c Allerton Press, Inc., 2015.
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2015, Vol. 59, No. 5, pp. 13–16. c I.A. Bikchantaev, 2015, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2015, No. 5, pp. 17–21.
Original Russian Text Inner Uniqueness Theorem for Second Order Linear Elliptic
Equation with Constant Coefficients
I. A. Bikchantaev1*
1
Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan, 420008 Russia
Received December 13, 2013
Abstract—We consider solution f to a linear elliptic differential equation of second order, and prove
that it vanishes if zeros of f condense to two points along non-collinear rays. The requirement of
non-collinearity of the rays is essential if the roots of the characteristic equation are distinct. In the
case of equal roots of the characteristic equation this property is valid if and only if the rays do not
belong to common straight line.
DOI: 10.3103/S1066369X15050023
Keywords: elliptic equation, uniqueness theorem.
In a domain D of complex variable z = x + iy we consider elliptic equation
2
k=0
ak
∂ 2 f (z)
=0
∂x2−k ∂y k
(1)
with constant complex coefficients a0 , a1 , a2 such that a2 = 0. The ellipticity means that characteristic
polynomial a0 + a1 s + a2 s2 has not real roots. We denote these roots by s1 and s2 . Real and imaginary
parts of solutions to Eq. (1) satisfy bi-harmonic equation (the main equation of plane isotropic theory
of elasticity) for s1 = s2 = i and the main equation of plane anisotropic theory of elasticity for s1 = s2 ,
Im s1 > 0, Im s2 > 0.
We say that E is a uniqueness set for solutions to Eq. (1), if for any solution f to the equation the
equality f |E = 0 implies f = 0. Clearly, any subset E of the domain D with non-empty interior is a
uniqueness set for solutions to Eq. (1). There are known also theorems on weak and strong uniqueness
for general linear elliptic differential equations of second order. Theorems on weak uniqueness show
that a solution equals zero if it vanishes in a neighborhood of origin. Theorems on the strong uniqueness
prove that a solution identically vanishes if it decreases more rapidly than any degree of |z| in a connected
neighborhood of the origin (see, e.g., [1, 2]). The goal of the present paper are minimal uniqueness sets
for the solutions to Eq. (1).
Let a and b be points of the domain D. We consider rays la,α = {z : z = a + eiα s, 0 ≤ s < +∞} and
lb,β = {z : z = b + eiβ s, 0 ≤ s < +∞}, and put
ξ :=
beiα cos β − aeiβ cos α + ei(α+β) Re(a − b)
.
i sin(α − β)
Theorem 1. Let s1 = s2 , and a, b be points of the domain D such that ξ ∈ D. Assume that nonsets Ea,α and Eb,β condensing to the points a and b,
collinear rays la,α and lb,β contain infinite
respectively. Then equality f | Ea,α Eb,β = 0 implies f = 0, excluding the case where values s1 ,
s2 , α and β satisfy the relation
(cos α + s2 sin α)(cos β + s1 sin β) n
= 1,
(cos α + s1 sin α)(cos β + s2 sin β)
where n > 1 is integer.
*
E-mail: [email protected].
13