Kernels of Trace Operators and Boundary Value Problems in Field Theory
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Journal of Mathematical Sciences, Vol. 251, No. 5, December, 2020
KERNELS OF TRACE OPERATORS AND BOUNDARY VALUE PROBLEMS IN FIELD THEORY Yu. A. Dubinskii National Research University “Moscow Power Engineering Institute” 14, Krasnokazarmennaya St., Moscow 111250, Russia julii [email protected]
UDC 517.956.25
We propose an approach to formulation and solution of a series of boundary value problems in field theory based on the use of kernels of the trace operators and functionals in Sobolev spaces. The boundary conditions are nonlocal and can contain the main first order operators (gradient, curl, and divergence) of field theory. The case of plane vector fields is separately studied. Bibliography: 4 titles.
1
Introduction
We continue the study of [1]–[3] and consider boundary value problems such that 1) the boundary conditions are nonlocal, 2) the boundary conditions contain the main first order operators of field theory (gradient, curl, and divergence). It is reasonable to distinguish the case of standard trace operators from the case of trace operators with values in the adjoint space. In the first case, the trace operator is the mapping 1/2
Tr : W21 (G) → W2 (Γ) which, by duality
(W21 (G), (W21 (G)∗ ),
(1.1)
determines the regular functional (Tr u, v|Γ )dγ, Γ
W21 (G)
where v(x) ∈ is an arbitrary vector-valued function. In the second case, the boundary value problem is determined by the kernel of an operator that does not have trace in the sense of (1.1), but possesses the trace as a functional on the 1/2 space W2 (Γ) and, simultaneously, is an extension of some regular trace functional. For an −1/2 (Γ) coinciding example of such an operator we can consider the operator DΓ : W21 (G) → W2 2 2 with the regular trace operator on the space W2 (G) for any u(x) ∈ W2 (G): ∂u 1/2 − [curl u, n] − div u · n ∈ W2 (Γ) DΓ u = ∂n Γ Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 73-89. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2515-0635
635
in the sense of trace theory for Sobolev spaces. Based on this fact, we can formulate boundary value problems (on the kernels of the operator DΓ and functional DΓ u) with first order operators of field theory in boundary conditions. The paper consists of two parts. In the first part, we define kernels of trace operators and functionals and consider the corresponding decompositions of Sobolev spaces. In the second part, we formulate boundary value problems and establish their solvability in the corresponding subspaces. The main results concerning the first part of the paper are presented in the following table, where Tr is the trace operator, Trnorm is the normal trace operator, Trtan is the tangent trace operator, and Tr⊥ tan is the orthogonal trace operator. A
Ker A
Tr
W
Trnorm
1 W2,tan
Trtan
1 W2,norm
Tr⊥ tan
1 W2,norm
DΓ
1 W2,0
0
1 2
(Au, v) (u, v)dγ
KerAu w : (u, w)dγ = 0
Γ
Γ
(u, n)(v, n)dγ
w:
Γ
w:
Γ
1 Ker Trnorm u ⊕ RN 2
([u, n], [w, n])dγ = 0
1 Ker Trtan u ⊕ RN 3
([u, n], w)dγ = 0
1 Ker Tr⊥ tan u ⊕ RN4
Γ
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