General Solution of a Second-Order Partial Differential Equation in a Banach Space with Potential Singular on Manifolds
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neral Solution of a Second-Order Partial Differential Equation in a Banach Space with Potential Singular on Manifolds T. N. Alikulov1* 1
National University of Uzbekistan named after M. Ulugbek, 4 Universitetskaya str., Tashkent, 100174 Republic of Uzbekistan Received November 23, 2019; revised May 16, 2020; accepted June 29, 2020
Abstract—In this work, we study the general solution of a second-order partial differential equation in a Banach space with a potential singular on the manifolds. DOI: 10.3103/S1066369X20100011 Key words: fractional power, elliptic operator, weakened solution, generalized solution, positive operator, Banach space.
1. INTRODUCTION Properties of fractional powers of differential operators are of importance for solution of various mathematical problems. A number of authors researched this subject (see, for instance, [1]– [4]). It should be emphasized that paper [1] by V.A. Il’in contains sufficiently complete research of representations of fractional powers of the Laplace operator as integral operators in Hilbert space. Sh.A. Alimov [2] by means of the fractional powers of elliptic differential operators proved isomorphism between space of summable functions and certain more complicated functional spaces. V.A. Kostin and M.N. Nebolsin [3] used the concept of fractional powers for proofs of theorems on correct resolvability of boundary-value problems for equations of second order. In addition, A.R. Khalmukhamedov [4] by means of fractional powers of the singular Schrödinger operator obtained theorems on convergence of spectral expansions. We consider in n-dimensional Euclidean space Rn (n > 3) smooth manifolds S1 , S2 , . . . , SM , dim Sk ≤ n − 3, such that any of them can be uniquely projected on certain hyperplane. This means that for every manifold Sk of dimension n − m, where 3 ≤ m < n, there exists an affine transformation Rn leading Sk to the appearance Sk = {x = (u, v) ∈ Rn : u = ϕk (v)}.
Here u ∈ Rm , v ∈ Rn−m , ϕk ∈ C 1 (Rn−m −→ Rm ), and inequalities ϕjk /∂vi ≤ const, where n−m . i = 1, . . . , n − m, j = 1, . . . , m and ϕk = (ϕ1k , . . . , ϕm k ) are valid uniformly on all v ∈ R Let S=
M
Sk , ρ = ρ(x) = dist(x, S) = inf |x − y|. y∈S
k=1
We put
m ρj = (uk − ϕkj (v)), k=1
*
E-mail: [email protected]
1
2
ALIKULOV
where uk and ϕkj are coordinates of vectors u ∈ Rm and ϕkj ∈ Rm , j = 1, 2, 3, . . . , M , respectively. In n-dimensional Euclidean space Rn , we consider elliptic Schrödinger operator with singular coefficient L(x, D) = −Δ + q(x)
(1)
with domain D(L) = Wp2 (Rn ), n > 3, 1 ≤ p < m, where potential q(x) ∈ C ∞ (Rn \S) admits singularity |D α q(x)| ≤ const[ρ(x)]−1−|α| {1 + [ρ(x)]−τ }. Here α is a multi-index,
0 ≤ |α| =
n i=1
(2)
αi ≤ n, 0 ≤ τ < 1.
In what follows we denote operator L(x, D) by L. We consider differential equation d2 u(t) = Lu(t), dt2
(3)
where t varies over the segment [0, T ]. Definition 1 ([5], p. 305). Solution of equation (3) is a function u(t) with values in D(L) which is twice continuously differentiable for all t ∈ [0, T ] an
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