Journal of Siberian Federal University. Mathematics & Physics 2021, 14(5), 554–565
DOI: 10.17516/1997-1397-2021-14-5-554-565
УДК 517.9
Energy Method for the Elliptic Boundary Value Problems
with Asymmetric Operators in a Spherical Layer
Valery V.Denisenko∗
Semen A.Nesterov†
Institute of Computational Modelling SB RAS
Krasnoyarsk, Russian Federation
Received 10.01.2021, received in revised form 13.03.2021, accepted 02.04.2021
Abstract. Three-dimensional elliptic boundary value problems arising in the mathematical modeling of
quasi-stationary electric fields and currents in conductors with gyrotropic conductivity tensor in domains
homeomorphic to the spherical layer are considered. The same problems are mathematical models of
thermal conductivity or diffusion in moving or gyrotropic media. The operators of the problems in the
traditional formulation are non-symmetric. New statements of the problems with symmetric positive
definite operators are proposed. For the four boundary value problems the quadratic energy functionals,
to the minimization of which the solutions of these problems are reduced, are constructed. Estimates of
the obtained quadratic forms are made in comparison with the form appearing in the Dirichlet principle
for the Poisson equation.
Keywords: mathematical modeling, energy method, elliptic equation, asymmetric operator.
Citation: V.V.Denisenko, S.A.Nesterov, Energy Method for the Elliptic Boundary Value Problems
with Asymmetric Operators in a Spherical Layer, J. Sib. Fed. Univ. Math. Phys., 2021, 14(5), 554–565.
DOI: 10.17516/1997-1397-2021-14-5-554-565.
Introduction
The tree-dimensional elliptic boundary value problems with asymmetric operators arise, for
example, in mathematical modeling of quasi-stationary electric fields and currents in the global
conductor, consisting of the Earth’s ionosphere and atmosphere, using the domain decomposition
method [1]. In such models as [2], a significant simplification of the description of the D-layer
of the Earth’s ionosphere, lying at heights of 50 − 90 km, is used. It is in this layer that the
two-dimensional model, which is adequate for higher layers, is inapplicable, and the conductivity
ceases to be a scalar, as in the underlying atmosphere. Therefore, in this layer, bounded by two
surfaces close to concentric spheres, it is necessary to solve a three-dimensional problem of current
continuity with a gyrotropic conductivity tensor. Within the framework of the decomposition
method [1], the boundary conditions arise at the selected boundaries of subdomains. In the case of
interest to us, they are mixed: the condition of ideal conductivity at the upper boundary of the D-
layer and the condition on the ideal insulator on the bottom. Both conditions are heterogeneous.
This boundary value problem is studied in detail in this work. For three problems which differ
from the main problem by the boundary conditions, the necessary changes in the formulations
∗denisen@icm.krasn.ru https://orcid.org/0000-0002-3024-3746
†twist3r0k@yandex.ru https://orcid.org/0000-0002-9409-5826
c⃝ Siberian Federal University. All rights reserved
– 554 –
Valery V.Denisenko, Semen A.Nesterov Energy Method for the Elliptic Boundary Value Problems . . .
and proofs are described. We use the approach previously used for the problem in a simply
connected domain with a given normal component of the current density on the boundary [3].
Note that mathematical models of thermal conductivity or diffusion in moving media can be
reduced to the same boundary value problems [4, 5], and the equations of heat conduction or
diffusion in stationary gyrotropic media differ from the electric current continuity problem only
in notation.
In the present work, the quadratic energy functionals are constructed, which makes it possible
to reduce the solution of the boundary value problems for a three-dimensional elliptic equation
with asymmetric coefficients to minimisation of functionals. It is shown that the energy norm
is equivalent to the norm of the space W (1)2 (Ω). The corresponding estimates are obtained with
specific values of the constants that will allow us to estimate the condition number of the matrix
of a system of linear algebraic equations, which will arise in the numerical solution of the problem.
1. The electric current continuity problem
In the three-dimensional domain Ω occupied by a conductor, the electric field strength E
and the current density J in the quasistationary approximation satisfy Faraday’s law, charge
conservation law and Ohm’s law:
rotE = G, divJ = d, J = .ˆE, (1)
where G ̸= 0, if there is a given magnetic field that varies with time, d ̸= 0, if there are external
currents, .ˆ is the conductivity tensor. We assume that the norms of the given functions, d and
the Cartesian components of G, are bounded in the space L2(Ω). All vectors in this paper are
considered as column vectors, which are transformed into row vectors by transposition, denoted
by the symbol ∗. The system (1) with proper boundary conditions is referred to as the electric
current continuity problem [6].
The conductivity of some substances in a magnetic field, for example, plasma in the Earth’s
ionosphere, is a gyrotropic tensor. In Cartesian coordinates x, y, z with the z axis directed along
the magnetic induction vector, the conductivity tensor has the form:
.ˆ =
 .P −.H 0.
H
.
P
0
0 0 .∥
 . (2)
Its components are called field-aligned (.∥), Pedersen (.P ) and Hall (.H) conductivities [6].
We also use the Cowling conductivity
.
C
= (.2
P
+ .2
H
)P.
P
.
In the article [3] a more general form of .ˆ is considered. Here we use the form (2), to
obtain more accurate estimates which are important for the numerical solution of the problems.
Since the passage of electric current is accompanied by dissipation of the electric energy with
density J∗E, the symmetric part of .ˆ is positive definite. For a tensor of the form (2), this
means the positiveness of the diagonal elements. Excluding ideal conductors and insulators
from consideration, we assume uniform in the domain Ω boundedness of all coefficients of .ˆ and
uniform positive definiteness. It is convenient to write these conditions in the form:
.1 6 .C 6 .2, .1 6 .∥ 6 .2, (3)
– 555 –
Valery V.Denisenko, Semen A.Nesterov Energy Method for the Elliptic Boundary Value Problems . . .
where .1, .2 are positive constants.
We assume that the domain Ω is bounded and homeomorphic to the spherical layer. Its outer
Γ and inner γ boundaries are twice continuously differentiable surfaces homeomorphic to the
sphere. Normal (positive outward direction) and tangent to the boundary components of vectors
are marked with indices n and / .
To carry out all the proofs by simple means, we assume the convexity of the surface Γ and the
boundedness of the curvature of γ. We also consider the differences of both surfaces to be limited
from two spheres with a common center and not too different radii. We will give a concrete form
to all these restrictions in Section 2.
As the main problem, consider a problem with boundary conditions arising in mathematical
modeling of the D-layer of the ionosphere [1]. If the domain under consideration borders on an
ideal insulator, then at the boundary, the normal component of J equals to zero. On the border
with the ideal conductor the tangent components of E equal to zero. The conditions can be
heterogeneous:
Jn|γ = q, Eτ |Γ = g, (4)
where q, g are given functions. Assume that the norms of these functions are bounded in the
spaces L2(Γ) and L2(γ), respectively, that is, the boundedness of the integrals of the squares of
the moduli of these functions over the surfaces on which they are given.
For the solvability of the problem (1), (4) the right-hand sides must satisfy some constraints.
First, it is necessary
divG = 0, (5)
since div of rot is identically equal to zero, and without (5) the first equation (1) cannot hold.
Let us calculate the components of rot normal to the boundary from the left and right sides
in the second boundary condition (4):
rotnEτ |Γ= rotn g |Γ .
The resulting left side also satisfies the first equation (1). This imposes on the given functions
the second condition necessary for the solvability of the problem
Gn |Γ= rotn g |Γ . (6)
In Section 4, a new problem will be formulated with an conjugately factorized operator by
definition [7], the solution of which is the solution to the original problem (1), (4). The existence
of the solution to the new problem implies the existence of a solution to the original one, but
the uniqueness must be proven independently. It does not differ from the proof given in [3] for
the problem in a domain homeomorphic to a ball. Heuristic considerations are also given there,
allowing us to propose a new formulation and to construct a quadratic energy functional.
2. The energy scalar product
We consider the set of pairs of smooth functions F , P, satisfying the conditions:
F |Γ= 0, cn |Γ= 0, Pτ |γ= 0. (7)
We denote with square brackets the symmetric bilinear form:[(
u
v
)
,
(
F
P
)]
=
∫ ((
gradu
rotv
)∗( 1
σ0
.ˆSˆ.ˆ∗ −.ˆSˆ
−Sˆ.ˆ∗ .
0
Sˆ
)(
gradF
rotP
)
+ divv divP
)
dΩ, (8)
– 556 –
Valery V.Denisenko, Semen A.Nesterov Energy Method for the Elliptic Boundary Value Problems . . .
where u,v and F,P are the pairs of smooth functions satisfying conditions (7), Sˆ is a symmetric
positive definite matrix, which we will choose later, as well as the value of the positive constant .
0
.
This bilinear form will be used as the energy scalar product. Let us check that it has the necessary
properties for this.
Consider the corresponding quadratic form. We start with the auxiliary integral∫
(gradF )∗rotP dΩ. (9)
We transform the integrand:∫
(div (F rotP)− F div rotP) dΩ.
The second term is identically zero. The remaining integral is transformed using the Gauss–
Ostrogradsky theorem: ∮
Γ
F rotnP dΓ +
∮
γ
F rotnP dγ.
Both integrands are equal to zero due to the first and third conditions (7), respectively.
Therefore, the integral (9) is equal to zero, and therefore it can be added with any coefficient to
the corresponding (8) quadratic form without changing its value.
The matrix appearing in (8) is degenerate, since its upper blocks are obtained from the lower
ones through multiplication by −.ˆP.
0
. By adding the doubled integral (9) the matrix of the
integrand quadratic form becomes equal to
K =
(
1
σ0
.ˆSˆ.ˆ∗ −.ˆSˆ + Iˆ
−Sˆ.ˆ∗ + Iˆ .
0
Sˆ,
)
,
where Iˆ is the identity matrix.
To get more accurate estimates than in [3], we use a special form .ˆ (2) and constraints (3),
taking
.0 =
√
.1.2, Sˆ
−1 = (.ˆ + .ˆ∗)P2. (10)
With this choice, Sˆ is a diagonal matrix, and the symmetric matrix K in the same local
Cartesian coordinates as (2) takes the form:
K =

σ
C
σ
0
0 0 0
σ
H
σ
P
0
0
σ
C
σ
0
0 −σHσ
P
0 0
0 0
σ∥
σ
0
0 0 0
0 −σHσ
P
0
σ
0
σ
P
0 0
σ
H
σ
P
0 0 0
σ0
σ
P
0
0 0 0 0 0
σ
0
σ∥

.
The eigenvalues of the matrix do not change with the simultaneous permutation of rows and
corresponding columns. This allows us to reduce the matrix K to a block-diagonal form with
blocks 
.
C
.
0
.
H
.
P
.
H
.
P
.
0
.
P
 ,

.
C
.
0
−.H
.
P
−.H
.
P
.
0
.
P
 , .∥
.0
,
.
0
.∥
. (11)
– 557 –
Valery V.Denisenko, Semen A.Nesterov Energy Method for the Elliptic Boundary Value Problems . . .
The eigenvalues of the two matrices are equal. The bigger of them λmax does not exceed the
matrix trace that is equal to
.
C
.
0
+
.
0
.
P
.
Due to (3), the eigenvalues of blocks and the last two numbers (11) do not exceed
λmax 6 2
√
.2P.1. (12)
The determinants of the blocks (11) are equal to one, which means that smaller eigenvalues
λmin = 1Pλmax >
√
.1P.2P2. (13)
Since the eigenvalues are not changed when the coordinate system is rotated, the parameter
(12) for all points of the domain Ω estimates the eigenvalues of the matrix K from above, and
the inverse value from below. Therefore, the condition number of the matrix K does not exceed
4.2P.1.
By adding an auxiliary integral (9), we have not changed the value of the quadratic form (8).
Therefore, from (12), (13), the upper and the lower estimations follow:[(
F
P
)
,
(
F
P
)]
6
√
4.2
.1
∫
((gradF )2 + (rotP)2 + (divP)2) dΩ, (14)
[(
F
P
)
,
(
F
P
)]
>
√
.1
4.2
∫
((gradF )2 + (rotP)2 + (divP)2) dΩ. (15)
Now consider the functions F and P separately in order to estimate the right-hand side (15)
from below. Since the function F is equal to zero on the surface Γ (7), it satisfies the Friedrichs
inequality ∫
F 2 dΩ 6 c0
∫
(gradF )2 dΩ, (16)
where the constant c0 is determined only by the shape of the domain and does not depend on
the specific function F . Usually this inequality is formulated for functions equal to zero on the
entire boundary, however, it suffices equality to zero on a segment of finite area (the theorem
on equivalent norms [8]). For the spherical layer under consideration, it is possible to obtain a
specific value of the constant c0 by the same method, which is used for the more complex case
of vector functions in [9].
In [9] for the functions P satisfying the boundary conditions (7), the following inequalities
are proved ∫
|P|2 dΩ 6 c1
∫
|gradP|2 dΩ, (17)
c2
∫
|gradP|2 dΩ 6
∫
((rotP)2 + (divP)2) dΩ, (18)
where |gradP|2 is the sum of the squared moduli of the gradients of all Cartesian components
of P, and the constants c1, c2 are determined only by the shape of the domain Ω and do not
depend on the specific function P.
The first inequality is similar to the Friedrichs inequality for the scalar functions which are
equal to zero at the boundary. If the entire vector P is equal to zero at the boundary, for each
of its Cartesian components one can use Friedrichs inequality to obtain the required inequality.
– 558 –
Valery V.Denisenko, Semen A.Nesterov Energy Method for the Elliptic Boundary Value Problems . . .
However, of interest are the functions with only normal or only tangent components equal to
zero at the boundary. In [10], both inequalities were proved for the functions with one of these
conditions posed at the entire boundary of an arbitrary multiply connected domain. For mixed
boundary conditions (7) new proofs are required.
To restrict ourselves to simple means, in [9] the additional restrictions are imposed on the
domain shape: convexity of the surface Γ and bounded curvature of γ, and also the limited
difference of both surfaces from two spheres with a common center and not too different radii.
Let R denote the minimum radius of curvature of γ. Let the surfaces Γ and γ be defined in
spherical coordinates using the functions RΓ(θ, φ) and Rγ(θ, φ). We assume that
0 < R1 6 Rγ(θ, φ) 6 R2, 0 < RΓ(θ, φ)−Rγ(θ, φ) 6 δR,
and the constants R,R1, R2, δR will be subject to one more general restriction.
Another condition limits the angle between the boundary normal and the radial direction at
each point. Let us write it down in a convenient form as a constraint on the radial component
of the unit vector of the outer normal:
n
Γ,r > ξ1, −nγ,r > ξ1.
We also require that the scalar product of the normals n
Γ
and n
γ
, calculated on one ray is
positive:
n
Γ
(θ, φ) · nγ (θ, φ) > ξ2 Q 0.
Then the constant obtained in [9],
c2 = 1− 2R
2
2δR
ξ1ξ22R
2
1R
. (19)
The inequality (18) makes sense only for positive c2, which gives one more general constraint
on the values of geometric parameters. In mathematical modeling of the D-layer of the iono-
sphere, this condition is satisfied by a large margin: the fraction in (19) is less than 0.02. If the
boundaries γ and Γ are the spheres with radii R1 and R1+ δR, then R2 = R = R1, ξ1 = ξ2 = 1,
and the only condition 2δR < R1 is enough.
Denoting by c4 the lesser of the constants 1Pc0, c2Pc1 in the inequalities (15)–(18) we obtain
an inequality that means positive definiteness of the bilinear form (8)[(
F
P
)
,
(
F
P
)]
> c4
√
4.2
.1
∫
(F 2 + |P|2) dΩ.
The same set of inequalities gives a lower estimate for (8) in terms of the sum of the squares
of the norms of F and the Cartesian components of P as elements of the space W (1)2 (Ω).
It is easy to prove the inequality:
(rotP)2 + (divP)2 6 3
3∑
i,j=1
|gradP|2. (20)
We expand the expressions on the left, immediately replacing the products with the sums of
squares, without decreasing the value of the entire expression. Bringing similar terms, we get
the sum, in which the squares of all derivatives of all components of the vector P enter with the
coefficients 2 or 3, while on the right-hand side (20) they all appear with coefficient 3.
– 559 –
Valery V.Denisenko, Semen A.Nesterov Energy Method for the Elliptic Boundary Value Problems . . .
With (14) and (20) the energy norm is estimated in terms of the same norms of F , P from
above.
The upper and lower estimates mean the equivalence of the introduced energy norm to the
norm of the space W (1)2 (Ω). In particular, this allows for the numerical solution of the problem
to use the same approximating functions, as for the Poisson equation.
3. The energy functional
In accordance with the energy method [11], we define the energy functional:
W (F,P) =
1
2
[(
F
P
)
,
(
F
P
)]
− 1
.0
∫
FddΩ−
∫
P∗G dΩ+
∫
Γ
P∗g dΓ +
1
.0
∫
γ
Fq dγ. (21)
We use the Cauchy-Bunyakovsky inequality to estimate the linear functionals:∣∣∣ ∫ FddΩ∣∣∣2 6 ∫ F 2 dΩ ∫ d2 dΩ, ∣∣∣ ∫ P∗G dΩ∣∣∣2 6 ∫ |P|2 dΩ ∫ |G|2 dΩ, (22)
∣∣∣ ∫
γ
Fq dγ
∣∣∣2 6 ∫
γ
F 2 dγ
∫
γ
q2 dγ,
∣∣∣ ∫
Γ
P∗g dΓ
∣∣∣2 6 ∫
Γ
|P|2 dΓ
∫
Γ
|g|2 dΓ. (23)
Due to the inequalities (16), (17), the right-hand sides (22) are estimated from above by
the energy norm with some factor independent of F,P. The second factors are bounded due to
belonging of d and Cartesian components of G to the space L2(Ω), specified in the formulation
of equations (1). Therefore, the first two linear functionals are bounded.
The integral |P|2 over the boundary Γ is estimated from above in terms of the integral
|gradP|2 over the domain Ω in [9]. It is not difficult to estimate in a similar way the integral of
F 2 over the boundary γ in terms of the integral of |gradF |2. Since the functions F , P from the
energy space are elements of W (1)2 (Ω), the right-hand sides of the inequalities (23) are estimated
from above by the energy norm of the pair of F,P with some factor independent of F,P. The
second factors in (23) are bounded due to q and Cartesian components of g belonging to the
spaces L2(Γ) and L2(γ), respectively, which was as agreed in the formulation of the boundary
conditions (4). This means that the last linear functionals in (21) are also bounded.
By F.Rees’s theorem, any bounded linear functional can be represented in the form of a scalar
product by some uniquely defined element of Hilbert space. Therefore, the energy functional (21)
can be written as
W (F,P) =
1
2
[(
F
P
)
,
(
F
P
)]
−
[(
t
b
)
,
(
F
P
)]
,
where t,b is some element of the energy space.
This expression can be transformed by highlighting the square of the difference:
W (F,P) =
1
2
[(
F − t
P− b
)
,
(
F − t
P− b
)]
− 1
2
[(
t
b
)
,
(
t
b
)]
. (24)
The second term does not depend on F,P, and the first is positive definite. Therefore the
minimum of W (F,P) is attained at F = t, P = b.
Since the element t,b belongs to the energy space and is defined uniquely, it is proved that
in the energy space there exists, and the only one, an element that supplies the energy functional
with a minimum value.
– 560 –
Valery V.Denisenko, Semen A.Nesterov Energy Method for the Elliptic Boundary Value Problems . . .
4. The generalized solution
Let us introduce the notation
E = − 1
.0
Sˆ.ˆ∗gradF + SˆrotP, J = .ˆE, (25)
where F,P are the functions which provide the minimum value to the energy functional. Let us
prove that these J,E are the solution to the original boundary value problem (1), (4).
The condition for the minimality of the energy functional, taking into account the nota-
tion (25), can be written as an identity valid for arbitrary smooth functions u,v satisfying the
conditions (7):∫ (− (gradu)∗JP.0 + (rotv)∗E+ divv divP− udP.0 − v∗G) dΩ+
+
1
.0
∫
γ
uq dγ +
∫
Γ
v∗g dΓ. (26)
If we additionally assume the smoothness of all functions, we can use integration by parts,
for transformation of this identity to the form:
1
.0
∫
u(divJ−d) dΩ+
∫
v∗(rotE−G) dΩ+
∫
divv divP dΩ +
+
1
.0
∫
γ
u(−Jn + q) dγ +
∫
Γ
v∗(−Eτ + g) dΓ = 0. (27)
Take u = 0 and the function v of the form
v = gradV, V |Γ = 0, V |γ = 0. (28)
Then the identity (27) takes the form:∫
(gradV )∗(rotE−G) dΩ+
∫
div gradV divP dΩ = 0. (29)
The first integral by the Gauss–Ostrogradskii theorem is∮
Γ
V (rotE−G)n dΓ +
∮
γ
V (rotE−G)n dγ −
∫
V (div rotE+ divG) dΩ.
The integrands are equal to zero, since V is equal to zero at the boundaries Γ, γ due to (28),
div rot is identically zero, and the function G satisfies the condition (5).
Therefore, from (29) we obtain the identity∫
div gradV divP dΩ = 0,
which is valid for any function V equal to zero on the boundary.
This identity allows us to prove the equality to zero divP in the usual way. Assuming the
opposite, take a point in Ω, where divP ̸= 0, and its neighborhood, where divP is sign-preserving.
We construct V as a solution to the Dirichlet problem for the Poisson equation with the right-
hand side, equal to 1 in the selected neighborhood, and zero for the rest of Ω. Substituting such
a function V , we obtain the nonzero value of the last integral, which contradicts the identity.
– 561 –
Valery V.Denisenko, Semen A.Nesterov Energy Method for the Elliptic Boundary Value Problems . . .
Proving that
divP = 0, (30)
the third integral in (27) can be eliminated. After that, using the arbitrariness of the functions
u,v, it is easy to prove that all factors for u,v are equal to zero. Since E and J are used in the
present section only for the abbreviated notation of the expression (25), the following equations
are fulfilled:
div
(
− 1
.20
.ˆSˆ.ˆ∗gradF +
1
.0
.ˆSrotP
)
= dP.0,
rot
(
− 1
.0
Sˆ.ˆ∗gradF + SˆrotP
)
= G, (31)
(
− 1
.20
.ˆSˆ.ˆ∗gradF +
1
.0
.ˆSˆ rotP
)
n
∣∣∣∣
γ
= q, (32)
(
− 1
.0
Sˆ.ˆ∗gradF + Sˆ rotP
)
τ
∣∣∣∣
Γ
= g. (33)
The converse statement is also easy to prove: the solution of the problem (30)–(33), (7) gives
the minimum value in the energy space to the energy functional. Indeed, let F,P be a solution
to this boundary value problem. Let us write down the energy functional for the sum of the
functions F + tu, P + tv, where t — an arbitrary number, u,v are smooth functions satisfying
the conditions (7):
W (F + tu,P+ tv) =W (F,P) +
t2
2
[(
u
v
)
,
(
u
v
)]
+ t
([(
u
v
)
,
(
F
P
)]
−
−
∫
(udP.0 + v
∗G) dΩ+
∫
γ
uqP.0 dγ +
∫
Γ
v∗g dΓ
)
. (34)
Since F,P is a solution to the boundary value problem, the identity (27) is satisfied for
arbitrary u,v. By integrating by parts from (27) we obtain the identity (26), which means
equality to zero of the factor of t in the square trinomial (34).
The coefficient of t2 is positive, since the positive definiteness of the energy quadratic form
has been proved, and we are interested in u,v, which are not identically zero. Therefore
W (F + tu,P+ tv) > W (F,P),
and equality is obtained only for t = 0, that is, the minimum of W is attained at the element
F,P.
As can be seen from (27), the constructed functions E and J satisfy equations and boundary
conditions of the original boundary value problem (1), (4). As already noted, the uniqueness of
the solution is proved in almost the same way as in [3].
Thus, we have proposed a new formulation of the problem (30)–(33), (7), which, unlike the
original problem, has a symmetric positive definite operator.
At the beginning of the section, an additional smoothness assumption was made. If this
condition is not satisfied, the identity (26) cannot be transformed into (27). In this case, the
pair of functions F,P, providing the energy functional the minimum value, we consider as the
generalized solution to the problem (30)–(33), (7), and the pair E, J constructed from them by
formulae (25) is the generalized solution of the original problem (1), (4).
– 562 –
Valery V.Denisenko, Semen A.Nesterov Energy Method for the Elliptic Boundary Value Problems . . .
The existence and uniqueness of the generalized solution of the problem (30)–(33), (7) have
been proved, therefore the existence of the generalized solution of the original boundary value
problem (1), (4) is proved, and the principle of the minimum of the energy functional is substan-
tiated for it.
5. Other boundary value problems
Consider three problems which differ from the main problem by the boundary conditions. In
the first problem, we swap the parts of the boundary γ and Γ in the boundary conditions (4).
Then, naturally, the condition of the form (6), that is necessary for the solvability of this first
problem, is moved to γ.
As described in [3] and demonstrated above when constructing the generalized solution (26),
the conditions for the set of pairs of smooth functions F , P, are the conjugate ones to the
boundary conditions of the original problem. Therefore, in the conditions (7) and in the energy
functional, all changes are limited to the permutation of γ and Γ. Note that the value of the
constant in the inequalities (17), (18) [9] varies.
As the second boundary value problem, consider the domain Ω surrounded by an insulator.
Then the first of the conditions (4) is set on both sections of the boundary γ and Γ. From
the charge conservation law, integrated over the entire domain, a necessary condition for the
solvability of this problem arises:∫
ddΩ−
∫
γ
q dγ −
∫
Γ
q dΓ = 0. (35)
The conditions for the pairs of functions F , P (7) take the form
Pτ |Γ= 0, Pτ |γ= 0,
∫
F dΩ = 0. (36)
The last condition is the conjugate one to the condition (35). It has to be added, since
without the equality F = 0 on the boundary that was in (7), the Friedrichs inequality cannot
be used for the function F . The inequality (16) with a different constant is now fulfilled as
Poincare’s inequality. The condition in question can be replaced by fixing some other average
value of the function F , for example, the average over γ or Γ. Then the inequality (16) will
be true as a consequence of the equivalent normalization theorem [8]. In fact, such a condition
should eliminate the ambiguity of the solution corresponding to the addition of an arbitrary
constant to F . Inequalities (17), (18) for the vector functions P whose tangent components are
equal to zero on the whole boundary (36), were proved in [10] for bounded multiply connected
domains of general form, however, without specifying the values of the constants.
The third boundary value problem describes the domain Ω surrounded by an ideal conductor.
Then, on both sections of the boundary γ and Γ, the second of the conditions (4) is set. In
contrast to the second boundary value problem, the additional solvability condition does not
arise. It would arise if the domain Ω is, for example, a torus, in which this problem has a nonzero
solution for zero right-hand sides. For isotropic constant conductivity in an axisymmetric torus,
this would be an azimuthal electric field, not changing with displacement in the direction of
the axis of symmetry and decreasing in inverse proportion to the distance to the axis, and the
current density proportional to it. Of course, the nonzero Joule dissipation of such a current
system must be compensated by some external sources. In this example, the energy source is a
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Valery V.Denisenko and Semen A.Nesterov Energy Method for the Elliptic Boundary Value Problems...
time-varying magnetic field in the domain encompassed by this torus. For multiply connected
domains of general form, the ambiguity of solutions to similar problems is analyzed in [10].
The conditions for the set of pairs of functions F , P in the third boundary value problem
have the form
F |γ,Γ= 0, cn |γ,Γ= 0.
The inequality (16) for the function F remains the Friedrichs inequality, and for the vector
functions P with zero normal component on the entire boundary the inequalities (17), (18) are
also proved in [10].
Thus, for three boundary value problems, new formulations of problems are proposed, which,
in contrast to the original problems, have symmetric positive definite operators, and the principles
of the minimum of energy functionals are justified for them.
Taking into account the notation (25), with the tensor Sˆ chosen in accordance with (10), the
quadratic form, corresponding to the energy scalar product (8) can be written in terms of the
original electric current continuity problem (1), (4):
1
.0
∫ (
E∗J+ (divP)2
)
dΩ. (37)
The product E∗J is the Joule dissipation density, that is, heat release accompanying the
passage of the electric current. Since the minimization of the energy functional means satisfaction
of the equality (30), the quadratic form (37) is equal to the total Joule dissipation in the domain
Ω, which justifies the name "energy". Thus, the introduced energy norm makes sense from the
point of view of nonequilibrium thermodynamics.
This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry
of Science and Higher Education of the Russian Federation in the framework of the establish-
ment and development of regional Centers for Mathematics Research and Education (Agreement
no. 075-02-2020-1631).
References
[1] V.V.Denisenko, Statements of the boundary value problems in mathematical simulation
of a quasistationary electric field in the atmosphere and ionosphere, Journal of Physics:
Conference Series, 1715(2021), no. 012016, 1–10. DOI: 10.1088/1742-6596/1715/1/012016
[2] V.V.Denisenko, M.J.Rycroft, R.G.Harrison, Mathematical Simulation of the Ionospheric
Electric Field as a Part of the Global Electric Circuit, Surveys in Geophysics, 40(1)(2019),
1–35.
[3] V.V.Denisenko, The energy method for three dimensional elliptical equations with asym-
metric tensor coefficients, Siberian Mathematical Journal, 38(6)(1997), 1099–1111.
[4] V.V.Denisenko, Energy method in problems of transfer in media moving in multiply
connected domains, Russian Journal of Numerical Analysis and Mathematical Modelling,
15(2)(2000), 127–143.
[5] V.V.Denisenko, Symmetric operators for transfer problems in three-dimensional moving
media, Siberian Journal of Industrial Mathematics, 4(2001), no. 1, 73–82.
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[6] J.K.Hargreaves, The Upper Atmosphere and Solar-terrestrial Relations, New York, Van
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[7] A.N.Konovalov, Conjugate-factorized models in mathematical physics problems, Novosi-
birsk, Computing Center SB RAS, Preprint 1095, 1997 (in Russian).
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[9] V.V.Denisenko, S.A.Nesterov, Inequalities for the norms of vector functions in a spherical
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[10] E.B.Bykhovsky, N.V.Smirnov, About the orthogonal decomposition of the space of the
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Энергетический метод для эллиптических краевых задач
с несимметричными операторами в шаровом слое
Валерий В.Денисенко
Семен А.Нестеров
Институт вычислительного моделирования СО РАН
Красноярск, Российская Федерация
Аннотация. Рассмотрены трехмерные эллиптические краевые задачи, возникающие при матема-
тическом моделировании квазистационарных электрических полей и токов в проводниках с гиро-
тропным тензором проводимости в областях, гомеоморфных шаровому слою. Аналогичные задачи
формулируются при моделировании теплопроводности или диффузии в движущихся или гиро-
тропных средах. Операторы задач в традиционной формулировке являются несимметричными.
Предложены новые формулировки задач с симметричными положительно определенными опера-
торами. Для четырех краевых задач построены квадратичные функционалы энергии, к миними-
зации которых сведено решение этих задач. Выполнены оценки полученных квадратичных форм
в сравнении с формой, фигурирующей в принципе Дирихле для уравнения Пуассона.
Ключевые слова: математическое моделирование, энергетический метод, эллиптическое уравне-
ние, несимметричный оператор.
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