Fractional-order elliptic problems are investigated in case of inhomogeneous
Dirichlet boundary data. The boundary integral form is proposed as a suitable
mathematical model. The corresponding theory is completed by sharpening the
mapping properties of the corresponding potential operators. Also a mild
condition is provided to ensure the existence of the classical solution of the
boundary integral equation

Izsák, Ferenc

Maros, Gábor

English

arXiv

Fractional-order elliptic problems are investigated in case of inhomogeneous Dirichlet boundary data. The boundary integral form is proposedas a suitable mathematical model. The corresponding theory is completedby sharpening the mapping properties of the corresponding potential operators. The existence-uniqueness result is stated also for two-dimensionaldomains. Finally, a mild condition is provided to ensure the existence of the classical solution of the boundary integral equation

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Fractional order elliptic problems with inhomogeneous Dirichlet boundary conditions

Fractional-order elliptic problems are investigated in case of inhomogeneous Dirichlet boundary data. The boundary integral form is proposed as a suitable mathematical model. The corresponding theory is completed by sharpening the mapping properties of the corresponding potential operators. The existence-uniqueness result is stated also for two-dimensional domains. Finally, a mild condition is provided to ensure the existence of the classical solution of the boundary integral equation

arXiv:1904.10734v2  [math.AP]  13 May 2020FRACTIONAL ORDER ELLIPTIC PROBLEMS WITHINHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONSFERENC IZSÁK AND GÁBOR MAROSAbstract. Fractional-order elliptic problems are investigated in case of inho-mogeneous Dirichlet boundary data. The boundary integral form is proposedas a suitable mathematical model. The corresponding theory is completedby sharpening the mapping properties of the corresponding potential opera-tors. Also a mild condition is provided to ensure the existence of the classicalsolution of the boundary integral equation.1. IntroductionAnalytic and numerical study of space-fractional diffusion problems became animportant research area in the past twenty years. A large number of real-life ob-servations confirmed the presence of the of the so-called anomalous (or fractional)diffusion. The fractional Laplacian, which can be given in many equivalent waysin Rd, seems to be the most adequate differential operator for modeling this phe-nomena. At the same time, on a bounded domain, the incorporation of boundaryconditions into a true mathematical model is by far not easy. Even to deal withhomogeneous Dirichlet boundary data is non-trivial [5]: the most meaningful [8]approach is given by the power of the Dirichlet-Laplacian. A similar approachcan be performed for the case of the homogeneous Neumann (no-flux) boundaryconditions.At the same time, only a few attempts [1] and [4] were made to incorporateinhomogeneous boundary conditions into a partial differential equation, which isgiven on a bounded domain (corresponding to the real-life situation).In any case, we should use fractional order differential operators, which are non-local. At the same time, in real-life situations, we do not have any data outsideof a physical domain. This basic difficulty motivates us to find an appropriateextension, which is formulated on Rd. Such an approach was successfully appliedin [11] but the corresponding extension can not be used for arbitrary domains inRd. Also, the simple choice in [7] and [3] can not be applied to inhomogeneousboundary data. Another difficulty in the analysis is that no generalization of theGreen formula is available for the fractional Laplacian.2000 Mathematics Subject Classification. Primary .Key words and phrases. space-fractional diffusion, boundary integral equation, inhomogeneousDirichlet boundary conditions, surface potential.This work was completed in the ELTE Institutional Excellence Program (1783-3/2018/FEKUTSRAT) supported by the Hungarian Ministry of Human Capacities.The project has been supported by the European Union, co-financed by the European SocialFund (EFOP-3.6.3-VEKOP-16-2017-00002).12 FERENC IZSÁK AND GÁBOR MAROSIt was pointed out in [6] that the fractional order elliptic equations with inho-mogeneous boundary conditions can be succesfully analyzed in the framework ofboundary integral equations. Our aim is to extend this result in the following sense:• the well-posedness is stated in two space dimensions,• the mapping properties of the corresponding single layer operator are gen-eralized,• a condition is given for the existence of a classical (pointwise) solution.An important motivation of this study is to prepare a numerical simulation in 2D,where the boundary integral form reduces the problem to compute one-dimensionalintegrals.1.1. Mathematical preliminaries. The main problem in this study is the precisestatement and the analysis of the elliptic boundary value problem(1.1){−(−∆)αu(x) = f(x) x ∈ Ωu(x) = g(x) x ∈ ∂Ω,where Ω ⊂ Rd is a bounded Lipschitz domain (d = 2, 3), α ∈ (12 , 1) and f, g are givenreal functions. At this stage, the differential operator (−∆)α is not yet defined. Inany case, it should be linear, such that u can be given as u = u1 + u2, where(1.2){−(−∆)αu1(x) = f(x) x ∈ Ωu1(x) = 0 x ∈ ∂Ωand(1.3){−(−∆)αu2(x) = 0 x ∈ Ωu2(x) = g(x) x ∈ ∂Ω.To deal with these problems, we recall that the fractional Dirichlet Laplacian(−∆D)α is defined as a power of−∆D : L2(Ω) → L2(Ω), Dom(−∆D) = H10 (Ω) ∩H2(Ω).This makes sense since (−∆D)−1 : L2(Ω) → L2(Ω) is compact, positive and self-adjoint.Accordingly, u1 in (1.2) can be given as u1 = −(−∆D)−αf . Therefore, we shallfocus to rewrite the problem in (1.3) for u2.The fractional Laplacian on Rd has many equivalent definitions [9]. This operatorcan be defined pointwise as(1.4) − (−∆Rd)αu(x) = limr→0+22αΓ(d2 + α)πd2 Γ(−α)∫B(0,r)Cu(x+ z)− u(x)|z|d+2α dz,whereB(x, r)C = Rd\B(x, r). Accordingly, the weak fractional Laplacian (−∆Rd)αucan be given as the function for which∫Rdv(−∆Rd)αu =∫Rdu(−∆Rd)αvis satisfied for all v ∈ C∞0 (Rd).We will make use of the fundamental solution φα of (−∆Rd)α, which is givenwithφα = F−11|Id|2α = 2α−n2Γ(α2)Γ(n−α2) |Id|2α−d.FRACTIONAL ORDER ELLIPTIC PROBLEMS WITH INHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS3In the analysis, we use the notation Hs(Ω) ⊂ L2(Ω) for the classical Sobolevspaces with arbitrary positive indices. Recall that the corresponding norms on Rdcan be defined by using the Fourier transform F as follows:(1.5) ‖u‖2Hs(Rd) =∫Rd(1 + |r|2)s|Fu|2(r) dr.For stating the well-posedness, we need also the Sobolev spaceḢs(Rd) ={u ∈ L2,loc(Ω) :∫Rd|r|2s|Fu|2(r) dr <∞}with the corresponding norm. If the underlying domain (Ω, ∂Ω or Rd) is obvious,simply the notation ‖ · ‖s will be used for the corresponding norms.An important tool in the analysis is the trace operator γ. For a bounded Lipschitzdomain Ω,(1.6) γ : Hs(Ω) → Hs− 12 (∂Ω)is continuous for s ∈ (12 , 32 ), see [10], Theorem 3.38. Also, one can define its Banachadjoint as a continuous operator with(1.7) γ∗ : H12−s(∂Ω) → H−s(Ω).We use the conventional notation 〈·, ·〉−β,β for the duality pairing betweenH−β(∂Ω)and Hβ(∂Ω) with some positive exponent β.We also recall that the Bessel function J0 : R+ → R of first kind is given with(1.8) J0(s) =12π∫ π−πe−is sin τ dτ.In the estimations, the relation K1 . K2 means that K1 ≤ CK2 for some domain-dependent constant C ∈ R+.1.2. The main objective, comparison with earlier achievements. In [6], theproblem in (1.3) for u2 was transformed to a boundary integral equation and thefollowing result was established.Theorem 1.1. For any bounded Lipschitz domain Ω ⊂ Rd with d ≥ 3 and g ∈Hα−12 (∂Ω) with α ∈ (12 , 1), the problem(1.9)−(−∆)αũ(x) = 0 x ∈ (∂Ω)Cũ(x) = g(x) x ∈ ∂Ω|ũ(x)| . |x|2α−d x ∈ B(0, 1)Chas a unique weak solution ũ ∈ Ḣα(Rd).Observe that (1.9) with the definition u2 := ũ|Ω delivers a precise setting forthe second problem in (1.3). Our aim is to sharpen this result and prove that theunique solution of (1.9)(i) exists also in case of d = 2 provided that α ∈ (12 , 34 ),(ii) satisfies −(−∆)αũ(x) = 0 also pointwise for any x ∈ (∂Ω)C under someweak condition.4 FERENC IZSÁK AND GÁBOR MAROS2. Main results2.1. Estimates for single layer potentials. We first investigate the fractionalversion Rα of the classical Newton potential, which is defined with(2.1) Rα(u)(x) =∫Rdφα(x− y)u(y) dy,and also called the Riesz potential.Lemma 2.1. Assume that α ∈ (0, 1) for d = 3 or α ∈ (12 , 34 ) for d = 2. Themapping Rα in (2.1) defines then a continuous linear map between Hs−2α(Ω) andHs(Ω), i.e. we have(2.2) ‖Rα(u)‖s . ‖u‖s−2α.Proof. We first define an extension of Rα(u) withR̃α(u)(x) =∫Rdλ(|x − y|)φα(x− y)u(y) dy,where λ ∈ C∞0 [0,∞) with λ|[0,2diam Ω] = 1 and 0 ≤ λ ≤ 1. Since R̃α(u) is anextension of Rα(u), we obviously have(2.3) ‖R̃α(u)‖s ≥ ‖Rα(u)‖s.Since this is a convolution, we can give its Fourier transform as(2.4)F[R̃α(u)](r)= F [u] (r) · F [λ(|z|)φα(z)] (r) =Cα4π∫Rde−i〈r,z〉λ(|z|)|z|2α−d dz.We estimate the second component, which, using polar coordinates in 3D, can berewritten asF [λ(|z|)φα(z)] (r) =Cα2∫ ∞0r2α−1λ(r)∫ π0e−ir|r| cos θ sin θ dθ dr=Cα2∫ ∞0r2α−1λ(r)sin(r|r|)r|r| dr.For |r| ≥ 1, we introduce w = r|r| and use that λ is compactly supported and2α− 2 ≤ 0 to get(2.5)F [λ(|z|)φα(z)] (r) =Cα2|r|2α∫ ∞0w2α−1λ(w|r|)sinwwdw .1|r|2α .1|1 + r2|α .For |r| ≤ 1, we only use again use that λ is compactly supported and 2α−1 > −1to have(2.6) F [λ(|z|)φα(z)] (r) =Cα2∫ ∞0r2α−1λ(r)sin(r|r|)r|r| dr . 1.In the 2D case, the polar transformation and the definition of J0 in (1.8) givesF [λ(|z|)φα(z)] (r) =Cα2∫ ∞0r2α−1λ(r)∫ 2π0e−ir|r| cos θ dθ dr= Cαπ∫ ∞0r2α−1λ(r)J0(r|r|) dr.FRACTIONAL ORDER ELLIPTIC PROBLEMS WITH INHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS5For |r| ≥ 1 with w = r|r|, we apply the estimate J0(w)√w . 1 (see 9.2.1 in [2]),which implies that(2.7)F [λ(|z|)φα(z)] (r) =Cα2|r|2α∫ ∞0w2α−1λ(w|r|)J0(w) dw .∫ ∞0w2α−32λ(w|r|)dw.1|1 + r2|α .provided that 2α− 32 ∈ (−1, 0], i.e. α ∈ (14 , 34 ].For |r| ≤ 1, we again only use 2α− 1 > −1 and the boundedness of the remainingcomponents to have(2.8) F [λ(|z|)φα(z)] (r) = Cαπ∫ ∞0r2α−1λ(r)J0(r|r|) dr . 1.Using (2.3) together with (1.5) and the estimates (2.5), (2.6), (2.7) and (2.8) in(2.4), we finally obtain that‖Rα(u)‖2s ≤ ‖R̃α(u)‖2s =∫Rd(1 + |r|2)s∣∣∣F[R̃α(u)](r)∣∣∣2dr=∫|r|≤1|F [u] |2(r)(1 + |r|2)s |F [λ(|z|)φα(z)]|2 (r) dr+∫|r|≥1|F [u] |2(r)(1 + |r|2)s |F [λ(|z|)φα(z)]|2 (r) dr≤∫|r|≤1|F [u] |2(r)(1 + r2)sC̃2R(r) dr+∫|r|≥1|F [u] |2(r)(1 + |r|2)s 2CR|1 + r2|2α dr≤ CR∫Rd|F [u] |2(r)(1 + |r|2)s−2α dr = ‖u‖2s−2α,as stated in the lemma. We need, however, the surface potential corresponding to Rα on ∂Ω, which isgiven for any x ∈ Rd with(2.9) Sα(u)(x) =∫∂Ωφα(x− y)u(y) dy.In precise terms, we state the following generalization of (4.1) in [6].Theorem 2.2. For any indices s, α satisfying the assumptions in Lemma 2.1 and2α − s ∈ (12 , 32 ), the mapping Sα defines a continuous linear operator betweenHs−2α+12 (∂Ω) and Hs(Ω), i.e. for all ψ ∈ Hs−2α+ 12 (∂Ω), we have(2.10) ‖Sα(ψ)‖s . ‖ψ‖s−2α+ 12,∂Ω.Proof. We first use the definitions in (2.1), (2.9) and the adjoint trace operator γ∗in (1.7) to rewrite Sα(ψ) asSα(ψ)(x) =∫∂Ωφα(x− y)ψ(y) dy =∫Ωφα(x− y)(γ∗ψ)(y) dy = Rαγ∗(ψ)(x).The smoothness of φα(x− ·) also implies that Sα(ψ) ∈ Hs(Ω). Accordingly, usingalso (2.2), (1.6) and (1.7), we have for any φ ∈ H−s(Ω) and ψ ∈ Hs−2α+ 12 (∂Ω) that|〈φ, Sα(ψ)〉−s,s| = |〈φ,Rαγ∗(ψ)〉−s,s| ≤ ‖Rαγ∗ψ‖s‖φ‖−s . ‖γ∗ψ‖s−2α‖φ‖−s. ‖ψ‖s−2α+ 12,∂Ω‖φ‖−s.6 FERENC IZSÁK AND GÁBOR MAROSTherefore, we arrive at the estimate‖Sα(ψ)‖s . ‖ψ‖s−2α+ 12,∂Ω,which completes the proof. Theorem 2.3. The map γSα is a coercive operator between H1/2−α(∂Ω) andHα−1/2(∂Ω) in the sense that(2.11) 〈u, γSαu〉 12−α,α− 12=∫∂Ω(Sαu)(x)u(x) dx & ‖u‖21/2−α,∂Ω.Proof. We first recall that according to the proof of Theorem 4.1. in [6], the lefthand side of (2.11) can be given as(2.12) 〈u, γSαu〉 12−α,α− 12=∫Rd|r|4−2α|F(S1u)|2(r) dr.In concrete terms, see (5.2) in [6]. Also, according to Theorem 4.1. in [6], for anyv ∈ H1(Rd) and u ∈ H1/2−α(∂Ω), we can rewrite 〈u, γv〉 12−α,α− 12as〈u, γv〉 12−α,α− 12=∫Rd∇S1u(x)∇v(x) dx.The basic step for proving our statement is to rewrite the fractional order norm asfollows:(2.13) ‖u‖1/2−α,∂Ω = supφ∈Hα−1/2(∂Ω)‖φ‖α−1/2=1|〈u, φ〉 12−α,α− 12| = supφ∈H1/2(∂Ω)‖φ‖α−1/2=1|〈u, φ〉 12−α,α− 12|,where in the second equality, we have used the density of H1/2(∂Ω) in Hα−1/2(∂Ω).Let ε : Hs(∂Ω) → Hs+1/2(Rd) denote the right inverse of the trace operator γ,which is continuous for s ∈ (0, 1), see [10], Theorem 3.37. Using (2.13), convert-ing everything to the Fourier space, applying the Cauchy–Schwarz inequality, theformula in (1.5) with the continuity of ε, we finally haveFRACTIONAL ORDER ELLIPTIC PROBLEMS WITH INHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS7‖u‖1/2−α,∂Ω = supφ∈H1/2(∂Ω)‖φ‖α−1/2=1∫Rd∇S1u(x)∇εφ(x dx)= supφ∈H1/2(∂Ω)‖φ‖α−1/2=1∫Rd|r|2F(S1u)(r)F(εφ)(r) dr= supφ∈H1/2(∂Ω)‖φ‖α−1/2=1∫Rd|r|2(1 + |r|)αF(S1u)(r) · (1 + |r|)αF(εφ)(r) dr≤ supφ∈H1/2(∂Ω)‖φ‖α−1/2=1[∫Rd|r|4(1 + |r|)2α |F(S1u)|2(r) dr]1/2 [∫Rd(1 + |r|)2αF(εφ)(r)(r) dr]1/2≤ supφ∈H1/2(∂Ω)‖φ‖α−1/2=1[∫Rd|r|4−2α|F(S1u)|2(r) dr]1/2‖εφ‖α,Rd. supφ∈H1/2(∂Ω)‖φ‖α−1/2=1[∫Rd|r|4−2α|F(S1u)|2(r) dr]1/2‖φ‖α−1/2,∂Ω=[∫Rd|r|4−2α|F(S1u)|2(r) dr]1/2.Therefore, using (2.12), we get〈u, γSαu〉 12−α,α− 12=∫Rd|r|4−2α|F(S1u)|2(r) dr & ‖u‖21/2−α,∂Ω,as stated in the theorem. We are ready now to prove the main statement of the article.Theorem 2.4. Assume that α ∈ (12 , 1) for d = 3 or α ∈ (12 , 34 ] for d = 2. Thenfor any g ∈ Hα− 12 (∂Ω), there is a unique function G ∈ H 12−α(∂Ω) such thatu = Sα(G) ∈ Ḣα(Rd) solves the problem in (1.9).If, we have additionally G ∈ L1(∂Ω), then the pointwise equality −(−∆)αũ = 0 in(∂Ω)C is also satisfied.Proof. Taking the special case α = s in Theorem 2.2, we have thatγSα : H12−α(∂Ω) → Hα− 12 (∂Ω).is continuous. Also, according to Theorem 2.3, γSα is coercive. Consequently, thisconstitutes a bijection between H12−α(∂Ω) and Hα−12 (∂Ω), so that for any giveng ∈ Hα− 12 (∂Ω), there is a unique function G ∈ H 12−α(∂Ω) such thatγSα(G) = g.Also, as pointed out in [6], Sα(G) ∈ Ḣα(Rd) and−(−∆Rd)αSα(G) = 0 on ΩC8 FERENC IZSÁK AND GÁBOR MAROSin weak sense, so that (1.9) with the conditions in the present theorem has a uniquesolution.To prove the pointwise identity −(−∆)αũ = 0 in (∂Ω)C , we use the definition in(1.4) so that−(−∆Rd)αSα(G)(x) = limr→0+22αΓ(d2 + α)πd2 Γ(−α)∫B(0,r)CSα(G)(x + z)− Sα(G)(x)|z|d+2α dz.Therefore, by the definition in (2.9), we have to verify that(2.14) limr→0+∫B(0,r)C1|z|d+2α∫∂Ω(φα(x + z− y)− φα(x− y))G(y) dy dz = 0.We may assume that here r ≤ r∗ := 12d(x,Ω) and note that |z| . |x − y + z| onB(0, r)C \B(y − x, r∗), which, along with the definition of φα, imply that∫B(0,r)Cφα(x+ z− y)|z|d+2α dz =∫B(0,r)C\B(y−x,r∗)φα(x− y + z)|z|d+2α dz+∫B(y−x,r∗)φα(x− y + z)|z|d+2α dz.∫B(0,r)C|z|2α−d|z|d+2α dz+∫B(0,r∗)φα(z0)|z0 − (x− y)|d+2αdz0.∫ ∞rs−2dsd−1 ds+∫ r∗0s2α−dsd−1 ds . K <∞,which can be used to get the following inequality:(2.15)∫B(0,r)C|φα(x+ z− y) − φα(x− y)||z|d+2α dz dy . K +∫ ∞rφα(r∗)sd−1sd+2αds. K̃ <∞,where K does not depend on y. Therefore, we also have the following estimate:∫∂Ω|G(y)|∫B(0,r)C|φα(x+ z− y) − φα(x− y)||z|d+2α dz dy . ‖G‖L1(Ω)K̃ <∞,such that we can apply Tonelli’s theorem in (2.14) together with (2.15) to obtain(2.16)∫B(0,r)C1|z|d+2α∫∂Ω(φα(x+ z− y)− φα(x− y))G(y) dy dz=∫∂ΩG(y)∫B(0,r)Cφα(x+ z− y) − φα(x− y)|z|d+2α dz dy.Using the equality (−∆)αφα(x− y) = 0, we have that herelimr→0G(y)∫B(0,r)Cφα(x+ z− y)− φα(x− y)|z|d+2α dy dz = 0and by (2.15), we also obtain that∣∣∣∣∣G(y)∫B(0,r)Cφα(x+ z− y)− φα(x− y)|z|d+2α dz∣∣∣∣∣ . K̃1|G(y)|such that we can use the dominated convergence theorem in (2.16) to obtain thedesired equality in (2.14). FRACTIONAL ORDER ELLIPTIC PROBLEMS WITH INHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS9AcknowledgmentsThe project has been supported by the European Union, co-financed by theEuropean Social Fund (EFOP-3.6.3-VEKOP-16-2017-00002). This work was com-pleted in the ELTE Institutional Excellence Program (1783-3/2018/FEKUTSRAT)supported by the Hungarian Ministry of Human Capacities.References[1] N. Abatangelo and L. Dupaigne. Nonhomogeneous boundary conditions for the spectral frac-tional Laplacian. Ann. Inst. H. Poincaré Anal. Non Linéaire, 34(2):439–467, 2017.[2] M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs,and mathematical tables. U.S. Government Printing Office, Washington, D.C., 1964.[3] G. Acosta and J. P. Borthagaray. A fractional Laplace equation: regularity of solutions andfinite element approximations. SIAM J. Numer. Anal., 55(2):472–495, 2017.[4] H. Antil, J. Pfefferer, and S. Rogovs. Fractional Operators with Inhomogeneous BoundaryConditions: Analysis, Control, and Discretization. arXiv e-prints, page arXiv:1703.05256,Mar. 2017.[5] B. Baeumer, M. Kovács, M. M. Meerschaert, and H. Sankaranarayanan. Boundary conditionsfor fractional diffusion. J. Comput. Appl. Math., 336:408 – 424, 2018.[6] T. Chang. Boundary integral operator for the fractional Laplace equation in a boundedLipschitz domain. Integral Equations Operator Theory, 72(3):345–361, 2012.[7] Q. Du, M. Gunzburger, R. B. Lehoucq, and K. Zhou. A nonlocal vector calculus, nonlo-cal volume-constrained problems, and nonlocal balance laws. Math. Mod. Meth. Appl. Sci.,23(03):493–540, 2013.[8] F. Izsák and B. J. Szekeres. Models of space-fractional diffusion: a critical review. Appl.Math. Lett., 71:38–43, 2017.[9] M. Kwaśnicki. Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl.Anal., 20(1):7–51, 2017.[10] W. McLean. Strongly elliptic systems and boundary integral equations. Cambridge UniversityPress, Cambridge, 2000.[11] B. J. Szekeres and F. Izsák. A finite difference method for fractional diffusion equations withNeumann boundary conditions. Open Math., 13:581–600, 2015.Department of Applied Analysis and Computational Mathematics & MTA ELTENumNet Research Group, Eötvös Loránd University, Pázmány P. stny. 1C, 1117 -Budapest, HungaryE-mail address: izsakf@caesar.elte.huDepartment of Applied Analysis and Computational Mathematics, Eötvös LorándUniversity, Pázmány P. stny. 1C, 1117 - Budapest, HungaryE-mail address: magabor42@gmail.com

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Fractional order elliptic problems with inhomogeneous Dirichlet boundary conditions

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