# On the Existence and Uniqueness ofRv-Generalized Solution for Dirichlet Problem with Singularity on All Boundary

Authors: V. Rukavishnikov; E. Rukavishnikova
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The existence and uniqueness of theRv-generalized solution for the first boundary value problem and a second order elliptic equation with coordinated and uncoordinated degeneracy of input data and with strong singularity solution on all boundary of a two-dimensional domain are established.


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## Body

## 1. Introduction

The singularity of solution for boundary value problems to two-dimensional closed domain can be due to the degeneration of the input data (coefficients and right-hand sides of equations and boundary conditions), availability of the reentrant corners, and change of the kind of the boundary conditions or by the internal properties of the solution. A boundary value problem is said to possess strong singularity if its solutionu(x) does not belong to Sobolev space W21 (H1) or, in other words, the Dirichlet integral of the solution u(x) diverges. In the case if the solution belongs to the space W21 (H1) but does not belong to the space W22 (H2), a boundary value problem is called the problem with a weak singularity.Boundary value problems with strong singularity are found in the physics of plasma and gas discharge, electrodynamics, nuclear physics, nonlinear optics, and other branches of physics. In particular cases, numerical methods for problems of electrodynamics and quantum mechanics with strong singularity were constructed, based on separation of singular and regular components, mesh refinement near singular points, multiplicative extraction of singularities, and so forth, (see, e.g., [1–6]).In [7], it was suggested to define the solution of boundary value problem for second-order elliptic equation with singularity on a finite set of points belonging to boundary of a two-dimensional domain as an Rν-generalized solution in the weighted Sobolev space. Such a new concept of solution led to the distinction of two classes of boundary value problems: problems with coordinated and uncoordinated degeneracy of input data; it also made it possible to study the existence and uniqueness of solutions as well as its coercivity and differential properties in the weighted Sobolev spaces (see [8, 9]).For boundary value problems for elliptic equations, Maxwell equations and Lamé system, we constructed the numerical methods with rate of convergence independent of the singularity based on the concept of anRν-generalized solution (see, e.g., [10–12]).In this paper, we consider the first boundary value problem for a second-order elliptic equation with strong singularity solution on all boundary of a two-dimensional domain. We distinguish two classes of the boundary value problems: problems with coordinated and uncoordinated degeneracy of input data. For this problem we define the solution as anRν-generalized one in a weighted Sobolev space H2,ν+β/21(Ω) and in a weighted set W2,ν+β/21(Ω,δ), respectively. We prove its existence and uniqueness in the corresponding weighted space and weighted set. It was established that, for all values of parameter ν for which the Rν-generalized solution exists, it is unique for all of these parameters.


## 2. Notation and Auxiliary Statements

We denote the two-dimensional Euclidean space byR2 with x=(x1,x2) and dx=dx1dx2. Let Ω⊂R2 be a bounded domain with sufficiently smooth boundary ∂Ω, and let Ω¯ be the closure of Ω; that is, Ω¯=Ω∪∂Ω. We denote by Ω′ the adjoining streak of the boundary ∂Ω of width δ>0 and Ω′⊂Ω.We introduce a weight functionρ(x) that coincides in Ω′ with the distance from point x to the boundary ∂Ω and is equal to δ for x∈Ω¯∖Ω′.LetH2,αk(Ω) and W2,αk(Ω) be the weighted spaces with norms:
(1)∥u∥H2,αk(Ω)=(∑|λ|≤k‍∫Ω‍ρ2(α+|λ|-k)|Dλu|2dx)1/2,(2)∥u∥W2,αk(Ω)=(∑|λ|≤k‍∫Ω‍ρ2α|Dλu|2dx)1/2,
where Dλ=∂|λ|/∂x1λ1∂x2λ2, λ=(λ1,λ2), and |λ|=λ1+λ2; λ1,λ2 are integer nonnegative numbers, α is some real nonnegative number, and k is an integer nonnegative number. For k=0 we use the notation H2,α0(Ω)=W2,α0(Ω)=L2,α(Ω).ByW2,α1(Ω,δ) for α>0, we denote a set of functions satisfying the following conditions:(a)
|Dku(x)|≤c1(δ/ρ(x))α+k for x∈Ω′, where k=0,1, c1 is positive constant independent of k,(b)
∥u∥L2,α(Ω∖Ω′)≥c2>0,and with the norm (2).The spacesH°2,αk(Ω)⊂H2,αk(Ω) and W°2,αk(Ω)⊂W2,αk(Ω) and the set W°2,α1(Ω,δ)⊂W2,α1(Ω,δ) are defined as the closures of the set of infinitely differentiable and finite in Ω functions in norms (1) and (2), respectively.LetH∞,-αk(Ω,c3)(k≥0,α∈R) be the set of functions with the norm satisfying the inequality
(3)∥u∥H∞,-αk(Ω,c3)=max⁡|λ|≤kesssup⁡x∈Ω|ρ-α+|λ|Dλu|≤c3,
with a positive constant c3 independent of u. For k=0, we have H∞,-α0(Ω,c4)=L∞,-α(Ω,c4).Lemma 1.
For each functionu in the set W2,α1(Ω,δ) and for any α*>α, the estimate
(4)∥u∥L2,α*-1(Ω′,δ)≤c5∥u∥L2,α*(Ω,δ)
holds, where c5=c6(δα/α*-α), c6=const>0.Proof.
Taking into account condition (a), one can show that, forα*>α, we have
(5)∥u∥L2,α*-1(Ω′,δ)2=∫Ω′ρ2(α*-1)u2dx≤c12δ2α∫Ω′‍ρ2(α*-1)ρ-2αdx≤c12δ2αc7δ2(α*-α)2(α*-α),
where c7 is a constant dependent of mesΩ′. Considering condition (b), we write the inequality for the function u as follows:
(6)∥u∥L2,α*(Ω)2≥∥u∥L2,α*(Ω∖Ω′)2=δ2(α*-α)∥u∥L2,α(Ω∖Ω′)2≥c22δ2(α*-α).
From inequalities (5) and (6) we get the estimate (4) with c6=(c1/c2)c7/2.


## 3. The Boundary Value Problem with Coordinated Degeneration of the Input Data on All Boundary of the Domain

In the domainΩ, we consider the differential equation
(7)-∑k,l=12‍akl(x)∂2u∂xk∂xl+∑k=12‍ak(x)∂u∂xk+a(x)u=f(x),x∈Ω,
with the boundary condition
(8)u=0,x∈∂Ω.Definition 2.
The boundary value problem (7) and (8) is called the Dirichlet problem with coordinated degeneration of the input data on all boundary of the domain or Problem A, if akl(x)=alk(x)    (k,l=1,2) and, for some real number β,
(9)akl∈H∞,-β1(Ω,c8),ak∈L∞,-(β-1)(Ω,c9),(k,l=1,2),a∈L∞,-(β-2)(Ω,c10),(10)∑k,l=12‍akl(x)ξkξl≥c11ρβ(x)∑k=12‍ξk2,(11)a(x)>c12ρβ-2(x)almosteverywhereon⁡Ω
and right-hand side of (7) satisfies
(12)f∈L2,μ(Ω),
where ci (i=8,…,12) are positive constants independent of x; ξ1 and ξ2 are any real parameters; μ is some nonnegative real number.Denote by(13)a(u,v)=∫Ω‍[∑k,l=12aklρ2ν∂u∂xk∂v∂xl+akl∂ρ2ν∂xk∂u∂xlv‍+∂akl∂xkρ2ν∂u∂xlv+akρ2ν∂u∂xkv+aρ2νuv∑k,l=12aklρ2ν∂u∂xk∂v∂xl+akl∂ρ2ν∂xk∂u∂xlv‍]dx,f(v)=∫Ω‍ρ2νfvdx
the bilinear and linear forms, respectively.Definition 3.
A functionuν from the space H°2,ν+β/21(Ω) is called an Rν-generalized solution of the Dirichlet problem with coordinated degeneration of the input data on all boundary of the domain or Problem A, if, for any v in H°2,ν+β/21(Ω), the identity
(14)a(uν,v)=f(v)
holds, where ν is arbitrary but fixed and satisfies the inequality
(15)ν≥μ+β2-1.For Problem A, we prove the main result.Theorem 4.
Let conditions (9)–(12) and (15) hold and let
(16)2(c8(2|ν|+1)+12c9)2<c11c12
be satisfied.
Then, theRν-generalized solution uν of the Dirichlet problem with coordinated degeneration of the input data on all boundary of the domain exists and is unique in the space H°2,ν+β/21(Ω) and the following estimate is valid:
(17)∥uν∥H2,ν+β/21(Ω)≤c13∥f∥L2,μ(Ω),
where c13 is a positive constant not depending on uν and f.Proof.
First, we show that the formsa(u,v) and f(v) are continuous on H°2,ν+β/21(Ω). In fact, by virtue of conditions (9), (12), and (15) and the Cauchy-Schwarz inequality, we have
(18)|a(u,v)|≤∫Ω‍|∑k,l=12‍aklρ2ν∂u∂xk∂v∂xl+∂akl∂xkρ2ν∂u∂xlv+akρ2ν∂u∂xkv+aρ2νuv∑k,l=12‍|dx+∫Ω′‍|akl∂ρ2ν∂xk∂u∂xlv|dx≤2c8(∫Ω‍ρ2ν+β∑k=12‍(∂u∂xk)2dx)1/2×(∫Ω‍ρ2ν+β∑k=12‍(∂v∂xk)2dx)1/2+22c8(∫Ω‍ρ2ν+β∑l=12‍(∂u∂xl)2dx)1/2×(∫Ω‍ρ2ν+β-2v2dx)1/2+2c9(∫Ω‍ρ2ν+β∑k=12‍(∂u∂xk)2dx)1/2×(∫Ω‍ρ2ν+β-2v2dx)1/2+c10(∫Ω‍ρ2ν+β-2u2dx)1/2×(∫Ω‍ρ2ν+β-2v2dx)1/2+42|ν|c8(∫Ω′‍ρ2ν+β∑l=12‍(∂u∂xl)2dx)1/2×(∫Ω′‍ρ2ν+β-2v2dx)1/2≤2c8|u|H2,ν+β/21(Ω)|v|H2,ν+β/21(Ω)+(22c8+2c9)|u|H2,ν+β/21(Ω)∥v∥L2,ν+β/2-1(Ω)+42|ν|c8|u|H2,ν+β/21(Ω′)∥v∥L2,ν+β/2-1(Ω′)+c10∥u∥L2,ν+β/2-1(Ω)∥v∥L2,ν+β/2-1(Ω),
or
(19)|a(u,v)|≤c14∥u∥H2,ν+β/21(Ω)∥v∥H2,ν+β/21(Ω),∀u,v∈H°2,ν+β/21(Ω)(20)|f(v)|≤(max⁡x∈Ωρ2ν-2μ-β+2)1/2(∫Ω‍ρ2μf2dx)1/2×(∫Ω‍ρ2ν+β-2v2dx)1/2≤c15∥f∥L2,μ(Ω)∥v∥H2,ν+β/21(Ω),(∫Ω‍ρ2ν+β-2v2dx)12∀v∈H°2,ν+β/21(Ω),∀f∈L2,μ(Ω).
Let us now prove theH°2,ν+β/21(Ω) ellipticity of the bilinear form a(u,v); that is,
(21)∃c16>0,∀u∈H°2,ν+β/21(Ω),a(u,u)≥c16∥u∥H2,ν+β/21(Ω)2.
We substitute v by u in (13), and by means of the Cauchy-Schwarz inequality, ɛ-inequality, and conditions (9), we estimate the absolute values of the second, third, and fourth terms of the form a(u,u):
(22)|∑k,l=12‍∫Ω‍akl∂ρ2ν∂xk∂u∂xludx|≤|∑k,l=12‍∫Ω‍akl2νρ2ν-1∂u∂xludx|≤22c8|ν|(ɛ1∑l=12‍∫Ω′‍ρ2ν+β(∂u∂xl)2dx+1ɛ1∫Ω′ρ2ν+β-2u2dxɛ1∑l=12‍∫Ω′‍ρ2ν+β(∂u∂xl)2dx),(23)|∑k,l=12‍∫Ω‍∂akl∂xkρ2ν∂u∂xludx|≤2c8(ɛ2∑l=12‍∫Ω‍ρ2ν+β(∂u∂xl)2dx+1ɛ2∫Ωρ2ν+β-2u2dxɛ2∑l=12‍∫Ω‍ρ2ν+β(∂u∂xl)2dx),(24)|∑k=12‍∫Ω‍akρ2ν∂u∂xkudx|≤2c9(ɛ32∑k=12‍∫Ω‍ρ2ν+β(∂u∂xk)2dxi+12ɛ3∫Ω‍ρ2ν+β-2u2dxɛ32∑k=12‍∫Ω‍ρ2ν+β(∂u∂xk)2dx).
Here, ɛ1, ɛ2, and ɛ3 are any positive numbers.
Using (10) and (11), we have
(25)∑k,l=12‍∫Ω‍aklρ2ν∂u∂xk∂u∂xldx≥c11∫Ω‍ρ2ν+β∑k=12‍(∂u∂xk)2dx,(26)∫Ω‍aρ2νu2dx≥c12∫Ω‍ρ2ν+β-2u2dx.
Then, from (22)–(26), we obtain
(27)a(u,u)≥(c11-2(c82|ν|ɛ1+c8ɛ2+12c9ɛ3))×∫Ω‍ρ2ν+β∑k=12‍(∂u∂xk)2dx+(c12-2(c82|ν|ɛ1-1+c8ɛ2-1+12c9ɛ3-1))×∫Ω‍ρ2ν+β-2u2dx.
Note that if condition (16) is satisfied, then there exists a positive constant ɛ such that
(28)c11-2(c8(2|ν|+1)+12c9)ɛ>0,c12-2(c8(2|ν|+1)+12c9)ɛ-1>0.
Supposing that ɛ1=ɛ2=ɛ3=ɛ in (27), we get
(29)a(u,u)≥c16(∫Ω‍ρ2ν+β∑k=12‍(∂u∂xk)2dx+∫Ω‍ρ2ν+β-2u2dx∫Ω‍ρ2ν+β∑k=12‍(∂u∂xk)2dx)=c16∥u∥H2,ν+β/21(Ω)
with constant
(30)c16=min⁡(c11-2(c8(2|ν|+1)+12c9)ɛ,c12-2(c8(2|ν|+1)+12c9)ɛ-1).
According to (19), (21), and (20), the bilinear form a(u,v) is continuous and H°2,ν+β/21(Ω)-elliptical, and the linear form f(v) is continuous on H°2,ν+β/21(Ω); then, the existence and uniqueness of an Rν-generalized solution of Problem A follow from the Lax-Milgram theorem (see [13]).
Taking into account that(31)c16∥uν∥H2,ν+β/21(Ω)2≤a(uν,uν)=f(v)≤c15∥f∥L2,μ(Ω)∥u∥H2,ν+β/21(Ω),
we get estimate (17).Corollary 5.
If there exists at least oneν for which there exists a unique Rν-generalized solution of the Problem A, then one can always define a half-open interval [ν1,ν2) such that, for each ν∈[ν1,ν2), there exists a unique Rν-generalized solution. Here,
(32)ν1=max⁡{μ+β2-1;(1-(c11c12)1/2-c9/2c8)+ɛ},ν2=(c11c12)1/2-c9/2c8-1,
where ɛ is a given sufficiently small positive number.Corollary follows from the proof of Theorem4.Theorem 6.
If the assumptions of Theorem4 are valid, then, for all ν in the interval [ν1,ν2), the Rν-generalized solution of the Problem A is unique.The proof of this statement is similar to that of Theorem 2 in [14].


## 4. The Boundary Value Problem with Uncoordinated Degeneration of the Input Data on All Boundary of the Domain

We consider the boundary value problem(33)-∑k=12‍∂∂xk(akk(x)∂u∂xk)+a(x)u=f(x),x∈Ω,(34)u=0,x∈∂Ω.Definition 7.
The boundary value problem (33) and (34) is called the Dirichlet problem with uncoordinated degeneration of the input data on all boundary of the domain or Problem B, if, for some real number β,
(35)akk∈H∞,-β1(Ω,c16),a∈L∞,-β(Ω,c17),(36)∑k=12‍akk(x)ξk2≥c18ρβ(x)∑k=12‍ξk2,(37)a(x)≥c19ρβ(x)almosteverywhereon⁡Ω,
and the right-hand side of the equation satisfies the condition
(38)f∈L2,μ(Ω,δ)
for some nonnegative real number μ. Here, ci, i=16,…,19, are positive constants not depending on x; ξ1 and ξ2 are arbitrary real parameters.Set(39)b(u,v)=∫Ω‍[∑k=12‍akkρ2ν∂u∂xk∂v∂xk+akk∂ρ2ν∂xk∂u∂xkv+aρ2νuv]dx,l(v)=∫Ω‍ρ2νfvdx.Definition 8.
A functionuν from the set W°2,ν+β/21(Ω,δ) is called an Rν-generalized solution of the Problem B if the identity b(uν,v)=l(v) holds for all v∈W°2,ν+β/21(Ω,δ) and for any given value of ν satisfying the inequality
(40)ν≥μ+β2.Theorem 9.
Let conditions (35)–(40) hold and
(41)ν+β2>0.
Then, for any ν satisfying conditions (40) and (41), there always exists parameter δ such that Rν-generalized solution uν of the Dirichlet problem with uncoordinated degeneration of the input data on all boundary of the domain exists and is unique in the set W°2,ν+β/21(Ω,δ). In this case, the following estimate is valid:
(42)∥uν∥W2,ν+β/21(Ω,δ)≤c20∥f∥L2,μ(Ω,δ),
where c20 is a positive constant independent of f and uν.Proof.
First of all, we note that the bilinear and linear forms are continuous on the setW°2,ν+β/21(Ω,δ) and the inequalities
(43)b(u,v)≤c21∥u∥W2,ν+β/21(Ω,δ)∥v∥W2,ν+β/21(Ω,δ),(44)l(v)≤c22∥f∥L2,μ(Ω,δ)∥v∥W2,ν+β/21(Ω,δ)
hold. The proofs of estimates (43) and (44) are established by analogy with (19) and (20), which we obtain by using conditions (35), (38), and (40) and Lemma 1.
Let us show that the bilinear form isW°2,ν+β/21-elliptical in Ω. We have
(45)b(u,u)=∑k=12‍∫Ω‍[akkρ2ν(∂u∂xk)2+akk∂ρ2ν∂xk∂u∂xku]dx+∫Ω‍aρ2νu2dx
for any u from W°2,ν+β/21(Ω,δ). By means of condition (35), we estimate the absolute value of the second term on the right-hand side in (45):
(46)|∑k=12‍∫Ω‍akk∂ρ2ν∂xk∂u∂xkudx|≤|∑k=12‍∫Ω′‍akk2νρ2ν-1∂u∂xkudx|≤ɛ∑k=12‍∫Ω′‍ρ2ν+β(∂u∂xk)2dx+2c162ν2ɛ∫Ω′‍ρ2ν+β-1u2dx.
From (45) and the last inequality we get
(47)b(u,u)≥∑k=12‍∫Ω‍akkρ2ν(∂u∂xk)2dx-ɛ∑k=12‍∫Ω′‍ρ2ν+β(∂u∂xk)2dx+∫Ω‍aρ2νu2dx-2c162ν2ɛ∫Ω′‍ρ2ν+β-1u2dx.
Supposing that α* and α equal ν and ν/2 in Lemma 1, respectively, we have
(48)∥u∥L2,ν+β/2-1(Ω′,δ)2≤2c62δνν∥u∥L2,ν+β/2(Ω,δ)2,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiic6=const>0.
Taking into account (36), (37), and (48), from estimate (47) we get
(49)b(u,u)≥(c18-ɛ)|u|W2,ν+β/21(Ω,δ)2+(c19-4c162c62νδνɛ)×∥u∥L2,ν+β/2(Ω,δ)2.
Obviously, we can always choose ɛ and δ such that the constants c18>ɛ, c19>4c162c62νδν/ɛ and the inequality
(50)b(u,u)≥c23∥u∥W2,ν+β/21(Ω,δ)2
are valid with constant c23=min⁡(c18-ɛ,c19-4c162c62νδν/ɛ). Therefore, bilinear form b(u,u) is W°2,ν+β/21-elliptical.
According to (43), (44), and (50), the bilinear form b(u,u) is continuous and W°2,ν+β/21-elliptical, and the linear form l(v) is continuous on W°2,ν+β/21(Ω,δ); then, the existence and uniqueness of an Rν-generalized solution of Problem B follow from the Lax-Milgram theorem (see [13]).
Taking into account that(51)c23∥uν∥W2,ν+β/21(Ω,δ)2≤b(uν,uν)=l(uν)≤c22∥uν∥W2,ν+β/21(Ω,δ)∥f∥L2,μ(Ω,δ),
we get estimate (42).Theorem 10.
If for someδ there is a set of values ν such that an Rν-generalized solution of the Problem B exists in the set W°2,ν+β/21(Ω,δ), then this solution is unique for all such ν.The proof of this statement is similar to that of Theorem 2 in [14].



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