)= a
ij 0
v(x)+ a
ij 1
v(q
−1
x
1
, px
2
)+ a
ij,−1
v(qx
1
,p
−1
x
2
),
p, q > 1, a
ij 0
,a
ij,±1
∈ C, i, j =1, 2, and the complex-valued function f (x) belongs to the space L
2
(Ω).
The subject matter of the paper is mostly related to the study of the well-known inequality
Re(A
R
u, u)
L
2
(Ω)
≥ c
1
‖u‖
2
H
1
(Ω)
− c
2
‖u‖
2
L
2
(Ω)
for all u ∈ C
∞
0
(Ω), (1.2)
called a G ˚ arding-type inequality, as well as to the coercitivity condition for the operator A
R
. If we
set a
ij,±1
=0, then the operator A
R
becomes a second-order linear differential operator with constant
coefficients and, in this case, estimate (1.2) is equivalent to the well-known strong ellipticity condition
Re
2
i,j =1
a
ij 0
ξ
i
ξ
j
≥ c
1
|ξ |
2
.
The question of the validity of inequality (1.2) for differential operators, including higher-order equations,
systems of equations, and variable coefficients, was solved in [1], [2] and, for difference-differential
operators in bounded domains, in [3]. Functional differential equations containing contractions and
dilations of the arguments of the unknown function in the leading part were considered in [4]–[7], where
it was assumed that the contraction (dilation) coefficient is the same for all variables:
v(x
1
,x
2
) → v(qx
1
,qx
2
). (1.3)
*
E-mail: lrossovskii@gmail.com
**
E-mail: atasevich@gmail.com
745