For α∈(1,2), we analyze a stationary superdiffusion equation in the right angle in the unknown u=u(x1,x2): Dx1αu+Dx2αu=f(x1,x2), where Dxα is the Caputo fractional derivative. The classical solvability in the weighted fractional Hölder classes of the associated boundary problems is addressed.
1. Introduction
Fractional partial differential equations (FPDE) play a key role in the description of the so-called anomalous phenomena in nature and in the theory of complex systems (see, e.g., [1]). In particular, these equations provide a more faithful representation of the long-memory and nonlocal dependence of many anomalous processes. The signature of an anomalous diffusion species 〈(Δx)2〉 scales as a nonlinear power law in time; i.e., 〈(Δx)2〉~tα, α>0. When α>1, this is referred to as superdiffusion.
Superdiffusion is used in modelling turbulent flow [2, 3], chaotic dynamics of classical conservative systems [4], model solute transport in underground aquifers [5–7], and rivers [8–10], biophysics [11], and physical and chemical models described by the Lévy processes [12, 13]. In the present paper, we focus on the boundary value problems to the stationary superdiffusion equation.
Let Ω={(x1,x2):x1∈(0,+∞),x2∈(0,+∞)} be the first quarter with a boundary ∂Ω=Γ1∪Γ2, Γ1={(x1,x2):x2=0,x1≥0}, Γ2={(x1,x2):x1=0,x2≥0}. For a fixed α∈(1,2), we consider the linear equation in the unknown function u=u(x1,x2):Ω→R,(1)Dx1αu+Dx2αu=fx1,x2,subject either to the Dirichlet boundary condition (DBC),(2)u=ψ1x1onΓ1andu=ψ2x2onΓ2,or to the Neumann boundary condition (NBC),(3)∂u∂x2=φ1x1onΓ1and∂u∂x1=φ2x2onΓ2,where the functions f, ψ1, ψ2, φ1, φ2 are prescribed.
Here, the symbols Dx1α and Dx2α stand for the Caputo fractional derivative of order α with respect to x1 and x2.
Introducing the function,(4)ωθy=yθ-1Γθ,θ>0,y≥0,we define the Caputo derivative Dyθv(·,y) as (see, e.g., (2.4.1) in [1]) (5)Dyθv·,y=∂θ∂yθ∫0yωθ-θy-z×v·,z-∑k=0θ-1∂kv·,0∂zkzkk!dz,where ⌈θ⌉ is the ceiling function of θ (i.e., the smallest integer greater than or equal to θ), Γ being the Euler Gamma-function. An equivalent definition in the case of u∈AC⌈θ⌉ reads (see (2.4.15) in [1]) (6)Dyθv·,y=∫0yωθ-θy-z∂θ∂zθv·,zdz.In the limit case θ=⌈θ⌉, the Caputo fractional derivatives of v(·,y) boils down (∂⌈θ⌉/∂y⌈θ⌉)v(·,y).
Elliptic boundary value problems, α=2, in domain with conical and dihedral singularities have been extensively studied, starting with Kondratiev’s famous paper [14]. In this field of researches, it should be also noted the works of Kondariev and Oleinik [15], Borsuk and Kondratiev [16], Grisvard [17], and Maz’ya and Plamenevskii [18]. The results of those investigations are formulated in both Wl,p and Cl+β spaces and corresponding weighted classes, where the desired functions together with their derivatives are bounded with some power weights in the Lp or Cl+β norms.
Recently, a great attention in the literature has been devoted to the study of boundary value problems to the fractional Laplacian operator -Δsu with order s∈(0,1) (see, e.g., [19–24] and references therein). Boundary value problems for the equation with the main part,(7)∂2∂x2ux,y+Dy2αux,y,α∈0,1,are studied with various approaches, such as spectral technique, method of potential theory, and Fourier method (see [25–28] and references therein). We also quote the works [29, 30], where the authors presented a Galerkin finite element approximation for variational solution to the steady state fractional advection dispersion equation:(8)-Dx1ap0Dx-ν+qxD1-νDx1u+bxDx1u+cxu=f,ν∈0,1,where 0Dx-ν, xD1-ν are left and right fractional integral operators, 0≤p,q,≤1, p+q=1.
The goal of the present paper is the proof of the well-posedness and the regularity of solutions to boundary value problems (1)-(3) in weighted Hölder classes. It is worth noting that these classes allow one to control the behavior of the solution near the boundary including the corner point and at the infinity.
Outline of the Paper. In the next section, we introduce necessary functional spaces and state the main results (Theorems 3 and 4) along with the general assumptions. In Section 3, we construct the integral representation to u(x1,x2) in the case of homogenous boundary conditions. To this end, we apply Mellin transform and reduce problem (1)-(3) to the linear nonhomogenous difference equation of the first-order with variables coefficients in the two-dimensional case. Section 4 is devoted to some auxiliary results which will play a key role in the investigation. In Section 5, we estimate the seminorms of the minor derivatives Dx1α1Dx2α2u, 0≤α1+α2≤1, and in Section 6 we evaluate the Hölder coefficients of the major derivatives Dx1α1Dx2α2u, 1<α1+α2≤α. In Section 7, using these estimates, we provide the proofs of Theorems 3 and 4. Moreover, in Remark 27 we show how results of Theorems 3 and 4 can be extend to the more general equation compared to (1).
2. Functional Setting and Main Results
Throughout this work, the symbol C will denote a generic positive constant, depending only on the structural quantities of the problem. We will carry out our analysis in the framework of the weighted Hölder spaces. Let β∈(0,1) be arbitrary fixed. We denote by r(x) and d(x) the distance from a point x(x1,x2)∈Ω to the origin (0,0) and to the boundary ∂Ω, correspondingly. Then for every x and x¯ from Ω we define r(x,x¯)=min{r(x),r(x¯)} and d(x,x¯)=min{d(x),d(x¯)}. Note that if x,x¯∈∂Ω, then r(x)=d(x) and r(x,x¯)=d(x,x¯).
For fixed s∈R, we introduce the Banach spaces Esβ(Ω¯) and Csβ(Ω¯) of the functions v with the norms(9)vEsβΩ¯=supΩ¯r-sxv+vx1,s-β,Ωβ+vx2,s-β,Ωβ,vCsβΩ¯=supΩ¯d-sxv+vx1,s-β,Ωβ+vx2,s-β,Ωβ,where(10)vy,s-β,Ωβ=supy,y¯∈Ω¯r-s+βy,y¯vy-vy¯y-y¯β,vy,s-β,Ωβ=supy,y¯∈Ω¯d-s+βy,y¯vy-vy¯y-y¯β,for y≠y¯.
Definition 1.
A function v=v(x1,x2) belongs to the class Msl+β(Ω¯), for l>0, if v∈Esβ(Ω¯), Dxilv∈Es-lβ(Ω¯), i=1,2, and the norms below are finite, (11)vMsl+βΩ¯=vEsβΩ¯+∑j=1l-1DxjvEs-jβΩ¯+∑i=12DxilvEs-lβΩ¯,if l≥1, and (12)vMsl+βΩ¯=vEsβΩ¯+∑i=12DxilvEs-lβΩ¯in the case of l∈(0,1).
Definition 2.
A function v=v(x1,x2) belongs to the class Nsl+β(Ω¯), for l>0, if v∈Csβ(Ω¯), Dxilv∈Cs-lβ(Ω¯), i=1,2, and the norms below are finite, (13)vNsl+βΩ¯=vCsβΩ¯+∑j=1l-1DxjvCs-jβΩ¯+∑i=12DxilvCs-lβΩ¯,if l≥1, and (14)vNsl+βΩ¯=vCsβΩ¯+∑i=12DxilvCs-lβΩ¯in the case of l∈(0,1).
In a similar way we introduce the spaces Esβ(∂Ω) and Msl+β(∂Ω). As for Csβ(∂Ω) and Nsl+β(∂Ω), these spaces concave with Esβ(∂Ω) and Msl+β(∂Ω).
Msl+β and Nsl+β defined above are Banach spaces. Indeed, the fact that they are normed spaces is easily seen, whereas the completeness follows from Theorem 2.7 [31] together with standard arguments (see, e.g., Remark 3.1.3 in [32]).
We begin to stipulate the general assumptions.
h1 (condition on the parameters): we require(15)α-1<β<1,s=α+β,0<s1<2-α.
h2 (conditions on the right-hand sides of boundary conditions): we demand(16)ψ1∈Msα+βΓ1∩M-s1α+βΓ1,ψ2∈Msα+βΓ2∩M-s1α+βΓ2,φ1∈Ms-11+βΓ1∩M-s11+βΓ1,φ2∈Ms-11+βΓ2∩M-s1-11+βΓ2.
h3 (conditions on the right-hand side of the equation): we suppose(17)eitherf∈Es-αβΩ¯∩E-s1-αβΩ¯(18)orf∈Cs-αβΩ¯∩C-s1-αβΩ¯.
We are now in the position to state our main results.
Theorem 3.
Let assumptions (h1), (h2), and (17) hold and moreover for every points x1∈Γ1 and x2∈Γ2(19)fx1,0-Dx1αψ1x1=0,f0,x2-Dx2αψ2x2=0,iftheDBC(2)holds;and(20)fx1,0-Dx1α-1φ1x1=0,f0,x2-Dx2α-1φ2x2=0,iftheNBC(3)holds.Then, (1) subject either to the DBC (2) or to the NBC (3), admits a unique classical solution u=u(x1,x2) on Ω, satisfying the regularity(21)u∈Msα+βΩ¯∩M-s1α+βΩ¯andDx1α1Dx2α2u=Dx2α2Dx1α1ufor 0<α1+α2<α. Besides, the following estimates hold:(22)uMsα+βΩ¯+uM-s1α+βΩ¯+uNs1+βΩ¯≤CfEs-αβΩ¯+fE-s1-αβΩ¯+∑i=12ψiMsα+βΓi+ψiM-s1α+βΓi≡FMf,ψ1,ψ2,(23)Dx1α1Dx2α2uEs-α1-α2βΩ¯+Dx1α1Dx2α2uE-s1-α1-α2βΩ¯≤FMf,ψ1,ψ2in the case of DBC (2), and(24)uMsα+βΩ¯+uM-s1α+βΩ¯+uCsβΩ¯+Dx1α1Dx2α2uEs-α1-α2βΩ¯+Dx1α1Dx2α2uE-s1-α1-α2βΩ¯≤CfEs-αβΩ¯+fE-s1-αβΩ¯+∑i=12φiMs-1α-1+βΓi+φiM-s1-1α-1+βΓiif NBC (3) hold.
Theorem 4.
Let assumptions (h1), (h2), and (18) hold. Then, boundary value problems (1)-(3) admit a unique classical solution u=u(x1,x2) on Ω, satisfying the regularity(25)u∈Nsα+βΩ¯∩N-s1α+βΩ¯andDx1α1Dx2α2u=Dx2α2Dx1α1ufor 0<α1+α2<α. Besides, the following estimates hold:(26)uNsα+βΩ¯+uN-s1α+βΩ¯≤CfCs-αβΩ¯+fC-s1-αβΩ¯+∑i=12ψiMsα+βΓi+ψiM-s1α+βΓi≡FNf,ψ1,ψ2,(27)Dx1α1Dx2α2uCs-α1-α2βΩ¯+Dx1α1Dx2α2uC-s1-α1-α2βΩ¯≤FNf,ψ1,ψ2,in the case of DBC (2), and(28)uNsα+βΩ¯+uN-s1α+βΩ¯+Dx1α1Dx2α2uCs-α1-α2βΩ¯+Dx1α1Dx2α2uC-s1-α1-α2βΩ¯≤CfCs-αβΩ¯+fC-s1-αβΩ¯+∑i=12φiMs-1α-1+βΓi+φiM-s1-1α-1+βΓiif NBC (3) hold.
Remark 5.
It is easily apparent that the functions,(29)f=x12+x22β/2x1x21+x12+x22s1+α/2x12+x22,ψi,φi≡0,i=1,2,satisfy conditions (h2) and (17), (19), and (20).
The remainder of the paper is devoted to the proof of Theorems 3 and 4 in the DBC case. The proof of these theorems for NBC is almost identical and is left to the interested reader.
3. Integral Representation for u(x1,x2) in the Special Case
We first dwell on the special case where ψi≡0 and f is a finite function. Namely (2), (17), and (19) are replaced by the simpler conditions(30)u∂Ω=0,(31)f≡0ifx12+x221/2>R0for some given positive R0 and(32)eitherf∈Es-αβΩ¯,fx1,0=f0,x2=0,orf∈Cs-αβΩ¯.
We denote by u⋆(p1,p2) the Mellin transform of the function u(x1,x2). Due to conditions (30)-(32) and assumptions of Theorem 3, we can apply, at least formally, the Mellin transformation to problem (1) and (2) (see for details § 1.4 and § 2.5 in [1]). Then simple calculations lead to the equation (33)u∗p1-α,p2+α+Γ1-p1Γ1+α-p1Γ1-p2Γ1-p2-αu∗p1,p2=Γ1-p1Γ1+α-p1f∗p1,p2+α.
Introducing new variables(34)p1=-αq1andp2=αq2,and new functions (35)Vq1,q2=u∗-αq1,αq2,Fq1,q2=Γ1+αq1Γ1+α+αq1f∗-αq1,αq2+α,Pq1,q2=Γ1+αq1Γ1+α+αq1Γ1-αq2Γ1-αq2-α,we rewrite the equation in the more compact form(36)Vq1+1,q2+1+Pq1,q2Vq1,q2=Fq1,q2.Thus, we transform problem (1) and (2) to the linear nonhomogeneous difference equation of the first-order with variable coefficients. In order to solve this equation, we adapt the technique from Section 3 in [33] to our case.
Proposition 6.
Let γ¯ denote Euler-Mascheroni constant (see, e.g., Definition in [34]),(37)Req1≠-m+1α,Req2≠m+1α,m=0,1,2,…,and let E1(q1) and E2(q2) be arbitrary analytic functions such that(38)E1q1=E1q1+1,E2q2=E2q2+1.Then the function(39)V0q1,q2=πeγ¯αeiπq1Γ1+αq1Γ1-αq2E1q1E2q2solves homogenous equation (36) (i.e., F≡0) and does not have any poles.
If, in addition, the functions E1(q1) and E2(q2) have no zeros, then the function 1/V0(q1,q2) does not have any poles and the following estimate holds:(40)E1q1E2q2V0q1,q2≤Cq1αReq1+1/2q21/2-αReq2×exp-πImq1-πα2Imq1-πα2Imq2for Imq1,Imq2→+∞ and for every fixed Req1 and Req2 satisfying inequalities above.
Proof.
In order to verify that the function V0(q1,q2) solves homogenous equation (36), it is enough to substitute V0 in (36) and take into account the properties of the Gamma-function: Γ(q+1)=qΓ(q). Besides, the properties of the function V0 are simple consequences of the well-known properties of Gamma function:
Γ(z) has simple poles in the points(41)Rez=-m,m=0,1,2,…
The Stirling asymptotic formula holds:(42)Γz=expz-12lnz-z+ln2π2+112z+Oz-3
as Imz→+∞ while Rez remains bounded
We are now in the position to construct the solution of nonhomogeneous equation (36).
Till the end of the paper, we assume that γ∈[0,1] is an arbitrary fixed quantity and define the contour Lγ in the complex plane ζ as
(43)Lγ=Reζ=-γ,Imζ∈-∞,+∞,ifγ∈0,1.
If γ=1, then L1 consists of three parts:
the half-circle ζ+1=γ0,Reζ>-1,
the intervals Reζ=-1,Imζ∈(-∞,-γ0) and Reζ=-1,Imζ∈(γ0,+∞)
with the small positive number γ0, γ0<γ/2.
The contour L0 (γ=0) is obtained from L1 after its shifting to the right on Reζ=1.
Introducing the periodic function K(ζ) with period 1,(44)Kζ=12icotπζ+isin2πasin2πζ+a,0<a-γ<1,we assert the following results.
Proposition 7.
Let ζ=-γ+iz, z∈R; conditions (32) and (31) hold, and let(45)Req1≠-m+1α+γ,Req2≠m+1α+γ-1,m=0,1,2,….Then the solution of inhomogeneous equation (36) is given by(46)Vq1,q2=V0q1,q22i∫L-γKζFq1+ζ,q2+ζV0q1+ζ+1,q2+ζ+1dζwhere V0(q1,q2) is defined in (39) with E1(q1), E2(q2)≡1.
Proof.
First, we prove this statement in the case γ=1; i.e., Lγ=L1. To this end, we will seek the solution of (36) as a product (47)Vq1,q2=V0q1,q2Yq1,q2,where the unknown function Y(q1,q2) solves the first-order difference inhomogeneous equation with the constant coefficients(48)Yq1+1,q2+1-Yq1,q2=Fq1,q2V0q1+1,q2+1.
Then adapting the technique from [35] to our case, we deduce that(49)Yq1,q2=12i∫L-1Fq1+ζ,q2+ζV0q1+1+ζ,q2+1+ζKζdζ,if condition (45) holds.
Indeed, substituting Y(q1,q2) to (48) and applying Cauchy’s residue theorem arrive at (50)Yq1+1,q2+1-Yq1,q2=12i∫L-1Fq1+1+ζ,q2+1+ζV0q1+2+ζ,q2+2+ζKζdζ-∫L-1Fq1+ζ,q2+ζV0q1+1+ζ,q2+1+ζKζdζ=πresζ=0Fq1+ζ,q2+ζV0q1+1+ζ,q2+1+ζKζ=Fq1,q2V0q1+1,q2+1.To reach these equalities, we essentially used the following properties of the functions F, V0, and K:
F(q1+ζ,q2+ζ)/V0(q1+1+ζ,q2+1+ζ) has no poles if Reζ∈(-1,γ0) and Req1, Req2 satisfy (45).
K(ζ) is periodic of period 1 with simple poles at ζ=0,±1 and multiple poles at ζ=-a.
Besides, the following inequality holds:(51)K-γ+iz≤Ce-4πz,z→+∞;e2πz,z→-∞,
for γ∈[0,1]
There is the asymptotic representation(52)KζFq1+ζ,q2+ζV0q1+1+ζ,q2+1+ζ→0as Imζ→0,
for the bounded Imq1 and Imq2 and if Req1, Req2 meet requirement (45).
Note that statements (i) and (ii) are simple consequences of Proposition 6 and (32). As for assessment (iii), it can be easily drawn from (i) and (ii).
After that, we return to the representation of V(q1,q2) and obtain solution (36) as(53)Vq1,q2=V0q1,q22i∫L-1KζFq1+ζ,q2+ζV0q1+1+ζ,q2+1+ζdζ.This proves Proposition 7 in the case Lγ≔L-1.
Recasting the arguments above in the case γ∈[0,1) and applying Cauchy theorem allow us to prove Proposition 7 in the case of arbitrary contour Lγ, γ∈[0,1). It completes the proof of Proposition 7.
We are now in the position to obtain integral representation of the solution u(x1,x2). Indeed, Propositions 6 and 7 provide(54)u⋆p1,p2=V0-p1/α,p2/α2i∫L-γKζ×F-p1/α+ζ,p2/α+ζV0-p1/α+1+ζ,p2/α+1+ζdζ=-12i×∫L-γKζe-iπζ×Γ1-p1+αζΓ1-p2-α-αζΓ1-p1Γ1-p2×f⋆p1-αζ,p2+αζ+αdζ,where(55)Rep1≠m+1-γα,Rep2≠m+1-α+αγ,m=0,1,2,…,γ∈0,1.Note that, conditions on Repi hold, for example, in the domain(56)Rep1<1-γαandRep2<1-α+γα.
At last, we carry out the inverse Mellin transform to derive an explicit integral representation for u(x1,x2),(57)ux1,x2=18πi∫0+∞dt1t1×∫0+∞dt2t2fx1t1,x2t2Lγt1,t2,x1,x2,with (58)Lγt1,t2,x1,x2=∫L-γKζx1t1-αζx2t2α+αζ×e-iπζ∫γ1-i∞γ1+i∞t1-p1Γ1-p1+αζΓ1-p1dp1×∫γ2-i∞γ2+i∞t2-p2Γ1-p2-α-αζΓ1-p2dp2dζ,for Reγ1∈(A1,B1) and Reγ2∈(A2,B2), where Ai and Bi, i=1,2, are chosen that(59)m+1-γα∉A1,B1andm+1-α+γα∉A2,B2,m=0,1,2,….
In the light of (56), we can pick up (A1,B1) and (A2,B2) as(60)A1,B1⊆Reγ1<1-γαandA2,B2⊆Reγ2<1-α+γα.
4. Some Technical Results
First we introduce some equivalent norms.
Proposition 8.
Let s and s1 be any nonnegative numbers. Then for any functions v and w, v∈Esβ(Ω¯)∩E-s1β(Ω¯), and w∈Csβ(Ω¯)∩C-s1β(Ω¯), we have the following norm equivalence: (61)C1vEsβΩ¯+vE-s1βΩ¯≤supΩ¯r-sx×1+r2xs1+s/2vx+supx,x¯∈Ω¯r-s+βx,x¯×1+r2x,x¯s1+s/2vx-vx¯x-x¯β≤C2vEsβΩ¯+vE-s1βΩ¯,C1wCsβΩ¯+wC-s1βΩ¯≤supΩ¯d-sx×1+dxs1+swx+supx,x¯∈Ω¯d-s+βx,x¯×1+dx,x¯s1+swx-wx¯x-x¯β≤C2wCsβΩ¯+wC-s1βΩ¯.
The proof of this statement follows with direct calculations. Let(62)a≥1,b∈0,2,z,r∈R.Next we represent certain estimates for the functions(63)I±1b,z,r=∫-∞+∞e±iprΓa±ip-zΓa+b±ipdp,which will be frequently used to evaluate the functions u(x1,x2) and Dxiαu, i=1,2.
Proposition 9.
Let ε be positive number, ε∈(0,1). Then there are estimates.(64)I±1b,z,r≤C1+z2coshπz2×e-εr,if b∈1,2;e-εr1+r-1,if b∈0,1;1+r-1,if b=0.
Proof.
For simplicity consideration, we put a=1. Here we prove this proposition for the function I+1(b,z,r). The case of I-1(b,z,r) is considered in the similar way. We start our consideration with the case of positive b (i.e., b≠0).
The Stirling asymptotic formula for the Gamma-function,(65)Γim+d≈Ce-π/2mim+dd-1/2×expimlnim+d-im-d×1+Oim+d3,asm→+∞,provides that the integrand Γ(1+i(p-z))/Γ(1+b+ip), b>0, has the order p-b at the infinity. Moreover, the function Γ(1+p¯-iz)/Γ(1+b+p¯) with p¯=Rep¯+ip has no poles in the domain Rep¯>-1. Thus, by the residue theorem, we get (66)I+1b,z,r=e-εr∫-∞+∞Γ1-ε+ip-zΓ1+b-ε+ipeiprdp,r≥0,eεr∫-∞+∞Γ1+ε+ip-zΓ1+b+ε+ipeiprdp,r<0=e-εr∫-∞+∞Γ1-εsgnr+ip-zΓ1+b-εsgnr+ipeiprdp.Then, following [33], we decompose the plane (p,z) into 15 subdomains, as shown in Figure 1(67)p,z=Ω0∪⋃i=17Ωi+∪⋃i=17Ωi-.In this decomposition M is a sufficiently large number such that the terms O((im+d)-3) in (65) can be neglected in all regions Ωi±.
Taking into account this decomposition, we represent the function I+1(b,z,r) as(68)I+1b,z,r=e-εr×∑i=27∫Ωi+Γ1-εsgnr+ip-zΓ1+b-εsgnr+ipeiprdp,z>2M,∑i=27∫Ωi-Γ1-εsgnr+ip-zΓ1+b-εsgnr+ipeiprdp,z<-2M,∫Ω1+∪Ω1-Γ1-εsgnr+ip-zΓ1+b-εsgnr+ipeiprdp+∫Ω0Γ1-εsgnr+ip-zΓ1+b-εsgnr+ipeiprdp,z<2M.It is apparent that, if z<-2M the function I+1(b,z,r) is estimated analogous to the function I+1(b,z,r) if z>2M. Thus, we consider here just case z>-2M. To this end, we apply Stirling formula (65) and the well-known properties of the Gamma-function. Introducing the functions (69)d1p,z=1-εsgnr+ip-z1+b-εsgnr+ip-εsgnr+1/2,d2p,z=expip-zln1-εsgnr+ip-z-ipln1+b-εsgnr+ip,we arrive at the representation to the function Γ(1-εsgnr+i(p-z))/Γ(1+b-εsgnr+ip) in every domain Ωi+, Ω1-, and Ω0,(70)Γ1-εsgnr+ip-zΓ1+b-εsgnr+ip=Φ0p,z,sgnr,in Ω0,d1p,zd2p,z1+b-εsgnr+ipbΦ1p,z,sgnr,in Ω1+,eπz/2d1p,zd2p,z1+b-εsgnr+ipbΦ2p,z,sgnr,in Ω2+,eπp/21+b-εsgnr+ipbΦ3p,z,sgnr,in Ω3+,eπp-z+πz/2d1p,zd2p,z1+b-εsgnr+ipbΦ4p,z,sgnr,in Ω4+,eπ/2p-z1-εsgnr+ip-εsgnr+1/2Φ5p,z,sgnr,in Ω5+,e-πz/2d1p,zd2p,z1+b-εsgnr+ipbΦ6p,z,sgnr,in Ω6+∪Ω7+,d1p,zd2p,z1+b-εsgnr+ipbΦ7p,z,sgnr,in Ω1-,where the functions Φi(p,z,sgnr), i=0,1,2,…,7, are uniformly bounded with respect to p and z together with (∂Φi/∂z)(p,z,sgnr).
Moreover,(71)∂Φ1∂p,∂Φ2∂p,∂Φ6∂p,∂Φ7∂p≤Cp3,for (p,z) belongs to one of the corresponding domains Ω1+, Ω2+, Ω7+, Ω1-. It is worth mentioning that representation (70) holds for each b∈[0,2].
If b∈(1,2], representation (70) with aid (65) and (71) arrives at the inequality(72)I+1b,z,r≤Ce-εr1+zcoshπz2,ifz>2M,e-εr1+z,ifz≤2M.These relations guarantee the first estimate in Proposition 9 for b∈(1,2].
Further, we verify the estimate I+1(b,z,r) if b∈(0,1]. Representation (70) and straightforward calculations lead to the inequality(73)∫Ω0Γ1-εsgnr+ip-zΓ1+b-εsgnr+ipeiprdp+∑i=36∫Ωi+Γ1-εsgnr+ip-zΓ1+b-εsgnr+ipeiprdp≤C1+zcoshπz2.To get the same estimate in the domains Ωi+, i=1,2,7, and Ω1-, it is enough to integrate by parts. Namely, let us consider the case Ω1+. Based on representation (70), we can write(74)∫3M+∞Γ1-εsgnr+ip-zΓ1+b-εsgnr+ipeiprdp=∫3M+∞d1p,zd2p,z1+b-εsgnr+ipbΦ1p,z,sgnr×eiprdp=iei3Mrrd13M,zd23M,z1+b-εsgnr+i3Mb×Φ13M,z,sgnr+ir×∫3M+∞eipr∂∂pd1p,zd2p,z1+b-εsgnr+ipb×Φ1p,z,sgnrdp.It is apparent that (75)d13M,z+d23M,z+Φ13M,z,sgnr≤C∂d1p,z∂p+∂d2p,z∂p≤C1+z2p.Thus, we can enhance estimate (74), taking into account (71), so as to get(76)∫3M+∞eiprΓ1-εsgnr+ip-zΓ1+b-εsgnr+ipdp≤C1+z2r.Recasting the arguments above in the case of domains Ω2+, Ω6+, and Ω7+ yields the same estimate to ∫Ω2+∪Ω7+∪Ω1-eiprΓ1-εsgnr+ip-z/Γ1+b-εsgnr+ipdp.
After that, we combine inequalities (73) and (76) and obtain the required estimate to I+1b,z,r if b∈(0,1].
At last, we are left to produce suitable estimate of I+10,z,r. Standard direct calculations lead to(77)d1p,z+d2p,z≤C1+zforp,z∈⋃i=26Ωi+∪Ω0andb∈0,2.In conclusion, we derive that(78)∫⋃i=36Ωi+∪Ω0Γ1+ip-zΓ1+ipeiprdp≤Ccoshπz21+z.
After that, we remark that the terms d1(p,z), d2(p,z), and Φi share the same properties in Ω1- and Ωi+, i=1,2,6,7. Thus, the estimate of ∫Ωi+(Γ(1+i(p-z))/Γ(1+ip))eiprdp is very similar and we will confine ourselves the consideration of the case Ω1+.
Simple conclusions draw to representations in Ω1+(79)d1p,z=1+g1z,p,p-zln1+p-z2-pln1+p2=-zlnp-z+g2p,z,where(80)g1≤C1+z2p2andg2≤Cp.Therefore, the main term in the integral ∫Ω1+(Γ(1+i(p-z))/Γ(1+ip))eiprdp is(81)∫3M+∞eirpe-iz-izlnpdp=e-iz∫0+∞eirpe-izlnpdp-∫03Meirpe-izlnpdp.Note that the second term here is a regular function of r and z. In order to evaluate the first term in this representation, we apply Proposition 7.1 from [33] and obtain (82)∫0+∞eirpe-izlnpdp=eizlnrzr-1g3z+1+zr-1g4z,where g3 and g4 are twice continuously differentiated and bounded functions for z<2M.
In summary, we obtain the estimate(83)∫Ω1+Γ1+ip-zΓ1+ipeiprdp≤C1+r-1.Finally, this inequality together with (78) completes the proof of Proposition 9.
Recasting the proof of the previous proposition, we state the results, which will be used to evaluate Dxiαiu, i=1,2, 0<αi<α, in Sections 5 and 6.
Proposition 10.
Let z∈(-2M,2M), a≥1, b∈[0,2], r∈R; then the inequalities are fulfilled (84)∂∂zI±1b,z,r≤Ccoshπz2×e-εr1,ifb∈1,2,1+r-1,if b∈0,1,∂∂z∫Ω0Γa±ip-zΓa±ipe±iprdp+∂∂ze∓izlnr∫Ω1+∪Ω1-Γa±ip-zΓa±ipe±iprdp≤C1+r-1.
The following results are related to the properties of the function(85)I±2ε1,ε2,ε,b,z,ρ≔∫-ε1ρε2ρdre-εr×∫-∞+∞Γ1±εsgnr±ip-zΓ1+b±εsgnr±ipe±irp-zdp,where ε1, ε2 are some positive constant, ε∈[0,1), b∈[0,2].
Proposition 11.
Let z∈R, ρ∈(0,+∞). Then there are the following estimates for every fixed ρ:
Besides, if ρ=+∞, then(88)I±2ε1,ε2,0,b,z,+∞=∫-∞+∞dr∫-∞+∞Γ1±ip-zΓ1+b±ipe±irp-zdp=πΓ1+b±iz,ifb∈0,2,I±2ε1,ε2,ε,0,z,+∞≔I±2ε,z≤C1+zcoshπz2.
Proof.
We will carry out the detailed proof in I+2(ε1,ε2,ε,b,z,ρ) case. The arguments for I-2(ε1,ε2,b,z,ρ) are almost identical and left to the interested reader.
Straightforward calculations lead to the representation(89)I+2ε1,ε2,ε,b,z,ρ=2∫-∞+∞Γ1+ip-zΓ1+b+ip×sinε1+ε2p-zρp-zeiε2-ε1p-zρ/2dp≡2∫-∞+∞i2dp.First, we consider case b∈(0,2]. Asymptotic representation (70) with the decomposition in Figure 1 provides estimate in R2∖(Ω0∪Ω3+∪Ω3-∪Ω5+∪Ω5-)(90)i2≤C1+zcoshπz/2p-zp2+1+b-ε2b/2with ε∈(0,1) and(91)i2≤Csinε1+ε2p-zρε1+ε2p-zρ1+coshπz2ε1+ε2ρin Ω0∪Ω3+∪Ω3-∪Ω5+∪Ω5-. Then, this inequality together with (89) leads to statement (ii).
Let us verify point (ii). One can easily check that (92)sinε1+ε2p-zρp-z≤Cε1+ε2ρ,ifp-z≤M,p-z-1,ifp-z>M.After that, recasting the proof of Proposition 9 with aid the last inequality arrives at the estimate(93)∑j=36∫Ωj±i2dp+∫Ω0i2dp≤C1+zcoshπz21+ρε1+ε2.
Then we are left to evaluate the terms ∫Ωj±i2dp, j=1,2,7. We restrict ourselves the estimate of ∫Ω2+i2dp, the remaining terms are evaluated the same way. Asymptotic (70) with b=0 and the change of variables, p-z=k, provide the representation(94)e-πz/2∫Ω2+i2dp=∫2M+∞d1k+z,zd2k+z,z2ikΦ2k+z,z×eiε2ρk-e-iε1ρkdk.After that, following the proof of Proposition 9, we integrate by parts and deduce (95)e-πz/2∫Ω2+i2dp≤C1ρε1+1ρε2×d12M+z,zd22M+z,z2M×Φ22M+z,z+∫2M+∞eiε2ρk+e-iε1ρk×∂∂kd1k+z,zd2k+z,zΦ2k+z,zkdk.The standard calculations lead to(96)∂∂kd1k+z,zd2k+z,zΦ2k+z,zk≤1+zk2.In conclusion, we reach the estimate(97)∫Ω2+i2dp≤Ce-πz/21+z1ρε1+1ρε2.Collecting this inequality with (93), we deduce the first estimate in statement (ii) of Proposition 11.
Further, to improve the estimate for I+2(ε1,ε2,ε,0,z,ρ), we consider two different cases:
(i) (98)2Mρε1+ε2>π2,
(ii) (99)2Mρε1+ε2≤π2.
It is apparent that, in the first case there is(100)1ε1ρ+1ε2ρ<4Mπε1-1+ε2-1ε1+ε2,and, therefore, keeping in mind the previous estimate of I±2(ε1,ε2,ε,0,z,ρ), we deduce that(101)I+2ε1,ε2,ε,0,z,ρ≤C1+zcoshπz2×1+ρε1+ε2+ε1ε2+ε2ε1.
Coming to the second case (i.e., ρ-1>4M(ε1-1+ε2-1)π-1), the change of the variable, k(ε1+ε2)ρ=v, leads to(102)e-πz/2∫Ω2+i2dp=∫2Mρε1+ε2+∞d1z+vρε1+ε2,z×d2z+vρε1+ε2,zsinvveiv2ε1+2ε2-1ε2-ε1dv=∫2Mρε1+ε2π/2d1z+vρε1+ε2,z×d2z+vρε1+ε2,zsinvveiv2ε1+2ε2-1ε2-ε1dv+∫π/2+∞d1z+vρε1+ε2,z×d2z+vρε1+ε2,zsinvveiv2ε1+2ε2-1ε2-ε1dv.The straightforward calculations ensure the uniformly boundedness of the first term in (102); moreover, this estimate is independent of ε1, ε2, and ρ.
Further, we treat the second term in (102). Integrating by parts leads to (103)∫π/2+∞d1z+vρε1+ε2,zd2z+vρε1+ε2,z×Φ2z+vρε1+ε2,zsinvveiv2ε1+2ε2-1ε2-ε1dv=-i8eiπε2-ε1ε1+ε2-1/44π-πε2-ε12ε2+ε1-2×d1π2ρε1+ε2+z,zd2π2ρε1+ε2+z,z×Φ2π2ρε1+ε2+z,z-44-ε2-ε12ε2+ε1-2×∫π/2+∞e-iv+ivε2-ε1/2ε2+ε1×∂∂vd1vρ-1ε1+ε2-1+z,zd2vρ-1ε1+ε2-1+z,z×Φ2vρ-1ε1+ε2-1+z,zv-1dv.After that, the properties of functions d1, d2, and Φ2 allow us to extend this estimate and conclude (104)∫π/2+∞d1z+vρε1+ε2,zd2z+vρε1+ε2,z×sinvveiv2ε1+2ε2-1ε2-ε1Φ2z+vρε1+ε2,zdv≤C44-ε2-ε12ε2+ε1-2≤C1+ε2ε1+ε1ε2.Hence, this estimate completes the proof of statement (ii).
Finally, we are left to verify statement (iii). As for the first equality in (iii), it is a simple consequence of Lemma 5.3 [36]. Let us check the second inequality in (iii). To this end, we rewrite I+2(ε,z) as (105)I+2ε,z=∫-∞+∞Γ1+ε+ip-zΓ1+ε+ipdpε-ip-z+∫-∞+∞Γ1-ε+ip-zΓ1-ε+ipdpε+ip-z≡i1+z+i2+z,where we set (106)i1+z=∫-∞+∞Γ1+ε+ip-zΓ1+ε+ip2εdpε2+p-z2,i2+z=-∫-11dq×∫-∞+∞ε2ε+ip-z∂∂1-qεΓ1+qε+ip-zΓ1+qε+ipdp.Then representation (70) with b=0 and direct calculations provide(107)i1+z+i2+z≤C1+zcoshπz2,which implies(108)I+2ε,z≤C1+zcoshπz2.Thus, the claim is proven.
Finally, we complete this preliminary section with two estimates that will be frequently used in the following sections.
Proposition 12.
Let z∈[-2M,2M], x1, x2∈[0,+∞), r,ρ∈R, ε∈(0,1). Then the following inequalities are fulfilled:
where the positive constant C is independent of ρ, ε1, ε2, ε, x1, and x2.
Proof.
To show the validity of statement (i), it is enough to recast the arguments in Propositions 11 and 10 if z∈[-2M,2M].
Next, we obtain the second inequality from this proposition. For the sake of clarity, we consider case x2r/x1ρ≥1. In the opposite case, we exchange the function eizqln(x2r/x1ρ) by e-izqln(x1ρ/x2r) and repeat the arguments below.
The direct calculations together with the change of variables, v=2Mqln(x2r/x1ρ), reduce the function J1 to the form(112)J1=∫02Mlnx2r/x1ρsinvvdv.At this point, we consider the two different cases for J1:
(i) (113)2Mlnx2rx1ρ>π2,
(ii) (114)0≤b2Mlnx2rx1ρ≤π2.
In the first case, we have (115)J1≤∫0π/2sinvvdv+∫π/22Mlnx2r/x1ρsinvvdv≤π2+-cos2Mlnx2r/x1ρ2Mlnx2r/x1ρ+∫π/22Mlnx2r/x1ρcosvv2dv≤π2+2π+∫π/2+∞dvv2≤C.Note that we apply integrating by parts in order to reach the last estimate.
Further, we obtain the same estimate in the second case. To this end, we use the easily verified inequality,(116)sinvv≤10<v<π2,and deduce the bound(117)J1≤∫02Mlnx2r/x1ρdv≤∫0π/2dv≤C.In summary, we complete the proof of Proposition 12.
5. Estimates on Minor Derivatives of u(x1,x2)
To evaluate the function u(x1,x2), we use representation (57) with γ∈(0,1) and γ1=-αγ, γ2=-α(1-γ). The change of variables(118)t1=er1andt2=er2,in the integrals in (57) yields(119)ux1,x2=eiπγx1αγx2α1-δi8πα∫-∞+∞dr1×∫-∞+∞dr2fx1e-r1,x2e-r2Lγr1,r2,x1,x2,where (120)Lγr1,r2,x1,x2=∫-∞+∞K-γ-izα×eπz/αx1-izx2izeizr1-r2∫-∞+∞e-ip1r1×Γ1-ip1-zΓ1+αγ-ip1dp1∫-∞+∞eip2r2×Γ1+ip2-zΓ1+α-αγ-ip2dp2dz,γ∈0,1.
5.1. Estimate on Maximum of ux1,x2
Before evaluating the functions u and Dxiαiu, i=1,2, 0<αi≤1, we describe the suitable properties of the kernel Lγ.
Lemma 13.
Let ε∈(0,1) and let r1, r2∈R. Then there are estimates.
If γ=0 or γ=1, the following representations hold:(122)L0r1,r2,x1,x2=exp-εr1-εr2×δr1+δ-r1L01r1,r2,x1,x2+L02r1,r2,x1,x2,L1r1,r2,x1,x2=exp-εr1-εr2×δr2+δ-r2L11r1,r2,x1,x2+L12r1,r2,x1,x2,
where δ(r) is a Dirac delta function and (123)L01r1,r2,x1,x2+L11r1,r2,x1,x2≤C,L02r1,r2,x1,x2≤C1+r1-11+r2,L12r1,r2,x1,x2≤C1+r2-11+r1.
Here the constant C is independent on x1, x2 and γ.
Proof.
First of all, we verify statement (i). It is easy to see that inequality (51) guarantees for γ∈[0,1](125)K-γ-izαeπz/α≤Ce-3πz/α,z→+∞,zK1-izα=zK-izα≤C,z≤2M,∂∂zzK1-izα=∂∂zzK-izα≤C,z≤2M. Then the results of Proposition 9 with aid of (125) provide statement (i).
As for statement (ii), we confirm ourselves the case of γ=0, due to the arguments in the case δ=1 are similar.
First, we represent Lγ, γ∈(0,1), as (126)Lγr1,r2,x1,x2=e-εr1-εr2∫-∞+∞dzK-γ-izα×eπz/αeizlnx2/x1eizr1-r2∫-∞+∞e-ip1r1×Γ1-εsgnr1-ip1-zΓ1+αγ-εsgnr1-ip1dp1∫-∞+∞eip2r2×Γ1+εsgnr2+ip2-zΓ1+α-αγ+εsgnr2+ip2dp2.Here we use the same reasons as in the proof of Proposition 9.
Then, as mentioned in the proof of Proposition 7 (see (ii) there) the function K(-γ-iz/α) has the simple poles if γ=0,1. Thus, following Cauchy’s residue theorem, we rewrite Lγ as(127)Lγr1,r2,x1,x2=e-εr1-εr2L¯01r1,r2+L¯02r1,r2,x1,x2,γ=0,where we set (128)L¯01r1,r2=C∫-∞+∞e-ip1r1∫-∞+∞eip2r2×Γ1+εsgnr2+ip2Γ1+α+εsgnr2+ip2dp2,L¯02r1,r2,x1,x2=∫-∞+∞K-izα×eπz/αeizlnx2r1/r2x1eizr1-r2×∫-∞+∞e-ip1r1e-izlnr1×Γ1-εsgnr1-ip1-zΓ1-εsgnr1-ip1dp1×∫-∞+∞eip2r2eizlnr2×Γ1+εsgnr2+ip2-zΓ1+α+εsgnr2+ip2dp2dz.Concerning the estimate of L¯01(r1,r2), we apply Lemma 5.3 from [36] and deduce(129)L¯01r1,r2=Cδ-r1+δr1×∫-∞+∞eip2r2Γ1+εsgnr2+ip2-zΓ1+α+εsgnr2+ip2dp2.Then the asymptotic representation of Gamma-function (see, e.g., (1.5.15) in [1]),(130)Γp+aΓp+b=pa-b1+Op-1,argp+a<π,p→+∞,provides the estimate(131)∫-∞+∞eip2r2Γ1+εsgnr2+ip2-zΓ1+α+εsgnr2+ip2dp2≤C.Hence, we are left to evaluate the function L¯02. To this end, we rewrite the term L02 in the form(132)L¯02=∑i=13li,where we put (133)Bz,r1,r2=zK-izαeizr2-r1∫-∞+∞e-ip1r1×Γ1-εsgnr1-ip1-zΓ1-εsgnr1-ip1eizlnr1dp1∫-∞+∞eip2r2×Γ1+εsgnr2+ip2-zΓ1+α+εsgnr2+ip2e-izlnr2dp2,l1=∫z>2MK-izαeizlnx2/x1eizr2-r1∫-∞+∞e-ip1r1×Γ1-εsgnr1-ip1-zΓ1-εsgnr1-ip1dp1∫-∞+∞eip2r2×Γ1+εsgnr2+ip2-zΓ1+α+εsgnr2+ip2dp2,l2=B0,r1,r2∫-2M2Meizlnx2r1/x1r2-1zdz,l3=∫-2M2Meizlnx2r1/x1r2Bz,r1,r2-B0,r1,r2zdz.Treating the first term in (132) via Proposition 9 and inequalities (125), we arrive at(134)l1≤C1+r1-1.Concerning l2, we get(135)B0,r1,r2=CL¯01r1,r2,and therefore(136)l2=CL¯01∫01dqq∫-2M2M∂∂zeizqlnx2r1/x1r2dz.After that, statement (ii) of Proposition 12 (with r=r1,ρ=r2) provides(137)∫01dqq∫-2M2M∂∂zeizqlnx2r1/x1r2dz≤C.In summary, we can conclude that(138)l2+L¯01=δ-r1+δr1×C1+∫01dqq∫-2M2M∂∂zeizqlnx2r1/x1r2dz×∫-∞+∞eip2r2Γ1+εsgnr2+ip2Γ1+εsgnr2+α+ip2dp2≡δ-r1+δr1L01r1,r2,x1,x2,L01r1,r2,x1,x2<C.Finally, coming to l3, we first rewrite it as(139)l3=∫01dq∫-2M2Meizlnx2r1/x1r2∂∂qzBqz,r1,r2dz.Then, the tedious calculations with Propositions 9 and 10 and estimates (125) entail(140)l3≤C1+1+r2r1.Hence, representations (127) and (132) together with estimates (134), (138), and (140) provide statement (ii) for the function L0 where(141)L02r1,r2,x1,x2=l1+l3.At last, the proof of statement (iii) is simple consequences of statements (i) and (ii) from this proposition. Thus, the claim is proven.
We are ready now to state estimates of the function u(x1,x2).
Lemma 14.
Let assumptions (h1), (31), and (32) hold. Then the function u(x1,x2) represented with (119) satisfies inequalities (142)supΩ¯r-α-βx1+r2xs1+α/2u+supΩ¯x1-α+x2-αr-βxu≤CR0fEββΩ¯,supΩ¯d-α-βx1+dxs1+αu≤CR0fCββΩ¯.
Proof.
At the beginning, we verify the first inequality in this lemma. To this end, we evaluate the term r-α-βxu. Putting γ∈(1/α,1) in (119), we conclude(143)r-α-βxu≤r-βx∫-∞+∞dr1×∫-∞+∞dr2fx1e-r1,x2e-r2-fx1e-r1,x2×Lγr1,r2,x1,x2+r-βx×∫-∞+∞dr1fx1e-r1,x2×∫-∞+∞dr2Lγr1,r2,x1,x2≡r-βxJ1+J2;here we use the simple inequality(144)x1αγx2α1-γ<rαγxrα1-γx=rαx.
Concerning J1, we apply statement (i) from Lemma 13 with ε>β to deduce(145)J1≤Cfx2,0,Ωβ∫0+∞dr1e-εr1×∫0+∞x2βe-r2-1βe-εr21+r2-1dr2≤Cx2βfx2,0,Ωβ≤Crβxfx2,0,Ωβ
with the constant is independent of x1, x2, and γ.
Coming to the term J2, applying Proposition 11 to the function(146)∫-∞+∞dr2∫-∞+∞e-izr2eip2r2Γ1+ip2-zΓ1+α-αγ+ip2dp2,
and keeping in mind Proposition 9 with inequality (125), we get(147)∫-∞+∞Lγr1,r2,x1,x2dr2≤Ce-εr1∫-∞+∞K-γ-izαeπz/α1+z×coshπz/2Γ1+α-αγ+izdz≤Ce-εr1.
In conclusion, we have(148)r-βxJ2≤r-βxfEββΩ¯∫0+∞x22e2r1+x12β/2e-εr1dr1≤CfEββΩ¯,if ε>β.
Then, collecting this inequality with (143) and (145) yields(149)supΩ¯r-α-βxu≤CfEββΩ¯.
Further, we estimate r-α-βx1+r2xs1+α/2u. Assumption (31) provides that(150)fx1,x2=F0x1,x21+x12+x22s1+α/2where the function F0 satisfies conditions (31) and (32). Thus, substituting this representation of f in (143) and recasting arguments above, we deduce(151)supΩ¯r-α-βx1+r2xs1+α/2u≤CR0fEββΩ¯.It is easy to see that the second estimate of Lemma 14 is proved with the similar arguments.
Finally, we are left to estimate xi-αr-βxu, i=1,2. For simplicity consideration, we evaluate x1-αr-βxu. Another case is studied with the same arguments. Using statement (ii) of Lemma 13 with γ=1, we rewrite u(x1,x2) as (152)x1-αux1,x2=C∫-∞+∞dr1∫-∞+∞dr2fx1e-r1,x2e-r2×e-εr1+r2L11r1,r2,x1,x2δr2+δ-r2+C∫-∞+∞dr1×∫-∞+∞dr2fx1e-r1,x2e-r2-fx1e-r1,x2×L12r1,r2,x1,x2e-εr1+r2+C∫-∞+∞dr1fx1e-r1,x2∫-∞+∞L12r1,r2,x1,x2×e-εr1+r2dr2.After that, statements (ii) and (iii) of Lemma 13 and Proposition 10 yield inequality (153)supΩ¯x1-αr-βxu≤CfEββΩ¯.This completes the proof of Lemma 14.
5.2. Estimate on Dx1α1Dx2α2ux1,x2, 0<α1+α2≤1
First, based on representation (57) and formula (2.1.17) [1], we obtain(154)Dx1α1Dx2α2u=Dx2α2Dx1α1u=eiπγ8πα×x1αγ-α1x2α1-γ-α2∫-∞+∞dr1×∫-∞+∞dr2fx1e-r1,x2e-r2-fx1,x2×Mγr1,r2,x1,x2+eiπγfx1,x28πα×x1αγ-α1x2α1-γ-α2∫-∞+∞dr1×∫-∞+∞dr2Mγr1,r2,x1,x2,where we set (155)Mγr1,r2,x1,x2=∫-∞+∞dzK-γ+izαeπzαeizr2-r1e-izlnx1x2×∫-∞+∞e-ip1r1Γ1-ip1-zΓ1+αγ-α1-ip1dp1×∫-∞+∞eip2r2Γ1+ip2-zΓ1+α1-γ-α2+ip2dp2,γ∈α1α,1-α2α.
After that, recasting the arguments of Lemma 14 with aid of Propositions 9–11, we can state the following results.
Lemma 15.
Let ε∈(β,1) and γ∈[α1/α,1-α2/α], and let the nonnegative numbers α1 and α2 satisfy 0<α1+α2≤1. Then the kernel Mγ(r1,r2,x1,x2) possesses the following properties:
Taking into account (154), we will carry out the detailed proof of the inequality(163)supΩ¯r-α-β+α1+α2x1+r2xs1+α/2Dx1α1Dx2α2u+supΩ¯x1-α+1Dx1u≤CR0fEββΩ¯. The remaining terms in (i) and (ii) are estimated in the same way and with the recasting of the corresponding arguments of Lemma 14.
We begin to evaluate the first term in (163). To this end, we represent Dx1α1Dx2α2u as(164)Dx1α1Dx2α2u=eiπγ8παx1αγ-α1x2α1-γ-α2∑j=19Qj,where we put (165)Q1=∫0+∞dr1∫-r1r1fx1er1,x2er2-fx1er1,x2×Mγr1,r2,x1,x2dr2,Q2=∫0+∞dr1fx1er1,x2-fx1,x2×∫-r1r1Mγr1,r2,x1,x2dr2,Q3=∫-∞0dr1∫r1-r1fx1er1,x2er2-fx1er1,x2×Mγr1,r2,x1,x2dr2,Q4=∫-∞0dr1fx1er1,x2-fx1,x2×∫r1-r1Mγr1,r2,x1,x2dr2,Q5=∫0+∞dr2∫-r2r2fx1er1,x2er2-fx1,x2er2×Mγr1,r2,x1,x2dr1,Q6=∫0+∞dr2fx1,x2er2-fx1,x2×∫-r2r2Mγr1,r2,x1,x2dr1,Q7=∫-∞0dr2∫r2-r2fx1er1,x2er2-fx1,x2er2×Mγr1,r2,x1,x2dr1,Q8=∫-∞0dr2fx1,x2er2-fx1,x2×∫r2-r2Mγr1,r2,x1,x2dr1,Q9=fx1,x2∫-∞+∞dr1∫-∞+∞Mγr1,r2,x1,x2dr2.Statement (ii) of Lemma 15 provides(166)Q1+Q3+Q5+Q7≤Cx2βfx2,0,Ωβ∫0+∞dr1×∫0r1er2-1βMγr1,r2,x1,x2dr2+x1βfx2,0,Ωβ∫0+∞dr2×∫0r2er1-1βMγr1,r2,x1,x2dr1≤Cx1β+x2βfx1,0,Ωβ+fx2,0,Ωβ.As for terms Q2, Q4, Q6, and Q8, we apply statement (iii) of Lemma 15 and deduce (167)Q2+Q4+Q6+Q8≤Cx1β+x2βfx1,0,Ωβ+fx2,0,Ωβ.
Finally, statement (i) of Lemma 16 guarantees the estimate to Q9(168)Q9≤CrβxsupΩ¯r-βxf. Hence, representation (164) and inequalities for Qj entail(169)supΩ¯r-α-β+α1+α2xDx1α1Dx2α2≤CfEββΩ¯.In order to complete the proof, we need a similar estimate to the term x1-α+1r-βxDx1u. We exchange Mγ in (154) by the function M1 from (iv) in Lemma 15. After that, recasting the arguments above together with statement (iv) in Lemma 15 derives(170)supΩ¯x1-α+1r-βxDx1u≤CfEββΩ¯.This finishes the proof of the lemma.
The following result is a simple consequence of Lemmas 14 and 16, interpolation inequalities, and Proposition 8.
Proposition 17.
Let assumptions of Lemma 14 hold. Then (171)uEsβΩ¯+uE-s1βΩ¯≤CR0fEββΩ¯,uCsβΩ¯+uC-s1βΩ¯≤CR0fCββΩ¯.
6. Estimates on Dx1α1Dx2α2u(x1,x2), 1<α1+α2≤α
We begin our consideration with the estimates of the function Dx1αu. Here, we use representation (119) to the function u(x1,x2) with γ=1; i.e., Lγ=L1.
Recasting the arguments leading to (127) arrives at (172)ux1,x2=∫-∞+∞∫-∞+∞fey1,ey2×x1αL¯11y1-lnx1,y2-lnx2e-εy1-lnx1dy1dy2+∫-∞+∞∫-∞+∞fey1,ey2×x1αL¯12y1-lnx1,y2-lnx2,x1,x2×e-εy1-lnx1-εy2-lnx2dy1dy2where we put(173)L¯11y1-lnx1,y2-lnx2=-14α×resz=0K-1+izα∫-∞+∞Γ1+ip2-zΓ1+ip2e-ip2y2-lnx2dp2×∫-∞+∞Γ1-ip1-zΓ1+α-ip1eip1y1-lnx1dp1eπz/αeizy2-y1,L¯12y1-lnx1,y2-lnx2,x1,x2=-18πi×∫-∞+∞dzeπz/αK-1+izαeizy2-y1×∫-∞+∞Γ1+εsgny1-lnx1-ip1-zΓ1+εsgny1-lnx1-ip1eip1y1-lnx1dp1×∫-∞+∞Γ1-εsgny2-lnx2+ip2-zΓ1-εsgny2-lnx2+ip2e-ip2y2-lnx2dp2.After that, straightforward calculations arrive at(174)L¯11y1-lnx1,y2-lnx2=Cδy2-lnx2+δ-y2+lnx2×∫-∞+∞Γ1-ip1eip1y1-lnx1Γ1+α-ip1dp1.
Thus, we can rewrite u(x1,x2) in the form(175)ux1,x2=C∫-∞+∞dy1fey1,x2×∫-∞+∞Γ1-ip1Γ1+α-ip1x1αeip1y1-lnx1dp1+C∫-∞+∞fey1,ey2×x1αL¯12y1-lnx1,y2-lnx2,x1,x2×e-εy1-lnx1-εy2-lnx2dy1dy2.
We are now in the position to calculate Dx1α1u(x1,x2). Indeed, formula (2.1.17) [1] and direct calculations provide the representation(176)Dx1αux1,x2=∑j=13Ujx1,x2 where we put (177)U1=∫-∞+∞fey1,x2R1x1,y1-lnx1dy1,U2=fx1,x2∫-∞+∞∫-∞+∞R2x1,x2,y1-lnx1,y2-lnx2dy1dy2,U3=∫-∞+∞∫-∞+∞fey1,ey2-fx1,x2×R2x1,x2,y1-lnx1,y2-lnx2dy1dy2,with (178)R1x1,y1-lnx1=C∫-∞+∞eip1y1-lnx1dp1,R2x1,x2,y1-lnx1,y2-lnx2=e-εy2-lnx2Dx1α1x1αe-εy1-lnx1L¯12y1-lnx1,y2-lnx2,x1,x2.After that, introducing new variables(179)r1=lnx1-y1,r2=lnx2-y2,and new functions (180)B1z,r1,r2=eπz/αzK-1+izα×eizr2-r1eizlnr2/r1∫-∞+∞e-ip1r1×Γ1-εsgnr1-ip1-zΓ1-εsgnr1-ip1dp1∫-∞+∞eip2r2×Γ1+εsgnr2+ip2-zΓ1+εsgnr2+ip2dp2,R3x1,x2,r1,r2=R2x1,x2,-r1,-r2-e-εr1-εr2B10,r1,r2∫z<2Mdzz×∫01∂∂qe-iqzlnx1r2/x2r1dq,we rewrite the functions R1, R2, and Uj in the more comfortable forms(181)R1x1,-r1=Cδr1+δ-r1,R2x1,x2,-r1,-r2=e-εr1-εr2∫-∞+∞B1z,r1,r2e-izlnx1r2/x2r1dzz,U1x1,x2=Cfx1,x2,U2x1,x2=fx1,x2×∫-∞+∞∫-∞+∞R2x1,x2,-r1,-r2dr1dr2,U3x1,x2=∫-∞+∞∫-∞+∞fx1e-r1,x2e-r2-fx1,x2×R3x1,x2,r1,r2dr1dr2.
In order to reach the last equality in (181), we recast the arguments of Lemma 13 which lead to representations (132), (134), and (138). Indeed,(182)B10,r1,r2=Cδ-r1+δr1δ-r2+δr2,and(183)∫z<2Mdzz∫01∂∂qe-iqzlnx1r2/x2r1dq=∫01dqq∫-2M2M∂∂ze-iqzlnx1r2/x2r1dz≤C,where the positive constant C is independent of xi and ri.
Hence, we obtain(184)∫-∞+∞∫-∞+∞fx1e-r1,x2e-r2-fx1,x2×B10,r1,r2∫z<2Mdzz×∫01∂∂qe-iqzlnx1r2/x2r1dqdr1dr2=0.
6.1. Estimates on Maximum Dx1αux1,x2
Further, in virtue of representations (176) and (181), estimates of Dx1αux1,x2 and Dx1αuxi,0,Ωβ follow from the corresponding estimates of the functions Uj(x1,x2).
First, we describe the properties of the kernels R2 and R3.
Lemma 18.
Let ε∈(β,1), x1, x2∈[0,+∞). Then there are the following estimates.
where the positive constant C is independent of ε, x1, and x2.
Proof.
First, applying Proposition 11 with b=0 and ρ=+∞ and using definition (85), we have(188)∫-∞+∞dr1∫-∞+∞dr2R2x1,x2,-r1,-r2=∫-∞+∞eπz/αK-1+izαe-izlnx1/x2I+2ε,zI-2ε,zdz=∫z>2Meπz/αK-1+izαe-izlnx1/x2I+2ε,zI-2ε,zdz+B0∫z<2Mdzz∫01∂∂qe-izqlnx1r2/x2r1dq+∫-2M2Me-izlnx1r2/x2r1z∫01∂∂qeπqz/αqzK-1+izqα×eiqzlnr2/r1I+2ε,qzI-2ε,qzdqdz,where(189)B0=eπz/αzK-1+izα×e-izlnx1/x2eiqzlnr2/r1I+2ε,zI-2ε,zz=0.Then, applying statement (iii) in Propositions 11 and 12 with aid (125) arrives at the first inequality of this proposition.
Further, we proceed with a detailed proof of statement (ii). The proof of statement (iii) is almost identical and is left to the interested reader.
At the beginning, we evaluate the first term in the left-hand side of the inequality in (ii)(190)J1≔∫0+∞dr1∫0r1e±r2-1βR3x1,x2,r1,r2dr2.Simple calculations lead to(191)R3x1,x2,r1,r2eεr1+εr2=∑j=13R3j, where we set (192)R31=∫z>2MB1z,r1,r2e-izlnx1/x2dzz,R32=∫01dq×∫z<2Me-izlnx1r2/x2r1∂∂qzeizqlnr2/r1×B1qz,r1,r2dz.
At this point, we estimate each term R3j separately.
By Proposition 9 and estimates (125),(193)R31≤C1+r1-11+r2-1×∫2M+∞1+z2e-3πz/αcoshπz2dz<C1+r1-11+r2-1where the positive constant C is independent of ri and xi.
Concerning R32, keeping in mind Propositions 9-10 and estimates (125), we deduce(194)R32≤C1+r1-11+r2-1∫01dq∫z<2Mdz<C1+r1-11+r2-1where the positive constant C is independent of ri and xi.
In summary, we can conclude(195)J1≤C∫0+∞dr11+r1-1e-εr1×∫0r1e±r2-1βe-εr21+r2-1dr2<C.In order to obtain the last inequality, we use the condition ε>β.
Then, we are left to evaluate the second term in inequality (ii)(196)J2≔∫0+∞dr1e±r1-1β∫-r1r1R3x1,x2,r1,r2dr2.
To this end, coming to representation (191), we deduce(197)J2≤J21+J22,with(198)J21=∫0+∞dr1e±r1-1β∫-r1r1R31e-εr1-εr2dr2,J22=∫0+∞dr1e±r1-1β∫-r1r1R32e-εr1-εr2dr2.Proposition 9, statement (ii) in Proposition 11 (with ε1=ε2=1, ρ=r1), and estimate (125) lead to(199)∫-r1r1R31e-εr1-εr2dr2≤C1+r1+r1-1×∫z>2MK-1+izαeπz/α1+z22×cosh2πz2dz<C1+r1+r1-1.Thus, we have(200)J21≤C∫0+∞e-εr1e±r1-1β1+r1+r1-1<C.
Finally, Propositions 9 and 10 provide inequality(201)∫-r1r1R32e-εr1-εr2dr2≤C1+r1+r1-1∫01dq∫02Mdz<C1+r1+r1-1,which guarantees that(202)J22≤C∫0+∞e-εr1e±r1-1β1+r1+r1-1dr1<C.Then, collecting this inequality with the estimate for J21 yields(203)J2≤Cwith the positive constant C independent of x1 and x2.
The last inequality together with the analogous estimate for J1 completes the proof of statement (ii). This finishes the proof of Lemma 18.
Now we estimate the function Dx1αu.
Lemma 19.
Let assumptions of Lemma 14 hold. Then(204)supΩ¯r-βx1+r2xs1+α/2Dx1αu≤CR0fEββΩ¯,supΩ¯d-βx1+dxs1+αDx1αu≤CR0fCββΩ¯.
Proof.
Here we prove in detail the estimate(205)supΩ¯r-βxDx1αu≤CfEββΩ¯.The rest inequalities are obtained with the same techniques and with the recasting the arguments in Lemma 14.
By representations (176) and (181), we have(206)Dx1αu≤C∑j=13Uj≤Cfx1,x2+U2+U3.
Further, we estimate Uj separately.
By statement (i) in Lemma 18,(207)U2≤CrβxsupΩ¯r-βxf.Concerning U3, we rewrite this function in the form analogous to (164) (208)U3≤∑j=18U3j,where we put (209)U31=∫0+∞dr1×∫-r1r1fx1e-r1,x2e-r2-fx1e-r1,x2×R3x1,x2,r1,r2dr2,U32=∫0+∞dr1fx1e-r1,x2-fx1,x2×∫-r1r1R3x1,x2,r1,r2dr2,U33=∫-∞0dr1×∫r1-r1fx1e-r1,x2e-r2-fx1e-r1,x2×R3x1,x2,r1,r2dr2,U34=∫-∞0dr1fx1e-r1,x2-fx1,x2×∫r1-r1R3x1,x2,r1,r2dr2,U35=∫0+∞dr2×∫-r2r2fx1e-r1,x2e-r2-fx1,x2e-r2×R3x1,x2,r1,r2dr1,U36=∫0+∞dr2fx1,x2e-r2-fx1,x2×∫-r2r2R3x1,x2,r1,r2dr1,U37=∫-∞0dr2×∫r2-r2fx1e-r1,x2e-r2-fx1,x2e-r2×R3x1,x2,r1,r2dr1,U38=∫-∞0dr2fx1,x2e-r2-fx1,x2×∫r2-r2R3x1,x2,r1,r2dr1.
It is apparent that the estimates of each U3j are simple consequence of Lemma 18. Thus, we deduce(210)U3≤Cfx1,0,Ωβ+fx2,0,Ωβx1β+x2β≤CrβxfEββΩ¯.
The last inequality with (206) and (207) provides(211)Dx1αu≤CrβxfEββΩ¯.This finishes the proof of Lemma 19.
6.2. Estimates on Hölder Seminorms of Dx1αu(x1,x2)
First of all, we return to representations (176) and (181) and evaluate each function Ui(x1,x2) separately.
Proposition 20.
Let assumptions of Lemma 14 hold. Then(212)∑i=12U1xi,0,Ωβ+U2xi,0,Ωβx1β+x2β≤CfEββΩ¯.
Proof.
It is apparent that (181) and properties of the function f ensure estimates of U1xi,0,Ω(β).
Next we evaluate the term U2x1,0,Ω(β). For simplicity consideration, we assume(213)x1,x¯1∈0,+∞andx1<x¯1,and put(214)Δ1x=x1-x¯1,Δ1U2=U2x1,x2-U2x¯1,x2.In virtue of representation (181) and Lemma 18, we have (215)Δ1U2≤fx1,x2-fx¯1,x2∫-∞+∞dr1×∫-∞+∞dr2R2x1,x2,-r1,-r2+fx¯1,x2×∫-∞+∞dr1∫-∞+∞dr2R2x1,x2,-r1,-r2-R2x¯1,x2,-r1,-r2≤Cfx1,0,ΩβΔx1β+fx¯1,x2∫-∞+∞dr1×∫-∞+∞dr2R2x1,x2,-r1,-r2-R2x¯1,x2,-r1,-r2.Then we are left to tackle the second term in the right-hand side of the last inequality. Simple calculations lead to (216)R2x1,x2,-r1,-r2-R2x¯1,x2,-r1,-r2=Δ1x∫01∂∂qΔ1x+x¯1R2qΔ1x+x¯1,x2,-r1,-r2dq=Δ1x∫01dqe-εr1-εr2×∫-∞+∞e-izlnqΔ1x+x¯1r2/x2r1B1z,r1,r2qΔ1x+x¯1dz.At this point, recasting the proof of statement (i) in Lemma 18 arrives at (217)∫-∞+∞dr1∫-∞+∞dr2R2x1,x2,-r1,-r2-R2x¯1,x2,-r1,-r2≤CΔ1x∫01dqqΔ1x+x¯1<CΔ1xx¯1β∫01dqqΔ1x1-β=CΔ1xβx¯1β.
In conclusion, we obtain the estimate (218)fx¯1,x2∫-∞+∞dr1∫-∞+∞dr2R2x1,x2,-r1,-r2-R2x¯1,x2,-r1,-r2≤Cfx¯1,x2-f0,x2Δ1xβx¯1β.In order to reach the last inequality we use property (32) to the function f.
Summarizing, we have inequality(219)Δ1U2≤Cfx1,0,ΩβΔ1xββ.The same arguments in the case of the difference Δ2U2=U2(x1,x2)-U2(x1,x¯2) guarantee estimate(220)Δ2U2≤Cfx2,0,Ωβx2-x¯2β.
Thus, collecting estimates for ΔiU2 we reach the conclusion(221)∑i=12U2xi,0,Ωβ≤CfEββΩ.This completes the proof of Proposition 20.
Next we obtain the same results to the function U3(x1). For each point x1, x¯1, x2, x¯2∈[0,+∞): x1>x¯1 and x2>x¯2, we put(222)Δ1x=x1-x¯1,Δ2x=x2-x¯2.Further, we describe the properties of the kernel R3(x1,x2,r1,r2).
Lemma 21.
Let ε∈(β,1) and εj, j=1,2,3,4, be some positive numbers, εj∈[0,2], and let(223)Δixx¯j<13,i,j=1,2.Then the kernel R3 possesses the following:
where the positive constant C is independent of xi, x¯i, εj.
Since the proof of this result is technically tedious and repeating certain steps in the proof of Lemma 18, we provide one in the Appendix.
Then, based on Lemma 21, we can conclude the following.
Lemma 22.
Let assumptions of Lemma 14 hold. Then there is the estimate(228)∑i=12U3xi,0,Ωβ≤CfEββΩ.
Proof.
We provide here the estimate of U3x1,0,Ω(β). The case of U3x2,0,Ω(β) is treated in similar arguments.
It should be noted that it is enough to evaluate U3x1,0,Ω(β) in case of (223). Indeed, in the opposite case, its estimate follows from Lemma 19. Thus, we assume that (223) hold. Introducing the domain(229)D¯1x1,x2=y1,y2∈R2:e-y1-x1≤2Δ1x,e-y2-x2≤2Δ2x,,we rewrite the function U3 in a more suitable form: (230)U3=∬D¯1x1,x2fey1,ey2-fx1,x2×R3x1,x2,lnx1-y1,lnx2-y2dy1dy2+∬R2∖D¯1x1,x2fey1,ey2-fx1,x2×R3x1,x2,lnx1-y1,lnx2-y2dy1dy2.Then we consider the difference(231)Δ1U3≔U3x1,x2-U3x¯1,x2=∑j=14Vj,where we set (232)V1=∬D¯1x1,x2fey1,ey2-fx1,x2×R3x1,x2,lnx1-y1,lnx2-y2dy1dy2,V2=-∬D¯1x1,x2fey1,ey2-fx¯1,x2×R3x¯1,x2,lnx¯1-y1,lnx2-y2dy1dy2,V3=∬R2∖D¯1x1,x2fx¯1,x2-fx1,x2×R3x1,x2,lnx1-y1,lnx2-y2dy1dy2,V4=∬R2∖D¯1x1,x2fey1,ey2-fx¯1,x2×R3x1,x2,lnx1-y1,lnx2-y2-R3x¯1,x2,lnx¯1-y1,lnx2-y2dy1dy2.Then putting(233)a1x1=ln1+2Δ1xx1,a1x¯1=ln1+Δ1xx-1,a3x1=-ln1-2Δ1xx1,a3x¯1=-ln1-3Δ1xx-1,a2=ln1+Δ1xx2,a4=-ln1-2Δ1xx2,and making change of variables (179), we reduce the domain D¯1(x1,x2) to either D1(x1,x2) in the case of y1=-r1+lnx1 or D1(x¯1,x2) if y1=-r1+lnx¯1: (234)D1x1,x2=r1,r2∈R2:-a3x1<r1<a1x1,-a4<r2<a2,D1x¯1,x2=r1,r2∈R2:-a3x¯1<r1<a1x¯1,-a4<r2<a2.At this point, we estimate each term Vj separately.
Change of variables (179) leads to (235)V1=∫0a1x1dr1×∫-a4/a1x1r1a2/a1x1r1fx1e-r1,x2e-r2-fx1,x2×R3x1,x2,r1,r2dr2+∫-a3x10dr1×∫a4/a3x1r1-a2/a3x1r1fx1e-r1,x2e-r2-fx1,x2×R3x1,x2,r1,r2dr2+∫0a2dr2×∫-a3x1/a2r2a1x1/a2r2fx1e-r1,x2e-r2-fx1,x2×R3x1,x2,r1,r2dr1+∫-a40dr2×∫a3x1/a4r2-a1x1/a2r2fx1e-r1,x2e-r2-fx1,x2×R3x1,x2,r1,r2dr1.Applying decomposition like (164) to each term in the representation above, then statements (i) and (ii) in Lemma 21 provide estimate(236)V1≤CΔ1xβfx1,0,Ωβ+fx2,0,Ωβ.
Indeed, let us verify this with the example of the first term in the representation to V1. Standard calculations lead to (237)∫0a1x1dr1∫-a4/a1x1r1a2/a1x1r1fx1e-r1,x2e-r2-fx1,x2R3x1,x2,r1,r2dr2=∫0a1x1dr1∫-a4/a1x1r1a2/a1x1r1fx1e-r1,x2e-r2-fx1e-r1,x2R3x1,x2,r1,r2dr2+∫0a1x1dr1fx1e-r1,x2-fx1,x2×∫-a4/a1x1r1a2/a1x1r1R3x1,x2,r1,r2dr2Then, properties of the function f and Lemma 21 with ε3=a1(x1), ε2=a2/a1(x1), ε1=a4/a1(x1) yield (238)∫0a1x1dr1×∫-a4/a1x1r1a2/a1x1r1fx1e-r1,x2e-r2-fx1,x2×R3x1,x2,r1,r2dr2≤Cfx1,0,Ωβ+fx2,0,Ωβx2βa2β+x2βa4β+x1βa1βx1a2a4+a4a2≤CΔ1xβfx1,0,Ωβ+fx2,0,Ωβ.To reach the last inequality we apply (223) to (233). The remaining terms in the representation of V1 are evaluated in the same way.
Concerning V2, we repeat the arguments above with changing D(x1,x2) by D(x¯1,x2) and obtain the estimate like (236) to the function V2.
Coming to V3, change of variables (179), Lemma 21, and condition (223) arrive at(239)V3=fx¯1,x2-fx1,x2×∬R2R3x1,x2,r1,r2dr1dr2+∬D1x1,x2R3x1,x2,r1,r2dr1dr2≤CΔ1xβfx1,0,Ωβ.Finally, we are left to estimate V4. Change of variables (179) and statement (iv) in Lemma 21 lead to(240)V4≤CΔ1xβfx1,0,Ωβ+fx2,0,Ωβ.Collecting this inequality with (236) and (239), we get the bound(241)Δ1U3≤CΔ1xβfEββΩ¯.This finishes the proof of Lemma 22.
At this point, we come back to Dx1αux1,0,Ω(β). Keeping in mind Proposition 20, Lemma 22, and representations (176) and (181), we deduce the inequality(242)Dx1αux1,0,Ωβ+Dx1αux2,0,Ωβ≤CfEββΩ¯.
Further, recasting arguments in Proposition 20 and Lemmas 22 and 14 and applying Proposition 8, we assert the result.
Lemma 23.
Let assumption of Lemma 14 hold. Then (243)Dx1αuEββΩ¯+Dx1αuE-s1-αβΩ¯≤CR0fEββΩ¯,Dx1αuCββΩ¯+Dx1αuC-s1-αβΩ¯≤CR0fCββΩ¯.
In order to obtain the same results to the functions Dx1α1Dx2α2u and Dx2α2Dx1α1u, 1<α1+α2≤α, it is enough to repeat the arguments from this section. Thus, we have.
Remark 24.
Under conditions of Lemma 23, the functions Dx1α1Dx2α2u, Dx2α2Dx1α1u, 1<α1+α2≤α, satisfy inequalities (23) and (27) and(244)Dx1α1Dx2α2u=Dx2α2Dx1α1u.
7. Proofs of Theorems 3 and 4
Under restrictions (30) and (31), the proof of Theorems 3 and 4 follows from the arguments of Sections 3–6. Indeed, Proposition 17 and Lemmas 16, 19, and 23 with Remark 24 provide existence of a solution u which satisfies estimates (22), (23) and (26), (27). Next, direct calculations with aid of representation (176) allow one to verify that the constructed solution u with (57) satisfies (1) and the corresponding boundary conditions. Further, the uniqueness of the solution follows from the coercive estimates. This completes the proof of Theorems 3 and 4 if (30) and (31) hold.
At this point, we remove restriction (30). To this end, we introduce the new function(245)W1=x1x12+x22ψ1x1+x2x12+x22ψ2x2.The following properties of the function W1 could be easily checked with aid (2.2) in [37] and properties of the functions ψi and f (see (h2), (19)): if β>α-1, then(246)eitherW1∈Msα+βΩ¯∩M-s1α+βΩ¯,orW1∈Nsα+βΩ¯∩N-s1α+βΩ¯,fx1,0-Dx1αW1x1,x2x2=0-Dx2αW1x1,x2x2=0-Dx1αψ1x1=0,f0,x2-Dx1αW1x1,x2x1=0-Dx2αW1x1,x2x1=0-Dx2αψ2x2=0.After that, we look for the solution to (1), (2) in the form(247)ux,t=W+W1,where the unknown function W solves the boundary value problem with homogenous boundary condition(248)Dx1αW-Dx2αW=f¯x1,x2inΩ,W=0onΓ1∪Γ2,with(249)f¯x1,x2=fx1,x2-Dx1αW1x1,x2-Dx2αW1x1,x2.It is apparent that the function f¯ satisfies conditions (h3), (19). Thus, arguments of Sections 3–6 guarantee the one-valued solvability to (248) and, besides, the function W satisfies properties (22), (23), and (27) and (26). Then, we return the function u and obtain the one-to-one solvability of the original problem, where the solution u possesses the same properties as W. Thus, Theorems 3 and 4 are proven in the absence of (30).
Finally, we are left to remove restriction (31). To this end, it is worth to remark that a function f∈Es-αβ(Ω¯) which satisfies condition (31) belongs to f∈E-s¯-αβ(Ω¯) with any positive s¯ and hence also with s¯=s1, s1∈(0,2-α). Further, it is enough to repeat the arguments from Sections 3–7 in order to prove Theorem 3, if f satisfies (17). If f meets requirement (18), the same reasons hold. This finishes the proof of Theorems 3 and 4.
Remark 25.
Requirement (30) could be removed without introducing the function W1. To this end, it is enough to recast the arguments of Sections 3–6 to the problems with the homogenous equation and nonhomogeneous boundary conditions (250)Dx1αu-Dx2αu=0inΩ,subject either to the Dirichlet boundary condition (DBC) (251)u=ψ1x1onΓ1andu=ψ2x2onΓ2,or to the Neumann boundary condition (NBC) (252)∂u∂x2=φ1x1onΓ1and∂u∂x1=φ2x2onΓ2.
Remark 26.
Arguments from Sections 3–6 guarantee that the requirement on the weight s in the assumption (h1) can be changed by(253)s≥α+β.
Remark 27.
Actually, with inessential modifications in the proofs, the very same results hold for the more general equation(254)Dx1α1u+Dx2α2u=fx1,x2inΩ,where 1<α1,α2<2. The details are left to the interested reader.
AppendixProof of Lemma 21
To prove statement (i), we estimate the first term in this inequality. The second one is evaluated with the same arguments. Let us rewrite the first term in the form(A.1)∫0ε3dr1∫-ε1r1ε2r1e±r2-1βR3dr2≤C∑j=12∫0ε3dr1∫0ε2r1r2βeβr2R3jdr2+∫0ε3dr1∫0ε1r1r2βeβr2R3jdr2,where R3j is defined in (191). After that, (191), (193), and (199) arrive at (A.2)∫0ε3dr1∫-ε1r1ε2r1e±r2-1βR3dr2≤C∫0ε3dr11+r1-1×∫0ε2r1r2βe-ε-βr21+r2-1dr2+∫0ε1r1r2βe-ε-βr21+r2-1dr2≤Cε3βε2β+ε1βwith the positive constant C being independent of εi, xj, ε.
Note that to reach the last inequality, we use the fact that ε>β. Thus, claim (i) is proven.
Concerning statement (ii), we apply inequalities analogous to (199) and (201) and obtain(A.3)∫-ε1r1ε2r1R3dr2≤∑j=12∫-ε1r1ε2r1R3jdr2≤Ce-εr11+r1-11+r1ε1+ε2+ε1ε2+ε2ε1.Then, keeping in mind this inequality, we deduce (A.4)∫0ε3dr1e±r1-1β∫-ε1r1ε2r1R3dr2≤C∫0ε3r1β-1e-r1ε-β1+r1ε1+ε2+ε1ε2+ε2ε1dr1≤Cε3β1+ε3ε1+ε2+ε1ε2+ε2ε1.Next, recasting this argument in the case of the term ∫0ε3dr2e±r2-1β∫-ε1r2ε2r2R3dr1 provides the analogous estimate to the second term in the left-hand side of (ii). This completes the proof of (ii).
It is apparent that statement (iii) follows from direct calculations and representation (A.1) and Propositions 11 and 12.
As for (iv), we prove the first inequality; the second one is obtained with the same arguments. Let us consider the difference(A.5)ΔR3=R3x1,x2,r1+ln1+Δ1xx1,r2-R3x¯1,x2,r1,r2=I1+I2,where we put(A.6)Ii=R3ix1,x2,r1+ln1+Δ1xx1,r2-R3ix¯1,x2,r1,r2for i=1,2.
Introducing new functions(A.7)K1z=zK-1+izαeπz/αeizr2-r1,Ψz,r2=∫-∞+∞Γ1+εsgnr2+ip2-zΓ1+εsgnr2+ip2eip2r2dp2,we rewrite the functions Ii in the suitable forms (A.8)I1=e-εr2ln1+Δ1xx1×∫z>2MK1zeizlnx2/x¯1Ψz,r2z×∫01dp∂∂r1e-εr1+pln1+Δ1x/x1×∫-∞+∞Γ1-ε-ip1-zΓ1-ε-ip1×e-ip1r1+pln1+Δ1x/x1dp1dz,I2=e-εr2ln1+Δ1xx1∫z<2Mdzeizlnx2/x¯1×∫01dq∂∂qzK1qzΨqz,r2∫01dp∂∂r1e-εr1+pln1+Δ1x/x1×∫01dp∂∂r1e-εr1+pln1+Δ1x/x1×∫-∞+∞Γ1-ε-ip1-qzΓ1-ε-ip1∫01dp∂∂r1e-εr1+pln1+Δ1x/x1×e-ip1r1+pln1+Δ1x/x1dp1Recasting the arguments of Propositions 9 and 10 arrives at (A.9)∂∂r1e-εr1+pln1+Δ1x/x1∫-∞+∞Γ1-ε-ip1-zΓ1-ε-ip1×e-ip1r1+pln1+Δ1x/x1dp1+∂∂qzeizqlnr1e-εr1+pln1+Δ1x/x1×∫-∞+∞Γ1-ε-ip1-qzΓ1-ε-ip1×e-ip1r1+pln1+Δ1x/x1dp1≤Ce-εr1r1+pln1+Δ1x/x12.By this inequality together with Propositions 9 and 10, we have(A.10)I1+I2≤Ce-εr2r121+r2-1.In conclusion, we obtain (A.11)∫ε3+∞dr1∫0ε2r1e±r2-1βΔR3dr2≤Cln1+Δ1xx1∫ε3+∞dr1r12∫0ε2r1r2β-1e-ε-βr2dr2≤Cε2βε31-βln1+Δ1xx1.
Finally, we estimate the term ∫ε3+∞dr1e±r1-1β∫-ε1r1ε2r1ΔR3dr2. To this end, we again represent ΔR3 via Ij and then using Propositions 11 and 12 deduce (A.12)∫ε3+∞dr1e±r1-1β∫-ε1r1ε2r1ΔR3dr2≤C×ln1+Δ1xx11+ε1ε2+ε2ε1∫ε3+∞e-ε-βr1dr1r12-β≤Cln1+Δ1xx11+ε1ε2+ε2ε1ε3β-1.Collecting this inequality with (A.11), we obtain the first inequality in (iv). This completes the proof of Lemma 21.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
KilbasA. A.SrivastavaH. M.TrujilloJ. J.2006204AmsterdamElsevier ScienceNorth-Holland Mathematics StudiesMR2218073CarrerasB. A.LynchV. E.ZaslavskyG. M.Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model2001812509651032-s2.0-003567986810.1063/1.1416180ShlesingerM. F.WestB. J.KlafterJ.Lévy dynamics of enchaced diffusion: application to turbulemce198758111100110310.1103/PhysRevLett.58.1100MR884850ZaslavskyG. M.StevensD.WeitznerH.Self-similar transport in incomplete chaos19934831683169410.1103/PhysRevE.48.1683MR13779152-s2.0-0001554548BensonD. A.WheatcraftS. W.MeerschaertM. M.Application of a fractional advection-dispersion equation20003661403141210.1029/2000WR9000312-s2.0-0034032484BensonD. A.WheatcraftS. W.MeerschaertM. M.The fractional-order governing equation of Lévy motion20003661413142310.1029/2000wr9000322-s2.0-0034113992BensonD. A.SchumerR.MeerschaertM. M.WheatcraftS. W.Fractional dispersion Lévy motion, and the MADE tracer tests2001421-221124010.1023/A:1006733002131MR1948593DengZ.-Q.SinghV. P.BengtssonL.Numerical solution of fractional advection-dispersion equation200413054224312-s2.0-214275549010.1061/(ASCE)0733-9429(2004)130:5(422)HuntB.Asymptotic solutions for one-dimensional dispersion in rivers2006132187932-s2.0-3144443151710.1061/(ASCE)0733-9429(2006)132:1(87)KimS.KavvasM. L.Generalized Fick’s law and fractional ADE for pollutant transport in a river: detailed derivation2006111808310.1061/(ASCE)1084-0699(2006)11:1(80)TanW.FuC.FuC.XieW.ChengH.An anomalous subdiffusion model for calcium spark in cardiac myocytes2007911818390110.1063/1.2805208ChechkinA. V.GoncharV. Y.KlafterJ.MetzlerR.TanatarovL. V.Lévy flights in a steep potential well20041155-6150515352-s2.0-354307410510.1023/B:JOSS.0000028067.63365.04Zbl1157.82305ChechkinA. V.SliusarenkoO. Y.MetzlerR.KlafterJ.Barrier crossing driven by Lévy noise: Universality and the role of noise intensity200775410.1103/PhysRevE.75.041101KondratievV. A.Boundary value problems for elliptic equations in domains with conical or angular points196716227313MR0226187Kondrat’evV. A.OleinikO. A.Boundary-value problems for partial differential equations in non-smooth domains19833821862-s2.0-8495606838310.1070/RM1983v038n02ABEH003470BorsukM.KondratievV.200669Amsterdam, NetherlandsElsevier Science B.V.10.1016/S0924-6509(06)80026-7MR2286361GrisvardP.198524Boston, Mass, USAPitmanMR775683Maz′yaV. G.PlamenevskiĭB. A.Estimates in Lp and an Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary198412315610.1090/trans2/123/01Zbl0554.35035BarlesG.GeorgelinC.JakobsenE. R.On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations201425641368139410.1016/j.jde.2013.11.001MR3145761Zbl1285.351222-s2.0-84890870326BarlesG.ChasseigneE.GeorgelinC.JakobsenE. R.On Neumann type problems for nonlocal equations set in a half space201436694873491710.1090/S0002-9947-2014-06181-3MR3217703BjorlandC.CaffarelliL.FigalliA.Nonlocal tug-of-war and the infinity fractional Laplacian201265333738010.1002/cpa.21379MR2868849Zbl1235.352782-s2.0-84355166599CaffarelliL.SilvestreL.Regularity theory for fully nonlinear integro-differential equations200962559763810.1002/cpa.20274MR2494809Zbl1170.450062-s2.0-68049121967Ros-OtonX.SerraJ.The Dirichlet problem for the fractional Laplacian: regularity up to the boundary2014101327530210.1016/j.matpur.2013.06.003MR3168912ServadeiR.ValdinociE.Weak and viscosity solutions of the fractional Laplace equation201458113315410.5565/PUBLMAT_58114_06MR3161511LopushanskaG. P.Basic boundary value problems for a certain equation with fractional derivatives1999511485910.1007/BF02591914MR1712756MasaevaO. K.Dirichlet problem for the generalized Laplace equation with the Caputo derivative201248344945410.1134/S0012266112030184MR3172907Zbl1258.352002-s2.0-84860625138TurmetovB. K.TorebekB. T.OntuganovaS.Some problems for fractional analogue of laplace equation20149445255322-s2.0-84905755695TurmetovB. K.TorebekB. T.On solvability of some boundary value problems for a fractional analogue of the Helmholtz equation20142012371251MR3291617ErvinV. J.RoopJ. P.Variational formulation for the stationary fractional advection dispersion equation200622355857610.1002/num.20112MR2212226ErvinV. J.RoopJ. P.Variational solution of fractional advection dispersion equation on bounded domain in ℝd200723225628110.1002/num.201692-s2.0-33847793586DiethelmK.2010Berlin, GermanySpringer10.1007/978-3-642-14574-2MR2680847KrylovN. V.199612AMSGraduate Studies in Mathematics10.1090/gsm/012MR1406091VasylyevaN.On the solvability of some nonclassical boundary-value problem for the Laplace equation in the plane corner2007121011671200MR2362267Zbl1153.350922-s2.0-84860917031LagariasJ. C.Euler's constant: Euler's work and modern developments201350452762810.1090/S0273-0979-2013-01423-XMR3090422Milne-ThomsonL. M.1951London, UKMacMillanMR0043339BazaliyB. V.FriedmanA.The Hele-Shaw problem with surface tension in a half-plane: a model problem2005216238743810.1016/j.jde.2005.03.007MR2162341Zbl1075.760222-s2.0-24144488378BrunnerH.HanH.YinD.The maximum principle for time-fractional diffusion equations and its application201536101307132110.1080/01630563.2015.1065887MR3402825Zbl1333.65093