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Globally existing solutions to the problem of Derichlet for the fractional 3D Poisson equation

A common approach is present concerning the problem of Dirichlet, both for bounded 3D domains and their (unbounded) complements, regarding the fractional (3D) Poisson equation.


Introduction
The topic for the fractional Laplacian has recently engaged an increasing interest (e.g. [4] and the big number of the relevant results cited there in). Nevertheless, the treatment for instance of the Dirichlet problem in the exterior of bounded, say 3D, domains seems to be not analysed for the fractional Laplace (and Poisson) equation. And similar remarks can be made concerning the (infinite) interior regularity of the fractional harmonic functions (satisfying the equation (−∆) α/2 u = 0), for a given domain D ⊂ R 3 . Here we deal with 3D bounded domains Ω ⊂ R 3 and their complements CΩ = R 3 \ (Ω ∪ Γ), with Γ = ∂Ω -the closed (2D) boundary surface of Ω, assumed of second order ( C 2 ) regularity. For the Laplacian degree α/2 in (−∆) α/2 we suppose here 1 < α ≤ 2 and the action (−∆) α/2 u is defined by its Fourier transform |ξ| αû (ξ), for u ∈ S ′ = S ′ (R 3 ) -the class of the Schwartz tempered distributions (e.g. [2]); |ξ| is the length of the vector ξ ∈ R 3 andû(ξ) = F[u](ξ) is the Fourier image of u. Concerning the Fourier transformation we proceed at the convention φ(ξ) = R 3 exp(−i x, ξ φ(x)dx and φ(x) = 1 (2π) 3 R 3 exp(i x, ξ φ (x)dx, φ ∈ S = S(R 3 ) -the Schwartz class of the fast decreasing (infinitely smooth) functions ( [2]), x, ξ is the scalar product of the vectors x, ξ. Thus we have (−∆) α/2 u(x) := F −1 [|ξ| αû (ξ)](x), u ∈ S ′ , x varying in R 3 , where F −1 is the inverse map of F. (The used here symbol (−∆) α/2 for the fractional Laplacian follows the introduced one in [4].) Before to sketch the essence of the approach used, let us note via the possible applications that it admits also to consider a relevant treatment of the fractional Helmholtz equation related for instance to electrostatics of heterogeneous material systems. Our approach to the problem of Dirichlet is based on exploring the simple but effective idea to act with the global Laplacian, i.e (−∆) α/2 u = F −1 [|.| α F], on such distributions u from S ′ that the action product (−∆) α/2 u coincides on Ω with a prescribed function (distribution) f , and possess traces (u| Γ ) on Γ (with prescribed values ϕ(x) of u| Γ , x ∈ Γ). Additionally noted, the way of looking for globally defined solutions (given a boundary value problem) has been firstly suggested from the 1D case of the Poisson equation (−∆) α/2 u = f , with ∆ = d 2 dx 2 , considered in unbounded intervals l 0 < x < ∞, l 0 ∈ R 1 . (We give in the Appendix some details to this case.) Slightly formalized, the above idea reads the following. Given a function f (x), x ∈ Ω, say bounded, i.e. f ∈ L ∞ (Ω), and boundary data ϕ(x), x ∈ Γ for instance ϕ ∈ L 2 (Γ), the point is to solve the (extended on R 3 ) equation (−∆) α/2 u = F 0 , with F 0 ∈ S ′ : F 0 | Ω = f , by a suitable distribution u ∈ S ′ , satisfying the condition u| Γ = ϕ. (As seen below, the proper choice for F 0 is: F 0 | CΩ = 0.) If assume found such u ∈ S ′ , we could get a globally existing (i.e. defined on R 3 ) solution of the problem under consideration, reformulated now in the form: a) (−∆) α/2 u| Ω = f, (u ∈ S ′ ); b) u| Ω = ϕ. (1.1) The above formulation actually gives the shortest illustration to the used here approach concerning the problem of Dirichlet (posed to given domain Ω, in particular -bounded), for the fractional Poisson equation. We are close now to the key question for existence of solutions in S ′ , of equation (1.1.a), at a sufficiently large class of boundary data. As a first accessory step to that goal, consider the volume type potential U β,f := , prompted from the analogy with the conventional case of (1.1.a), where α = 2, β = 1. Following [4], we will call such potentials Riesz (volume) potentials, andenlarging the terminology -the functions of the type V β,g := Γ g(y)ds y |x − y| β , (x ∈ R 3 ) will be called surface (or single layer) Riesz potentials (here g ∈ L 2 (Γ), by assumption, and ds y is the known surface differential element). The right value of β , namely β = 3 − α, is however directly shown from the well known (e.g. [8], [3], and also [4]) Fourier transform relation F : . And taking the precise expression instead , which satisfies the equation and f 0 [f ](y) = 0, when y ∈ CΩ, U * V is the convolution (see [2] for details) of U , V ∈ S ′ .
These calculations evidently remain valid also for the generalized expression of u α,f , Above and from now on we shall assume fulfilled the following requirements for the function f (x), when defined for x ∈ D: It is not difficult to establish that the given in (1.2) potential u α,f (x) is a continuous function in D ∪ Γ, therefore the trace ϕ α,f (x), x ∈ Γ, ϕ α,f := u α,f | Γ is continuous on Γ. And via the problem of Dirichlet, we found that u α,f is a solution of the equation (−∆) α/2 u| D = f , with u| Γ = ϕ α,f . The next step is to seek solutions in S ′ with arbitrary prescribed data, assumed in L 2 (Γ). Going to this direction, suppose u ∈ S ′ is a solution of the above equation. Then is a distribution supported on the surface Γ. We shall deal here with the case: we are interesting in solutions u ∈ S ′ of the equation is the supported on Γ delta function of Dirac, with a density function g = g(x) ∈ L 2 (Γ). (As known, e.g. [2], the action ( δ Γ [g], φ) of δ Γ [g] on an arbitrary φ ∈ S is defined by the next surface integral, (δ Γ [g], φ) := Γ g(y)φ(y)ds y .) The important partial case f = 0 ( u α,f = 0) concerns these distributions w ∈ S ′ , solving the equation below (for g varying in L 2 (Γ)): We call BF harmonic (basic fractional harmonic) functions in R 3 such solutions w , and the family S ′ α,f -of all solutions u ∈ S ′ to (1.4) (with g varying in L 2 (Γ)), can be called GS (global solutions) family.
Remarks: 1) Clearly for each two distributions u 1 , u 2 ∈ S ′ α,f the difference u 2 − u 1 is a BF harmonic function.

The zero kernel of the boundary integral operator
It turns out that the kernel of the operator B α,Γ (acting from L 2 (Γ) into L 2 (Γ)) consists only of the zero element g = 0, i.e. the unique solution of the equation The key to this very important property lies in a simple but essential relation in the form I α (∞) = const.J Γ,α , where I α (∞) = lim r→∞ I α (r) and the terms I α (r), J Γ,α present respectively the integrals: Clearly the above relation (when holds) means in particular that the integral I α (∞) = .) The mentioned equality shall be found as a specific consequence of the well known Parseval equality (e.g. [2], [6]). To this goal we shall begin by considering a complement to Parseval's equality idea. (2.1)

Proof:
Note firstly that ψ is the complex conjugated quantity to ψ and recall that the notation (δ Γ [g], ψ) expresses the action of δ Γ [g], as a distribution in S ′ , on the function ψ -as an arbitrary element of S. Thus (δ Γ [g], ψ) = Γ g(x)ψ(x)ds x , and by analogy about the notation The proof uses the approximation approach to (2.1) following the two step scheme: obtain is the space of the compactly supported infinitely smooth functions. And apply afterwards an approximation procedure with w n (w n ∈ C ∞ 0 (R 3 ), n = 1, 2, . . . ) tending to δ Γ [g], at n → ∞. The first step is done in the given lemma.
Below we add a consequence of (2.1), useful for the basic result in this section.
In addition (δ Γ [g], ψ g ) evidently equals to the right hand integral above. Clearly, it is not difficult to check the two assumptions regarding ψ g : as a first, it is directly seen thatψ g ∈ L 1 (R 3 ), secondly, from , the validation whether ψ g ∈ C ∞ (R 3 ) gets obvious. Thus the proof of (2.4) follows directly from (2.1). Now the basic result in Section 2 can be presented.
The zero is not an eigen value of the boundary integral operator i.e. the only solution of the equation B α,Γ [g] = 0 is g = 0.

Main results
The found property of the operator B α,Γ is certainly of an essential importance in our approach for solving the problem of Dirichlet. It is in a direct relation with the well known Hilbert-Schmidt theorem (e.g. [7], [6]) and, as a first step below, we recall a selected formulation of this theorem. (For the proof we refer to the known literature on Functional Analysis.) Here . , . is the scalar product in H (and ||h|| = h, h is the norm of h).
Preparing to apply Theorem 3.1 concerning the operator B α,Γ , we will start with the next two initial properties -the first one follows from the classical theory of the weakly singular integral equations, and the second -from Theorem 3.1.
(ii*) Each function µ ∈ L 2 (Γ) can be uniquely expressed by the below given decomposition formula where {ζ k,α } is the complete orthogonal system of eigen functions for B α,Γ and γ k are the Fourier coefficients of µ, γ k := Γ µ(x)ζ k,α (x)ds x . In our basic result below we shall use the already mentioned subspace H 1 α (Γ) ⊂ L 2 (Γ). Definition 3.1 Let us set Note that the inverse operator B −1 α,Γ of B α,Γ is correctly defined on H 1 α (Γ), by the evident In the first theorem below, except results on existence, uniqueness and continuous data dependence of solutions, additional ones are also included concerning the asymptotic (at |x| → ∞ ) and L loc 1 (R 3 ) approximation of solutions (by globally defined continuous functions). As a specific moment, the approximation process is uniquely generated by the corresponding boundary one in L 2 (Γ). Consider now the central result of our study.
The above function u is the unique solution of (1.7) in the family S ′ α,f , contained in the class L loc 1 (R 3 ) and continuous in the two domain components of R 3 \Γ. Solution (3.2) is additionally characterized by the next conventional but essential properties.
(P1) In case of f (x) with a compact support in D, when D = CΩ, the asymptotic relation below holds for u(x): there exist constants C 0 K , C * K so that the difference u 2 − u 1 of the solutions, corresponding to the above data, satisfies an estimate in the form:

Proof:
Recall that the verification whether function u(x) from (3.2) satisfies the equation (−∆) α/2 u| D = f has been actually done in Section 1: by the notations v α,g (x), u α,f (x)respectively for the already introduced single layer and volume Riesz potential, with g = g f [ϕ], formula (3.2) is rewritten as u = v α,g + u α,f , where v α,g is a BF harmonic function, while Thus the existence assertion is proved (i.e. u is a solution of the problem (1.7) in S ′ α,f ). For the uniqueness of solution (3.2) in S ′ α,f , assuming existence of two ones, u 1 , u 2 ∈ S ′ α,f which satisfy (1.7) (with identical data ϕ, f ), it is directly seen that the difference U = u 2 − u 1 is a BF harmonic function, i.e. (−∆) α/2 U = δ Γ [g] in S ′ , with a density g ∈ L 2 (Γ). To resolve this equation regarding U (recall the analogous comments about (1.4)) we have evidently to act by the operation c 3,α |.| 3−α * , finding thus the expression Next, looking at formula (3.2) (i.e. u = v α,g + u α,f ), it is directly seen that v α,g , u α,f ∈ L loc 1 (R 3 ), and the same for u(x). More over, as in the proof of (3.4) (below), it follows the estimate (Here K ⊂ R 3 is an arbitrary fixed compact, and a choice of constants C 0 K , C * K shall be given concerning (3.4)). The property u(x) ∈ C 0 (R 3 \ Γ) (C 0 -the space of the continuous functions) in both the cases D = Ω and D = CΩ is also an automatic consequence from the clear relations v α,g ∈ C 0 (K), u α,f ∈ C 0 (K), valid for each compact K ⊂ R 3 \ Γ.
Consider now the properties (P1) -(P3).The asymptotic relation (3.3) is actually evident (as a slight consequence of the standard inequality |x − y| ≥ |x| − |y| > 0, valid at |x| → ∞ and y varying in a compact). For the proof of (3.4) let us firstly rewrite (3.2) with u 2 − u 1 , ϕ 2 − ϕ α,f 2 − (ϕ 1 − ϕ α,f 1 ), f 2 − f 1 , respectively instead of u, ϕ − ϕ α,f , f . For the sake of convenience we will use below the notations ∆ f ϕ = ϕ − ϕ α,f , ((∆ f ϕ) i = ϕ i − ϕ α,f i , i = 1, 2). After integration of |u 2 − u 1 | on a compact K ⊂ R 3 it easily follows: In order to rework suitably the above inequality we shall take into account the estimate where b * α,Γ is the norm of the operator c 3,α B −1 α,Γ . This way we come to the next relation for the difference u 2 − u 1 : Afterwards it remains to set: . Thus (3.7) takes the form of (3.4). Remark: The partial case f 1 = f 2 could be practically more valuable (then the accent is paid on the boundary data dependence). Now the estimates (3.7), (3.4) take respectively the forms: Consider finally the proof of (P3). Via the remark above, when the boundary problem (1.7) is used in a model, the contour L 2 data ϕ can be preferably changed by suitable continuous approximations {ϕ n } in order to simplify say a numerical procedure. In our approach the boundary operator pair {B α,Γ , B −1 α,Γ } suggests to seek {ϕ n } by the map B α,Γ [ψ n ], given an arbitrary sequence {ψ n } ⊂ H 1 α (Γ) of continuous L 2 approximations to g f [ϕ]. In the framework of problem (1.7) (considered now at boundary data ϕ n , regarding an unknown solution u n ) the basic formula (3.2) serves the answer both, for ϕ n and u n , namely ϕ n = B α,Γ [ψ n ] + ϕ α,f and u n as follows: The property u n ∈ C 0 (R 3 ) follows (by the integral above) from the continuous assumption for ψ n , and the same for ϕ n . (We have taken again into account that the single layer Riesz potentials possess, at 3 − α < 2, the same continuous properties as in the classical case of 3 − α = 1.) And the announced estimate (3.5) is actually proved by the already shown (3.9). In a conclusion let us comment how to construct approximating systems {ψ n } ⊂ H 1 α (Γ) of continuous functions: introducing an arbitrary system of (real) numbers {τ k }, k = 1, 2, . . . : ∞ k=1 τ 2 k λ −2 k,α < ∞, we get an element g of the space H 1 α (Γ), g(x) := ∞ k=1 τ k ζ k,α (x) and for an obviously convenient approximating (to g) sequence we have to take ψ n (x) := n k=1 τ k ζ k,α (x), n = 1, 2, . . . . Recall that the eigen functions ζ k,α of B α,Γ are continuous (i.e. {ζ k,α } ⊂ C 0 (Γ)), according to the known classical theorem for the continuous L 2 solutions of weakly singular integral equations (e.g. [5], [10], [8]). Our next result concerns regularity properties of the solution (3.2), in the interior of R 3 \Γ, as a consequence from these of f (x). We consider below two cases for the regularity of f (x), assumed with a compact support: f ∈ C m 0 (D), m = 1, 2, . . . , and f ∈ L ∞ (D) ∩ L ′ (D), at Here, as usually, C m 0 (D) is the space of the functions smooth up to order m in D, with compact supports; H s (R 3 ) are the known Sobolev classes, related to R 3 (e.g. [2]), and L ′ (D) is the space of the Lauren Schwartz distributions, defined on D ([2], [7], [6]), possessing compact supports therein. Clearly the elements of L ′ (D) are automatically extended (on the whole R 3 ) as zeros out of their supports, presenting thus distributions from S ′ (R 3 ).
For analyzing the H s properties of u(x) we will use the following accessory assertion (e.g. [9]). (The given proof of the lemma is due to the university lectures [1].) S -an arbitrary fixed function, is a continuous operator, acting from H s into itself, for each (fixed) real s. (Here S = S(R 3 ) and the same for H s .) Proof: As a necessary initial step, recall the very useful representation for the Fourier image (Φv)(ξ), ξ ∈ R 3 , of the (generalized) function (Φv)(x): (3.11) From (3.11), taking into account the known Peetre inequality (e.g. [9]), we get: Applying afterwards the Young inequality ( [2], [3], [8]) in the integral term above, we find the sought estimate: Here ||.|| s is the norm in the space H s , ||Φv|| s := ||(1 + |ξ)| 2 ) s/2 (Φv)(ξ)|| L 2 (R 3 ) , and . We can now formulate and proof our result concerning the regularity of the solution in (3.2).