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

Abstract

The existence and uniqueness of the 𝑅_𝑣-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.

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 solution 𝑢(𝑥) does not belong to Sobolev space 𝑊¹₂ (𝐻¹) or, in other words, the Dirichlet integral of the solution 𝑢(𝑥) diverges. In the case if the solution belongs to the space 𝑊¹₂ (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 𝑅_𝜈-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 an 𝑅_𝜈-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 an 𝑅_𝜈-generalized one in a weighted Sobolev space 𝐻¹_{2,𝜈+𝛽/2}(Ω) and in a weighted set 𝑊¹_{2,𝜈+𝛽/2}(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 𝑅_𝜈-generalized solution exists, it is unique for all of these parameters.

2. Notation and Auxiliary Statements

We denote the two-dimensional Euclidean space by ℝ² 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 function 𝜌(𝑥) that coincides in Ω^′ with the distance from point 𝑥 to the boundary 𝜕Ω and is equal to 𝛿 for 𝑥∈Ω⧵Ω^′.

Let 𝐻^𝑘_2,𝛼(Ω) and 𝑊^𝑘_2,𝛼(Ω) be the weighted spaces with norms: ‖𝑢‖_𝐻^𝑘_2,𝛼(Ω)=_{|𝜆|≤𝑘Ω}𝜌^{2(𝛼+|𝜆|−𝑘)}𝐷^𝜆𝑢²𝑑𝑥^1/2,‖𝑢‖_𝑊^𝑘_2,𝛼(Ω)=_{|𝜆|≤𝑘Ω}𝜌^2𝛼𝐷^𝜆𝑢²𝑑𝑥^1/2, (1)(2) where 𝐷^𝜆=𝜕^|𝜆|/𝜕𝑥^𝜆1₁𝜕𝑥^𝜆2₂, 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 𝐻⁰_2,𝛼(Ω)=𝑊⁰_2,𝛼(Ω)=𝐿_2,𝛼(Ω).

By 𝑊¹_2,𝛼(Ω,𝛿) for 𝛼>0, we denote a set of functions satisfying the following conditions:(a)|𝐷^𝑘𝑢(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 spaces 𝐻^∘𝑘_2,𝛼(Ω)⊂𝐻^𝑘_2,𝛼(Ω) and 𝑊^∘𝑘_2,𝛼(Ω)⊂𝑊^𝑘_2,𝛼(Ω) and the set 𝑊^∘1_2,𝛼(Ω,𝛿)⊂𝑊¹_2,𝛼(Ω,𝛿) are defined as the closures of the set of infinitely differentiable and finite in Ω functions in norms (1) and (2), respectively.

Let 𝐻^𝑘_∞,−𝛼(Ω,𝑐₃)(𝑘≥0,𝛼∈𝑅) be the set of functions with the norm satisfying the inequality ‖𝑢‖_𝐻^𝑘_{∞,−𝛼(Ω,𝑐3)}=max_{|𝜆|≤𝑘}esssup_𝑥∈Ω𝜌^{−𝛼+|𝜆|}𝐷^𝜆𝑢≤𝑐₃, (3) with a positive constant 𝑐₃ independent of 𝑢. For 𝑘=0, we have 𝐻⁰_∞,−𝛼(Ω,𝑐₄)=𝐿_∞,−𝛼(Ω,𝑐₄).

Lemma 1. For each function 𝑢 in the set 𝑊¹_2,𝛼(Ω,𝛿) and for any 𝛼^∗>𝛼, the estimate ‖𝑢‖_{𝐿∗2,𝛼−1(Ω′,𝛿)}≤𝑐₅‖𝑢‖_{𝐿∗2,𝛼(Ω,𝛿)} (4) holds, where 𝑐₅=𝑐₆(𝛿^𝛼/√𝛼^∗−𝛼), 𝑐₆=𝑐𝑜𝑛𝑠𝑡>0.

Proof. Taking into account condition (a), one can show that, for 𝛼^∗>𝛼, we have ‖𝑢‖²_{𝐿∗2,𝛼−1(Ω′,𝛿)}=_Ω′𝜌^{2(𝛼∗−1)}𝑢²𝑑𝑥≤𝑐²₁𝛿^2𝛼_Ω′𝜌^{2(𝛼∗−1)}𝜌^−2𝛼≤𝑐𝑑𝑥₂₁𝛿^2𝛼𝑐₇𝛿^{2(𝛼∗−𝛼)}2(𝛼^∗),−𝛼 (5) where 𝑐₇ is a constant dependent of mesΩ^′. Considering condition (b), we write the inequality for the function 𝑢 as follows: ‖𝑢‖²_{𝐿∗2,𝛼(Ω)}≥‖𝑢‖²_{𝐿∗2,𝛼(Ω⧵Ω′)}=𝛿^{2(𝛼∗−𝛼)}‖𝑢‖²_{𝐿2,𝛼(Ω⧵Ω′)}≥𝑐²₂𝛿^{2(𝛼∗−𝛼)}. (6) 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 domain Ω, we consider the differential equation −²_𝑘,𝑙=1𝑎_𝑘𝑙(𝑥)𝜕²𝑢𝜕𝑥_𝑘𝜕𝑥_𝑙+²_𝑘=1𝑎_𝑘(𝑥)𝜕𝑢𝜕𝑥_𝑘+𝑎(𝑥)𝑢=𝑓(𝑥),𝑥∈Ω, (7) with the boundary condition 𝑢=0,𝑥∈𝜕Ω. (8)

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 𝑎_𝑘𝑙(𝑥)=𝑎_𝑙𝑘(𝑥)    (𝑘,𝑙=1,2) and, for some real number 𝛽, 𝑎_𝑘𝑙∈𝐻¹_∞,−𝛽Ω,𝑐₈,𝑎_𝑘∈𝐿_{∞,−(𝛽−1)}Ω,𝑐₉,(𝑘,𝑙=1,2),𝑎∈𝐿_{∞,−(𝛽−2)}Ω,𝑐₁₀,_2𝑘,𝑙=1𝑎_𝑘𝑙(𝑥)𝜉_𝑘𝜉_𝑙≥𝑐₁₁𝜌^𝛽(𝑥)²_𝑘=1𝜉²_𝑘,𝑎(𝑥)>𝑐₁₂𝜌^𝛽−2(𝑥)almosteverywhereonΩ (9)(10)(11) and right-hand side of (7) satisfies 𝑓∈𝐿_2,𝜇(Ω), (12) where 𝑐_𝑖 (𝑖=8,…,12) are positive constants independent of 𝑥; 𝜉₁ and 𝜉₂ are any real parameters; 𝜇 is some nonnegative real number.

Denote by 𝑎(𝑢,𝑣)=_Ω²_𝑘,𝑙=1𝑎_𝑘𝑙𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑘𝜕𝑣𝜕𝑥_𝑙+𝑎_𝑘𝑙𝜕𝜌^2𝜈𝜕𝑥_𝑘𝜕𝑢𝜕𝑥_𝑙𝑣+𝜕𝑎_𝑘𝑙𝜕𝑥_𝑘𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑙𝑣+𝑎_𝑘𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑘𝑣+𝑎𝜌^2𝜈𝑓(𝑣)=𝑢𝑣𝑑𝑥,_Ω𝜌^2𝜈𝑓𝑣𝑑𝑥 (13) the bilinear and linear forms, respectively.

Definition 3. A function 𝑢_𝜈 from the space 𝐻^∘1_{2,𝜈+𝛽/2}(Ω) is called an 𝑅_𝜈-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^∘1_{2,𝜈+𝛽/2}(Ω), the identity 𝑎𝑢_𝜈,𝑣=𝑓(𝑣) (14) holds, where 𝜈 is arbitrary but fixed and satisfies the inequality 𝛽𝜈≥𝜇+2−1. (15)

For Problem A, we prove the main result.

Theorem 4. Let conditions (9)–(12) and (15) hold and let 2𝑐₈(2|𝜈|)+1+12𝑐₉²<𝑐₁₁𝑐₁₂ (16) be satisfied.
Then, the 𝑅_𝜈-generalized solution 𝑢_𝜈 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^∘1_{2,𝜈+𝛽/2}(Ω) and the following estimate is valid: 𝑢_{𝜈𝐻12,𝜈+𝛽/2(Ω)}≤𝑐₁₃𝑓_{𝐿2,𝜇(Ω)}, (17) where 𝑐₁₃ is a positive constant not depending on 𝑢_𝜈 and 𝑓.

Proof. First, we show that the forms 𝑎(𝑢,𝑣) and 𝑓(𝑣) are continuous on 𝐻^∘1_{2,𝜈+𝛽/2}(Ω). In fact, by virtue of conditions (9), (12), and (15) and the Cauchy-Schwarz inequality, we have |𝑎(≤𝑢,𝑣)|_{Ω2𝑘,𝑙=1}𝑎_𝑘𝑙𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑘𝜕𝑣𝜕𝑥_𝑙+𝜕𝑎_𝑘𝑙𝜕𝑥_𝑘𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑙𝑣+𝑎_𝑘𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑘𝑣+𝑎𝜌^2𝜈+𝑢𝑣𝑑𝑥_Ω′𝑎_𝑘𝑙𝜕𝜌^2𝜈𝜕𝑥_𝑘𝜕𝑢𝜕𝑥_𝑙𝑣𝑑𝑥≤2𝑐_8Ω𝜌^22𝜈+𝛽_𝑘=1𝜕𝑢𝜕𝑥_𝑘²𝑑𝑥^1/2×_Ω𝜌^22𝜈+𝛽_𝑘=1𝜕𝑣𝜕𝑥_𝑘²𝑑𝑥^1/2+2√2𝑐_8Ω𝜌^22𝜈+𝛽_𝑙=1𝜕𝑢𝜕𝑥_𝑙²𝑑𝑥^1/2×_Ω𝜌^{2𝜈+𝛽−2}𝑣²𝑑𝑥^1/2+√2𝑐_9Ω𝜌^22𝜈+𝛽_𝑘=1𝜕𝑢𝜕𝑥_𝑘²𝑑𝑥^1/2×_Ω𝜌^{2𝜈+𝛽−2}𝑣²𝑑𝑥^1/2+𝑐_10Ω𝜌^{2𝜈+𝛽−2}𝑢²𝑑𝑥^1/2×_Ω𝜌^{2𝜈+𝛽−2}𝑣²𝑑𝑥^1/2+4√2|𝜈|𝑐_8Ω′𝜌^22𝜈+𝛽_𝑙=1𝜕𝑢𝜕𝑥_𝑙²𝑑𝑥^1/2×_Ω′𝜌^{2𝜈+𝛽−2}𝑣²𝑑𝑥^1/2≤2𝑐₈|𝑢|_𝐻¹_{2,𝜈+𝛽/2(Ω)}|𝑣|_𝐻¹_{2,𝜈+𝛽/2(Ω)}+2√2𝑐₈+√2𝑐₉|𝑢|_𝐻¹_{2,𝜈+𝛽/2(Ω)}‖𝑣‖_{𝐿2,𝜈+𝛽/2−1(Ω)}+4√2|𝜈|𝑐₈|𝑢|_{𝐻12,𝜈+𝛽/2(Ω′)}‖𝑣‖_{𝐿2,𝜈+𝛽/2−1(Ω′)}+𝑐₁₀‖𝑢‖_{𝐿2,𝜈+𝛽/2−1(Ω)}‖𝑣‖_{𝐿2,𝜈+𝛽/2−1(Ω)}, (18) or |𝑎(𝑢,𝑣)|≤𝑐₁₄‖𝑢‖_𝐻¹_{2,𝜈+𝛽/2(Ω)}‖𝑣‖_𝐻¹_{2,𝜈+𝛽/2(Ω)},∀𝑢,𝑣∈𝐻^∘1_{2,𝜈+𝛽/2}(Ω)𝑓(𝑣≤)max_𝑥∈Ω𝜌^{2𝜈−2𝜇−𝛽+21/2}_Ω𝜌^2𝜇𝑓²𝑑𝑥^1/2×_Ω𝜌^{2𝜈+𝛽−2}𝑣²𝑑𝑥^1/2≤𝑐₁₅𝑓_{𝐿2,𝜇(Ω)}‖𝑣‖_𝐻¹_{2,𝜈+𝛽/2(Ω)},∀𝑣∈𝐻^∘1_{2,𝜈+𝛽/2}(Ω),∀𝑓∈𝐿_2,𝜇(Ω). (19)(20)
Let us now prove the 𝐻^∘1_{2,𝜈+𝛽/2}(Ω) ellipticity of the bilinear form 𝑎(𝑢,𝑣); that is, ∃𝑐₁₆>0,∀𝑢∈𝐻^∘1_{2,𝜈+𝛽/2}(Ω),𝑎(𝑢,𝑢)≥𝑐₁₆‖𝑢‖²_{𝐻12,𝜈+𝛽/2(Ω)}. (21) We substitute 𝑣 by 𝑢 in (13), and by means of the Cauchy-Schwarz inequality, 𝜀-inequality, and conditions (9), we estimate the absolute values of the second, third, and fourth terms of the form 𝑎(𝑢,𝑢): ²_{𝑘,𝑙=1Ω}𝑎_𝑘𝑙𝜕𝜌^2𝜈𝜕𝑥_𝑘𝜕𝑢𝜕𝑥_𝑙≤𝑢𝑑𝑥_{2𝑘,𝑙=1Ω}𝑎_𝑘𝑙2𝜈𝜌^2𝜈−1𝜕𝑢𝜕𝑥_𝑙√𝑢𝑑𝑥≤22𝑐₈|𝜈|𝜀₁²_𝑙=1Ω′𝜌^2𝜈+𝛽𝜕𝑢𝜕𝑥_𝑙²+1𝑑𝑥𝜀_1Ω^′𝜌^{2𝜈+𝛽−2}𝑢²𝑑𝑥_{2𝑘,𝑙=1Ω},𝜕𝑎_𝑘𝑙𝜕𝑥_𝑘𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑙≤√𝑢𝑑𝑥2𝑐₈𝜀₂²_𝑙=1Ω𝜌^2𝜈+𝛽𝜕𝑢𝜕𝑥_𝑙²+1𝑑𝑥𝜀_2Ω𝜌^{2𝜈+𝛽−2}𝑢²𝑑𝑥_2𝑘=1Ω,𝑎_𝑘𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑘≤√𝑢𝑑𝑥2𝑐₉𝜀₃2²_𝑘=1Ω𝜌^2𝜈+𝛽𝜕𝑢𝜕𝑥_𝑘²+1𝑑𝑥2𝜀_3Ω𝜌^{2𝜈+𝛽−2}𝑢²𝑑𝑥. (22)(23)(24) Here, 𝜀₁, 𝜀₂, and 𝜀₃ are any positive numbers.
Using (10) and (11), we have ²_{𝑘,𝑙=1Ω}𝑎_𝑘𝑙𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑘𝜕𝑢𝜕𝑥_𝑙𝑑𝑥≥𝑐_11Ω𝜌^22𝜈+𝛽_𝑘=1𝜕𝑢𝜕𝑥_𝑘²𝑑𝑥,_Ω𝑎𝜌^2𝜈𝑢²𝑑𝑥≥𝑐_12Ω𝜌^{2𝜈+𝛽−2}𝑢²𝑑𝑥. (25)(26) Then, from (22)–(26), we obtain 𝑎()≥𝑐𝑢,𝑢₁₁−√2𝑐₈2|𝜈|𝜀₁+𝑐₈𝜀₂+12𝑐₉𝜀₃×_Ω𝜌^22𝜈+𝛽_𝑘=1𝜕𝑢𝜕𝑥_𝑘²+𝑐𝑑𝑥₁₂−√2𝑐₈2|𝜈|𝜀₁⁻¹+𝑐₈𝜀₂⁻¹+12𝑐₉𝜀₃⁻¹×_Ω𝜌^{2𝜈+𝛽−2}𝑢²𝑑𝑥. (27)
Note that if condition (16) is satisfied, then there exists a positive constant 𝜀 such that 𝑐₁₁−√2𝑐₈(2|𝜈|)+1+12𝑐₉𝑐𝜀>0,₁₂−√2𝑐₈(2|𝜈|)+1+12𝑐₉𝜀⁻¹>0. (28) Supposing that 𝜀₁=𝜀₂=𝜀₃=𝜀 in (27), we get 𝑎(𝑢,𝑢)≥𝑐_16Ω𝜌^22𝜈+𝛽_𝑘=1𝜕𝑢𝜕𝑥_𝑘²𝑑𝑥+_Ω𝜌^{2𝜈+𝛽−2}𝑢²𝑑𝑥=𝑐₁₆‖𝑢‖_𝐻¹_{2,𝜈+𝛽/2(Ω)} (29) with constant 𝑐₁₆=min𝑐₁₁−√2𝑐₈(2|𝜈|)+1+12𝑐₉𝑐𝜀,₁₂−√2𝑐₈(2|𝜈|)+1+12𝑐₉𝜀⁻¹. (30)
According to (19), (21), and (20), the bilinear form 𝑎(𝑢,𝑣) is continuous and 𝐻^∘1_{2,𝜈+𝛽/2}(Ω)-elliptical, and the linear form 𝑓(𝑣) is continuous on 𝐻^∘1_{2,𝜈+𝛽/2}(Ω); then, the existence and uniqueness of an 𝑅_𝜈-generalized solution of Problem A follow from the Lax-Milgram theorem (see [13]).
Taking into account that 𝑐₁₆𝑢_𝜈²_{𝐻12,𝜈+𝛽/2(Ω)}≤𝑎𝑢_𝜈,𝑢_𝜈=𝑓(𝑣)≤𝑐₁₅𝑓_{𝐿2,𝜇(Ω)}‖𝑢‖_𝐻¹_{2,𝜈+𝛽/2(Ω)}, (31) we get estimate (17).

Corollary 5. If there exists at least one 𝜈 for which there exists a unique 𝑅_𝜈-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 𝑅_𝜈-generalized solution. Here, 𝜈₁𝛽=max𝜇+2𝑐−1;1−¹¹𝑐₁₂^1/2−𝑐₉/2𝑐₈,𝜈+𝜀₂=𝑐₁₁𝑐₁₂^1/2−𝑐₉/2𝑐₈−1, (32) where 𝜀 is a given sufficiently small positive number.

Corollary follows from the proof of Theorem 4.

Theorem 6. If the assumptions of Theorem 4 are valid, then, for all 𝜈 in the interval [𝜈₁,𝜈₂), the 𝑅_𝜈-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 −²_𝑘=1𝜕𝜕𝑥_𝑘𝑎_𝑘𝑘(𝑥)𝜕𝑢𝜕𝑥_𝑘+𝑎(𝑥)𝑢=𝑓(𝑥),𝑥∈Ω,𝑢=0,𝑥∈𝜕Ω. (33)(34)

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, 𝑎_𝑘𝑘∈𝐻¹_∞,−𝛽Ω,𝑐₁₆,𝑎∈𝐿_∞,−𝛽Ω,𝑐₁₇,_2𝑘=1𝑎_𝑘𝑘(𝑥)𝜉²_𝑘≥𝑐₁₈𝜌^𝛽(𝑥)²_𝑘=1𝜉²_𝑘,𝑎(𝑥)≥𝑐₁₉𝜌^𝛽(𝑥)almosteverywhereonΩ, (35)(36)(37) and the right-hand side of the equation satisfies the condition 𝑓∈𝐿_2,𝜇(Ω,𝛿) (38) for some nonnegative real number 𝜇. Here, 𝑐_𝑖, 𝑖=16,…,19, are positive constants not depending on 𝑥; 𝜉₁ and 𝜉₂ are arbitrary real parameters.

Set 𝑏(𝑢,𝑣)=_Ω²_𝑘=1𝑎_𝑘𝑘𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑘𝜕𝑣𝜕𝑥_𝑘+𝑎_𝑘𝑘𝜕𝜌^2𝜈𝜕𝑥_𝑘𝜕𝑢𝜕𝑥_𝑘𝑣+𝑎𝜌^2𝜈𝑙(𝑣)=𝑢𝑣𝑑𝑥,_Ω𝜌^2𝜈𝑓𝑣𝑑𝑥. (39)

Definition 8. A function 𝑢_𝜈 from the set 𝑊^∘1_{2,𝜈+𝛽/2}(Ω,𝛿) is called an 𝑅_𝜈-generalized solution of the Problem B if the identity 𝑏(𝑢_𝜈,𝑣)=𝑙(𝑣) holds for all 𝑣∈𝑊^∘1_{2,𝜈+𝛽/2}(Ω,𝛿) and for any given value of 𝜈 satisfying the inequality 𝛽𝜈≥𝜇+2. (40)

Theorem 9. Let conditions (35)–(40) hold and 𝛽𝜈+2>0. (41) Then, for any 𝜈 satisfying conditions (40) and (41), there always exists parameter 𝛿 such that 𝑅_𝜈-generalized solution 𝑢_𝜈 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^∘1_{2,𝜈+𝛽/2}(Ω,𝛿). In this case, the following estimate is valid: 𝑢_{𝜈𝑊12,𝜈+𝛽/2(Ω,𝛿)}≤𝑐₂₀𝑓_{𝐿2,𝜇(Ω,𝛿)}, (42) where 𝑐₂₀ is a positive constant independent of 𝑓 and 𝑢_𝜈.

Proof. First of all, we note that the bilinear and linear forms are continuous on the set 𝑊^∘1_{2,𝜈+𝛽/2}(Ω,𝛿) and the inequalities 𝑏(𝑢,𝑣)≤𝑐₂₁‖𝑢‖_𝑊¹_{2,𝜈+𝛽/2(Ω,𝛿)}‖𝑣‖_𝑊¹_{2,𝜈+𝛽/2(Ω,𝛿)},𝑙(𝑣)≤𝑐₂₂𝑓_{𝐿2,𝜇(Ω,𝛿)}‖𝑣‖_𝑊¹_{2,𝜈+𝛽/2(Ω,𝛿)} (43)(44) 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 is 𝑊^∘1_{2,𝜈+𝛽/2}-elliptical in Ω. We have 𝑏()=𝑢,𝑢_2𝑘=1Ω𝑎_𝑘𝑘𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑘²+𝑎_𝑘𝑘𝜕𝜌^2𝜈𝜕𝑥_𝑘𝜕𝑢𝜕𝑥_𝑘𝑢𝑑𝑥+_Ω𝑎𝜌^2𝜈𝑢²𝑑𝑥 (45) for any 𝑢 from 𝑊^∘1_{2,𝜈+𝛽/2}(Ω,𝛿). By means of condition (35), we estimate the absolute value of the second term on the right-hand side in (45): ²_𝑘=1Ω𝑎_𝑘𝑘𝜕𝜌^2𝜈𝜕𝑥_𝑘𝜕𝑢𝜕𝑥_𝑘𝑢𝑑𝑥≤²_𝑘=1Ω′𝑎_𝑘𝑘2𝜈𝜌^2𝜈−1𝜕𝑢𝜕𝑥_𝑘𝑢𝑑𝑥≤𝜀²_𝑘=1Ω′𝜌^2𝜈+𝛽𝜕𝑢𝜕𝑥_𝑘²+𝑑𝑥2𝑐²₁₆𝜈²𝜀_Ω^′𝜌^{2𝜈+𝛽−1}𝑢²𝑑𝑥. (46) From (45) and the last inequality we get 𝑏(𝑢,𝑢)≥²_𝑘=1Ω𝑎_𝑘𝑘𝜌^2𝜈𝜕𝑢𝜕𝑥_𝑘²𝑑𝑥−𝜀²_𝑘=1Ω′𝜌^2𝜈+𝛽𝜕𝑢𝜕𝑥_𝑘²𝑑𝑥+_Ω𝑎𝜌^2𝜈𝑢²𝑑𝑥−2𝑐²₁₆𝜈²𝜀_Ω^′𝜌^{2𝜈+𝛽−1}𝑢²𝑑𝑥. (47) Supposing that 𝛼^∗ and 𝛼 equal 𝜈 and 𝜈/2 in Lemma 1, respectively, we have ‖𝑢‖²_{𝐿2,𝜈+𝛽/2−1(Ω′,𝛿)}≤2𝑐²₆𝛿^𝜈𝜈‖𝑢‖²_{𝐿2,𝜈+𝛽/2(Ω,𝛿)},𝑐₆=const>0. (48) Taking into account (36), (37), and (48), from estimate (47) we get 𝑏(𝑢,𝑢)≥𝑐₁₈−𝜀|𝑢|²_{𝑊12,𝜈+𝛽/2()Ω,𝛿}+𝑐₁₉−4𝑐²₁₆𝑐²₆𝜈𝛿^𝜈𝜀×‖𝑢‖²_{𝐿2,𝜈+𝛽/2(Ω,𝛿)}. (49) Obviously, we can always choose 𝜀 and 𝛿 such that the constants 𝑐₁₈>𝜀, 𝑐₁₉>4𝑐²₁₆𝑐²₆𝜈𝛿^𝜈/𝜀 and the inequality 𝑏(𝑢,𝑢)≥𝑐₂₃‖𝑢‖²_{𝑊12,𝜈+𝛽/2(Ω,𝛿)} (50) are valid with constant 𝑐₂₃=min(𝑐₁₈−𝜀,𝑐₁₉−4𝑐²₁₆𝑐²₆𝜈𝛿^𝜈/𝜀). Therefore, bilinear form 𝑏(𝑢,𝑢) is 𝑊^∘1_{2,𝜈+𝛽/2}-elliptical.
According to (43), (44), and (50), the bilinear form 𝑏(𝑢,𝑢) is continuous and 𝑊^∘1_{2,𝜈+𝛽/2}-elliptical, and the linear form 𝑙(𝑣) is continuous on 𝑊^∘1_{2,𝜈+𝛽/2}(Ω,𝛿); then, the existence and uniqueness of an 𝑅_𝜈-generalized solution of Problem B follow from the Lax-Milgram theorem (see [13]).
Taking into account that 𝑐₂₃𝑢_𝜈²_{𝑊12,𝜈+𝛽/2(Ω,𝛿)}≤𝑏𝑢_𝜈,𝑢_𝜈=𝑙𝑢_𝜈≤𝑐₂₂𝑢_{𝜈𝑊12,𝜈+𝛽/2(Ω,𝛿)}𝑓_{𝐿2,𝜇(Ω,𝛿)}, (51) we get estimate (42).

Theorem 10. If for some 𝛿 there is a set of values 𝜈 such that an 𝑅_𝜈-generalized solution of the Problem B exists in the set 𝑊^∘1_{2,𝜈+𝛽/2}(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].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.