The existence of a generalized solution of elliptic boundary value problems and its character of singularities depending on power of data singularities are established

Lopushanska, H.P.

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

GENERALIZED SOLUTIONS OF ELLIPTIC BOUNDARY VALUEPROBLEMS WITH STRONG POWER SINGULARITIESc© H.P.LopushanskaThe existence of a generalized solution of elliptic boundary value problems and itscharacter of singularities depending on power of data singularities are established.In [1-3] and other articles the behavior of the generalized solutions to the ellipticboundary value problems with power singularities on the right-side was studied. It wasestablished that the generalized solutions in the sense [4] exist if the power growth ofthe problem data has the order λ > −n inside the domain and λ > max{1 − n,−n +2m−1−m′} on its boundary (m′ denotes the maximum order of the normal derivativesin the boundary conditions) and that these data require a regularization if their powergrowth is more strong [1,2].We trace the behavior of generalized solutions (in specific sense) for any power sin-gularities on the right-side without any using of the data regularization. We start froma representation of the solution.Consider the following problemA(x,D) = F0, x ∈ Ω, Bj(x,D)u |S = Fj , j = 1,m, (1)where Ω denotes a bounded domain in Rn, with a closed boundary S of class C∞,A(x,D) is an elliptic operator of order 2m, {Bj(x,D)}mj=1 are some normal system ofboundary differential expressions satisfying Lopatinsky’s condition. We assume that thecoefficients of operators are infinitely differentiable.Let Bˆj , Tj , Tˆj be such boundary differential operators with the infinitely differentiablecoefficients that Green’s formula∫Ω(Auv − uA∗v)dx =m∑j=1∫S(BjuTˆjv − TjuBˆjv)dSholds.We define now the following function spaces: D(Ω) = C∞(Ω), D(S) = C∞(S),X(Ω) = {ϕ ∈ D(S) : Bˆjϕ |S = 0, j = 1,m} and D′(Ω), D′(S), X ′(Ω) as spaces of linearcontinious functionals defined respectively on D(Ω), D(S), X(Ω). Let (ϕ,F ) denotethe action of F ∈ D′(Ω)(X ′(Ω)) onto ϕ ∈ D(Ω)(X(Ω)) and < ϕ,F >–the action ofF ∈ D′(S) onto ϕ ∈ D(S).For the case F0 ∈ X ′(Ω), Fj ∈ D′(S), j = 1,m, we define the solution of the problem(1) as such generalized function u ∈ D′(Ω) that the equality(A∗ψ, u) = (ψ,F0) +m∑j=1< Tˆjψ, Fj > (2)is fulfilled for any ψ ∈ X(Ω).In [4,6] there is established the existence and is studied the properties of Green’svector-function (G0(x, y), G1(x, y), . . . , Gm(x, y)) of the problem (1) on the class of thefunctions u(x) which are orthogonal to the kernel N of the problem (1) (i.e. Pu = 0).If(ψ,F0) +m∑j=1< Tˆjψ, Fj >= 0 (3)for any ψ ∈ N∗ (N∗ is the kernel of the adjoint problem), the solution of the problem(1) in the sense (2) exists in D′(Ω)/N . It is defined by the formula(ϕ, u) = (∫Ωϕ(x)G0(x, y)dx, F0) +m∑j=1<∫Ωϕ(x)Gj(x, y), Fj >,ϕ ∈ D(Ω).(4)The function with strong power singularities doesn’t belong to D′(Ω) or X ′(Ω) but(4) can be extend to this case also. We consider special function spaces for it.Let x0 denote a given point in Ω, %(x, x0) = %0(x − x0) be nonnegative compactlysupported infinitely differentiable function in Ω, wich has order d(x, x0) = |x − x0| inneighborhood of the point x0 ∈ Ω, %(x0, x0) = 0.For k ∈ R1 we define spaces Zk(Ω, x0) = {ϕ ∈ C∞(Ω \ x0) : %|α|(x, x0)Dαϕ(x) =%k(x, x0)ϕα(x), ϕα(x) ∈ C(Ω) for arbitrary multi-index α}.We shall say that the sequence ϕν → 0 in the space Zk(Ω, x0), if for all multi-indexα the sequence ϕαν(x) = %−k+|α|(x, x0)Dαϕν(x) uniformly tends to zero in Ω underν →∞.We now notice the following main properties of the functions of these spaces:1) Z0(Ω, x0) ⊂ C(Ω), C∞(Ω) ⊂ Zk(Ω, x0) for all k ≤ 0;2) if ϕ ∈ Zk(Ω, x0), than, for all λ ∈ R1, |x− x0|λϕ ∈ Zk+λ(Ω, x0);3) if ϕ ∈ Zk(Ω, x0), than Dγϕ ∈ Zk−|γ|(Ω, x0) for any multi-index γ;4) if ϕ ∈ Zk(Ω, x0), ψ ∈ Zp(Ω, x0), than ϕψ ∈ Zk+p(Ω, x0);5) Zk2(Ω, x0) ⊂ Zk1(Ω, x0) for k1 < k2;6) Zk(Ω, x0) ⊂ C [k](Ω) for k ≥ 0 ( here [k] denotes the integral part of the numberk).We denote the spaces of linear continuous functionals defined on Zk(Ω, x0) by Z ′k(Ω, x0).Then1) Z ′k1(Ω, x0) ⊂ Z ′k2(Ω, x0) for k1 < k2; if k ≥ 0, than (C [k](Ω))′ ⊂ Z ′k(Ω, x0); sinceC∞(Ω) = D(Ω) ⊂ Z−k(Ω, x0) for k ≥ 0, Z ′−k(Ω, x0) ⊂ D′(Ω) for k ≥ 0.2) If F ∈ Z ′k(Ω, x0), than D|α|F ∈ Z ′k+|α|(Ω, x0) for any multi-index α.3) Z−k(Ω, x0) ⊂ Z ′k(Ω, x0).Indeed, fα(x) = %k+|α|D|α|f ∈ C(Ω) ⊂ L1(Ω), if f ∈ Z−k(Ω, x0), and then the fol-lowing linear continuous functionals fα on Zk(Ω, x0) are defined: (ϕ, fα) =∫Ωϕαfαdx =∫Ω%−k+|α|Dαϕfαdx =∫Ω%−k+|α|Dαϕ%k+|α|Dαfdx =∫ΩDαϕ%2|α|Dαfdx, ϕ ∈ Zk(Ω, x0),for any multi-index α, in particular (ϕ, f) =∫Ωϕ0f0dx =∫Ω%−kϕ%kfdx =∫Ωϕfdx, ϕ ∈Zk(Ω, x0). Hence if f ∈ Z−k(Ω, x0), f will be a regular generalized function of Z ′k(Ω, x0).4) Let gα ∈ L1(Ω) for any multi-index α, then for any ϕ ∈ Zk(Ω, x0) the expressions∫Ω%−k+|α|Dαϕgαdx exist and for any natural number N we have that the functiong(x) =∑|α|≤NDα((−1)|α|gα%−k+|α|) (derivatives are regarded in generalized sense) isa linear continuous functional on Zk(Ω, x0): (ϕ, g) =∑|α|≤N∫Ω%−k+|α|gαDαϕdx. Inparticular g(x) = g0(x)(x− x0)−κ ∈ Z ′|κ|(Ω, x0).5) For any multi-index α, ϕ ∈ Zk(Ω, x0), bounded functions gα(x) in Ω and anynumbers pα > −n, the expression∫Ωgα%pα+|α|−kDαϕdx exists. Then g(x) =∑|α|≤NDα(gα%pα+|α|−k) ∈ Z ′k(Ω, x0).In particular,g(x) = g0(x)(x− x0)−κ ∈ Z ′|κ|−n+ε(Ω, x0)for bounded function g0(x) in Ω and any ε > 0.Notice, that g(x) ∈ D′(Ω) for |κ| < n and g(x) /∈ D′(Ω) for |κ| ≥ n.Let f0(x) be a bounded function in Ω, F0(x) = f0(x)(x − x0)κ, |κ| ≥ 0,F1 = · · · =Fm = 0, N∗ = {0}. We shall obtaine that the solution u(x) of the problem (1), suchas Pu = 0, belongs to Z ′|κ|−2m−n+ε(Ω, x0) for any ε > 0. It follows from the followingtheorem.Theorem 1 Let F0 ∈ Z ′p(Ω, x0), p > 2m− n, F1 = · · · = Fm = 0, N∗ = {0}, u(x) besuch solution of the problem (1) that Pu(x) = 0. Then u(x) ∈ Z ′p−2m(Ω, x0).This conclusion is exact in the sense that there exists F0(x) = Au(x) ∈ Z ′p(Ω, x0),for the solution u(x) ∈ Z ′p−2m−ε(Ω, x0) ⊂ Z ′p−2m(Ω, x0), ε > 0, of the problem, and itis possibly, that doesn’t exist such F0 = Au(x) ∈ Z ′p(Ω, x0), for the solution u(x) ∈Z ′p−2m+ε(Ω, x0). Really, for u(x) ∈ Z ′p−2m+ε(Ω, x0), we have Au(x) ∈ Z ′p+ε(Ω, x0), andZ ′p(Ω, x0) ⊂ Z ′p+ε(Ω, x0), for ε > 0.Proof. It is shown in [5] that ψ(y) = (G∗0ϕ)(y) =∫Ωϕ(x)G(x, y)dx ∈ X(Ω) ⊂ D(Ω)for any ϕ ∈ D(Ω). Let us study its properties for ϕ ∈ Zk(Ω, x0), x0 ∈ Ω.Let h(x) ∈ D(Ω), 0 ≤ h(x) ≤ 1, h(x) ={1, |x− x0| < η0, |x− x0| > 2η,ψ(y) = (G∗0ϕ)(y) =∫Ωh(x)ϕ(x)G(x, y)dx+∫Ω(1− h(x))ϕ(x)G(x, y)dx = ψ1(y) + ψ2(y).The function (1 − h(y))ψ(y) = 0, if |y − x0| < η, therefore (1 − h(y))ψ(y) ∈Zk+2m(Ω, x0) for any k ∈ R1, ϕ ∈ Zk(Ω, x0).The function (1 − h(x))ϕ(x) = 0, if |x − x0| < η, therefore also h(y)ψ2(y) ∈Zk+2m(Ω, x0) for any ϕ ∈ Zk(Ω, x0), k ∈ R1.The function h(y)ψ1(y) = O(%2m+k(y, x0)), if ϕ ∈ Zk(Ω, x0), k > −n. We shall ob-taine that, for any multi-index α, the function vα(y) = %|α|(y, x0)Dα∫Ωh(x)ϕ(x)G0(x, y)h(y)dx = wα(y)%2m+k(y, x0), where wα ∈ C(Ω).Indeed, we assume the function vα(y) in the form of the sum v1α(y)+v2α(y)+v3α(y)of three items respectively to the partition of the domaine Ω into Ω1 = {x ∈ Ω :|x − x0| < |y−x0|2 }, Ω2 = {x ∈ Ω : |x − y| < |y−x0|2 }, Ω3 = Ω \ (Ω1 ∪ Ω2). Later theestimates of the derivatives of G0(x, y) are using.As a result, (G∗0ϕ)(y) ∈ Zk+2m(Ω, x0), if ϕ ∈ Zk(Ω, x0) and k > −n, or (G∗0ϕ)(y) ∈Zp(Ω, x0), for ϕ ∈ Zp−2m(Ω, x0) and p > 2m− n.Then the map F0 → u, defined by (ϕ, u) = (G∗0ϕ,F0), determines u ∈ Z ′p−2m(Ω, x0)for any F0 ∈ Z ′p(Ω, x0), if p > 2m − n. Since G∗0(A∗ψ) = ψ, defined by that mapfunction u satisfies the condition (2) for any ψ ∈ Zp−2m(Ω, x0), and, therefor, anyψ ∈ Zp−2m(Ω, x0) ∩X(Ω).The spaces Zk(S, x0), x0 ∈ S, are defined similarly and we obtaine that ψj(y) =(G∗jϕ)(y) ∈ Zk+mj+1(S, x0), j = 1,m, if ϕ ∈ Zk(Ω, x0), k > −n.Therefore, the map Fj → uj , defined by (ϕ, uj) =<∫Ωϕ(x)Gj(x, y)dx, Fj >, deter-mines uj ∈ Z ′p−mj−1(Ω, x0) for any Fj ∈ Z ′p(S, x0), if p > mj + 1− n.Let F0 ∈ Z ′p(Ω, x0), Fj ∈ Z ′pj (S, xj), x0 ∈ Ω, xj ∈ S, j = 1,m, p = (p, p1, . . . , pm),Xp =Xp(Ω, x0, x1, . . . , xm) = {ϕ(x) ∈ Zp(Ω, x0)∩ (∩mj=1Zpj (S, xj)) : Bˆjϕ |S = 0} (if xi = xj ,the space Zpi ∩ Zpj remaines by Zk(S, xi), k = max{pi, pj}).It can be shown that, in the case N∗ = {0}, the function (4) satisfies the equality (2)for any ψ ∈ Xp(Ω, x0, x1, . . . , xm), if p > 2m − n(x0 ∈ Ω), p > max{2m − n,m′′}(x0 ∈S), pj > max{1 − n +mj ,mj + 1 − 2m +m′′}, j = 1,m, m′′ denotes the maximum ofthe degrees of Bˆl, l = 1,m.So, we obtaine that, in the case N∗ = {0},F0 = f0(x)(x− x0)−κ, Fj = fj(x)(x− xj)−κj ,f0(x) ∈ L∞(Ω), fj(x) ∈ L∞(S)(5)(then F0 ∈ Z ′|κ|−n+ε|(Ω, x0), Fj ∈ Z ′|κj |−n+1+εj (S, xj), ε, εj > 0), j = 1,m, |κ| >2m − ε(x0 ∈ Ω), |κ| > max{2m,n +m′′} − ε(x0 ∈ S), |κj | > max{mj , n − 2m +mj +m′′} − εj , j = 1,m, the formula (4) determines the solution of the problem (1)u(x) = u0(x) +m∑j=1uj(x),u0(x) ∈ Z ′|κ|−n+ε−2m(Ω, x0), uj(x) ∈ Z ′|κj |−n−mj+εj (Ω, xj),in the sense of fulfilment (2) for any ψ ∈ Xp, where p = |κ|−n+ε, pj = |κj |−n+1+εj , j =1,m. The existence of such function ψ may be proved.This solution can have the power singularities of the order |κ| − 2m + ε inside Ω,|κj | −mj + 1 + ε on its boundary S,j = 1,m.We obtaine the similar results in the case of the power singularities of the right-handside data on any smooth closed manifold S1 inside Ω.There are similar properties of the solutions of some boundary value problems forthe elliptic operators in fractional derivativesAu(x) =1(2pi)−n∫Rna(x, ξ)Fu(ξ)ei(x,ξ)dξ,where Fu denotes Fourier transform of the function u(x),a(x, ξ) =N∑j=1∑|α|=sj≤saα(x)(−iξ)α,α = (α1, . . . , αn), αi, sj , s are nonnegative numbers, fractional in general, aα(x) ∈C∞(Rn).We assume the existence of the normal fundamental solution ω(x, y) ∈ C∞(x 6= y)satisfying the following estimate|ω(x, y)| ≤{C|x− y|s−n, n is oddC|x− y|s−n(ln|x|+ 1), n is even .In the case of constant coefficients such fundamental solution exists.References1. .., , . . . 20 (1968), no. 3, 412-417.2. Roitberg Y., Elliptic Boundary Value Problems in the Spaces of Distributions. Kluver AcademicPublishers. (1996).3. .., , . ... 36 (1971), no. 1, 202-209.4. .., , . ... 33 (1969), no. 1, 109-137.5. .., .., , . 22 (1987), no. 3, 518-521.6. .., .., ., . . . 19 (1967), no. 5, 3-32.Ivan Franko Lviv State UniversityUniversitetskaya str., 1290001 Lviv, UkraineE-mail: diffeq@ franko.lviv.uaTel: 794-593

Generalized solutions of elliptic boundary value problems with strong power singularities

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Generalized solutions of elliptic boundary value problems with strong power singularities

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)