Regularity of solutions of elliptic problems with a curved fracture

Accepted Manuscript Regularity of solutions of elliptic problems with a curved fracture

S. Ariche, C. De Coster, S. Nicaise

PII: DOI: Reference:

S0022-247X(16)30607-2 http://dx.doi.org/10.1016/j.jmaa.2016.10.021 YJMAA 20796

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Journal of Mathematical Analysis and Applications

Received date:

11 July 2016

Please cite this article in press as: S. Ariche et al., Regularity of solutions of elliptic problems with a curved fracture, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2016.10.021

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REGULARITY OF SOLUTIONS OF ELLIPTIC PROBLEMS WITH A CURVED FRACTURE S. ARICHE*, C. DE COSTER** AND S. NICAISE** Abstract. We consider the Laplace equation with a right-hand side concentrated on a curved fracture of class C m+2 for some non negative integer m (i.e., a sort of Dirac mass). We show that the solution belongs to a weighted Sobolev space of order m, the weight being the distance to this fracture. Our proof relies on a priori estimates in a dihedron or a cone with singularities for elliptic operators with variable coefficients. In both cases, such an estimate is obtained using a dyadic covering of the domain.

Key Words Laplace equation, Dirac measure, regularity. AMS (MOS) subject classification 35R06, 35B65. 1. Introduction We consider the boundary value problem  −Δu = q δσ , (1.1) u = 0,

in on

O, ∂O,

where O is a bounded domain of R3 , the fracture σ is a one-dimensional curve strictly included in O of class C m+2 for some m ∈ N (see Figure 1) and q belongs to L2 (σ).

Figure 1. A model situation. Such models are, for instance, used in fluid mechanics to save computational resources when the original system is too complex, see [15, 16]. Darcy’s law in fractured domains is a typical example, where σ corresponds to a one-dimensional fracture. Some finite element method approximations of such problems can be found in [2, 3, 30] (in dimension two) and in [15, 16] (in dimension three). ˚1 (O) since the The boundary value problem (1.1) is nonstandard because its solution cannot be in H right-hand side of this problem is not in its dual. Nevertheless it enters in the framework of problems set in spaces of distributions, that were studied in [5, 23, 28, 29], but in our case the datum belongs to H −s (O), for all s > 1, and therefore the shift theorem yields only a solution u ∈ H 2−s (O), for all s > 1, that is not really satisfactory for numerical purposes. In [15], the author shows that a solution exists in a weighted Sobolev space, the weight being the distance to the fracture. More precisely [15, Corollary ˚1 (O; σ) with 0 < β < 1 (see Section 2 for the 2.2] shows that problem (1.1) has a weak solution u ∈ H β ˚1 (O; σ) such that definition of this space), i.e., u is the unique function in H β   ˚1 (O; σ), ∇u · ∇v = qγσ v, ∀v ∈ H (1.2) −β O

σ 1

2

S. ARICHE, C. DE COSTER AND S. NICAISE

˚1 (O; σ) to L2 (σ) (that is well-defined due to [16, Theorem 4.2]). where γσ is the trace operator from H −β Moreover we have (1.3)

uH ˚1 (O;σ)  q0,σ . β

In [15, 16] a Galerkin approximation of problem (1.2) is proposed. Further under an improved regularity of the solution in a scale of weighted Sobolev spaces (the weight being the distance to σ), an error estimate is obtained in [15, Corollary 3.8]. Since in [15] this improved regularity is taken for granted, our main goal is to prove such a regularity result. This regularity result in a scale of weighted Sobolev spaces suggests us to see problem (1.1) as a boundary value problem set in the non-smooth domain O \ σ, the interior of the fracture playing the rule of an edge, while its extremities correspond to the corners. The literature on boundary value problems in non-smooth domains is very large. A first technique to prove some regularity results in weighted spaces is based on localization and the use of Mellin and Fourier transform, see [17, 20, 21]. A second technique consists in the use of a dyadic partition technique (i.e. a combination of the use of a partition of unity and a reduction to the compact and smooth case). This technique is a powerfull tool to prove natural regularity shift results, namely from the basic regularity of the solution in a lower order weighted Sobolev space and higher-order regularity in a weighted Sobolev space of the datum, one deduces an improved regularity of the solution in the natural weighted Sobolev space. For two- or three-dimensional domains with corners and/or edges, we refer to [1, 6, 8, 10, 12, 14, 18, 20, 24, 25, 27]; for an elliptic problem with degenerate coefficients near the boundary, see [22]. The first technique was used in [4] to treat the case where the fracture is a bounded segment strictly included in the domain. Here we consider the case where the fracture is a curve strictly included in O, for which the Fourier or Mellin transformation technique is no more valid. Indeed by localization and straightening the curve, we can obtain a problem with a fracture that is a full line or a half-line but with an operator with variable coefficients. Hence we replace the Fourier or Mellin transformation technique by an a priori method that consists in proving a priori estimates in a dihedron or a cone with singularities. In both cases, such an estimate is obtained using dyadic partition technique. In this paper we shall prove the following result (see the next section for the notations). Theorem 1.1. Let m ≥ 2, O a bounded domain of R3 of class C m and σ a one-dimensional submanifold ˚1 (O; σ) be the solution of (1.1) given by of class C m+2 with σ ⊂ O. Let q ∈ L2 (σ), 0 < β < 1 and u ∈ H β m [15, Corollary 2.2]. Then u ∈ Vβ+m−1 (O; σ) and satisfies the estimate m uVβ+m−1 (O;σ)  q0;σ .

Remark 1.2. We can equally well consider less regular domain O or fracture σ, see the remarks of the last section. This paper is organized as follows. In the second section, we recall or prove some technical results concerning the weighted Sobolev spaces. In Section 3, we localize and straighten the curve. In Section 4 and 5 we give two natural shift theorems, the first one on a dihedron, the second one on cones with singularities. These results are applied in the last section to prove our main result. 2. Preliminaries Let us first recall some basic definitions and give some properties used in the whole paper. 2.1. Notations. For a bounded domain O with a Lipschitz boundary, real parameters s ≥ 0 and p > 1, the usual norm and semi-norm of W s,p (O) are denoted by  · s,p,O and | · |s,p,O , respectively. In the case p = 2 (resp. s = 0), we drop the index p (resp. s). We define also ˚1 (O) = {u ∈ H 1 (O) | u = 0 on ∂O}. H The notations A  B and A ∼ B mean the existence of positive constants C1 and C2 , which are independent of A and B such that A ≤ C2 B and C1 B ≤ A ≤ C2 B respectively. We denote by Bk (x0 , δ) the open ball in Rk of center x0 and radius δ.

REGULARITY OF SOLUTIONS OF ELLIPTIC PROBLEMS WITH A CURVED FRACTURE

3

2.2. Weighted Sobolev spaces. Definition 2.1. Let α be an arbitrary real number, O be a domain with Lipschitz boundary of Rn with n ≥ 1, σ ⊂ O and r be the distance to σ. We denote by L2α (O; σ) the Hilbert space made of measurable functions u such that  u2L2α (O;σ) =

For γ ∈ Nn , we define

∂γ u ∂xγ

O

|u(x)|2 r2α (x)dx < ∞.

as the element of D (O \ σ) that satisfies, for all ϕ ∈ D(O \ σ),   ∂γ u ∂γ ϕ |γ| ϕ = (−1) u . γ γ O ∂x O ∂x For m ≥ 1, we define the weighted Sobolev space ∂γ u ∈ L2α (O; σ)}. Hαm (O; σ) = {u ∈ L2α (O; σ) | ∀|γ| ≤ m, ∂xγ This is a Hilbert space with norm   12 m |u|2Hαk (O;σ) , uHαm (O;σ) = k=0

where the semi-norm on

Hαk (O; σ)

is defined by    γ 2 ∂ u   |u|Hαk (O;σ) =  ∂xγ  2 |γ|=k

 12 .

Lα (O;σ)

Similarly the weighted Sobolev space of Kondratiev’s type is defined by ∂γ u Vαm (O; σ) = {u ∈ L2α−m (O; σ) | ∀|γ| ≤ m, r|γ|+α−m γ ∈ L2 (O)}. ∂x It is a Hilbert space for the norm      γ 1/2  ∂ u 2 2(|γ|+α−m)   (x)dx . uVαm (O;σ) =  γ (x) r O ∂x |γ|≤m

Its corresponding semi-norm is defined by      γ 1/2  ∂ u 2 2α   . |u|Vαm (O;σ) =  ∂xγ (x) r (x)dx |γ|=m

O

˚1 (O; σ) (resp. V ˚1 (O; σ)) defined as the space of functions u ∈ H 1 (O; σ) We need also the space H α α α 1 (resp. Vα (O; σ)) such that γu = 0 on ∂O where γu denotes the trace of u. As a direct consequence of the definition of the spaces Vαm (O; σ), we notice that for any m ∈ N and all α ∈ R, we have (2.1)

m+1 Vα+1 (O; σ) → Vαm (O; σ).

Lemma 2.2. Let Q := B2 (0, ) × (− , ) and σ = {0} × (− , ). Then, for every β > 0, we have Hβ1 (Q ; σ) = Vβ1 (Q ; σ). Proof. The embedding Vβ1 (Q ; σ) → Hβ1 (Q ; σ) is trivial since Q is bounded. Observe that the embedding in the other direction holds if, for all u ∈ Hβ1 (Q ; σ), we have rβ−1 u ∈ L2 (Q ). On Q , we have r = x2 + y 2 ≤ . Fix a cut-off function η ∈ C 1 (R3 ) such that  1, on {(x, y, z) ∈ R3 | r ≤ 2 }, η(x, y, z) = 0, on {(x, y, z) ∈ R3 | r ≥ }. Denote by (r, θ) the polar coordinates in the x − y plane. Then for almost all z ∈ (− , ) and θ ∈ (0, 2π) we can write 1 ∂ (ηu)(reiθ , z) = −L( (ηu)(· eiθ , z))(r), r ∂r

4

S. ARICHE, C. DE COSTER AND S. NICAISE

where, using the notations from [19, p. 28], the operator L is defined by  1 ∞ v(t) dt. (Lv)(r) = r r Hence, by continuity of L in L2β+ 1 (R+ 0 ; 0) for β > 0 (called Hardy’s inequality, see [19, observations before 2 Theorem 1.4.4.3]), we have, for almost all z ∈ (− , ) and θ ∈ (0, 2π), 2 2  ∞  ∞ 1 ∂    (ηu)(reiθ , z) r2β+1 dr ≤ 1  (ηu)(reiθ , z) r2β+1 dr. r    2 β 0 ∂r 0 Integrating in θ and in z and applying Leibniz’s rule, we obtain 2 2    2π  ∞     2π  ∞  1 ∂  2β+1  2β iθ iθ β−1 2     ηuL2 (Q ) = drdθdz  r  r (ηu)(re , z) r  ∂r (ηu)(re , z) r rdrdθdz. − 0 0 − 0 0  2    2π    ∂u   (|u|2 +  )r2β rdrdθdz  u2H 1 (Q ;σ) , β ∂r − 0 0 which allows to conclude that (2.2)

ηuVβ1 (Q ;σ)  uHβ1 (Q ;σ) .

On the other hand, since η = 1 on {(x, y, z) ∈ R3 | r ≤ 2 } then (1 − η)u ∈ H 1 (Q ), or equivalently (1 − η)u ∈ Vβ1 (Q ; σ) with (2.3)

(1 − η)uVβ1 (Q ;σ)  (1 − η)uHβ1 (Q ;σ)  uHβ1 (Q ;σ) . 

The conclusion follows from (2.2) and (2.3). Lemma 2.3. Let > 0 and σ = {(0, 0, z) | 0 < z < }. Then, for every β > 0, we have Hβ1 (B3 (0, ); σ) = Vβ1 (B3 (0, ); σ).

Proof. This can be easily deduced from a localization argument together with Lemma 2.2 and [4, Proposition 2.15].  3. Localisation and straightening of the curve In this section we show how problem (1.1) is transformed when we localize it and straighten the fracture. As σ is a one-dimensional submanifold of class C m+2 of R3 , we deduce (see [7, Theorem 2.1.2]) that, for all x0 ∈ σ, there exist δ  > 0, an open neighbourhood V  of 0 in R3 and a C m+2 −diffeomorphism ϕ : B3 (x0 , δ  ) → V  : x → x ˆ such that ϕ(x0 ) = 0 and (3.1)

ϕ(B3 (x0 , δ  ) ∩ σ) = {(0, 0, x ˆ3 ) | x ˆ3 ∈ R} ∩ V  =: σ ˆ , in case x0 is interior to σ,  ˆ3 ) | x ˆ3 ∈ R+ } ∩ V  =: σ ˆ , in case x0 is an extremity of σ. ϕ(B3 (x0 , δ ) ∩ σ) = {(0, 0, x

For a fixed δ < δ  , we will denote by V = ϕ(B3 (x0 , δ)), see Figures 2 and 3. In order to study the local regularity of the solution, we need to consider the boundary value problem satisfied by ηu where η is a truncated function near x0 . To this aim we need the following Lemma. Lemma 3.1. Let σ be a one-dimensional submanifold of class C m+2 of R3 , x0 ∈ σ, 0 < δ < δ  , β > 0, u ∈ Hβ1 (B3 (x0 , δ  ); σ) and η ∈ D(R3 ) such that η = 1 in B3 (x0 , δ) and supp(η) ⊂ B3 (x0 , δ  ). Then, for 1 all v ∈ H−β (B3 (x0 , δ  ); σ), we have   u∇η · ∇v = − v div(u∇η). B3 (x0 ,δ  )

B3 (x0 ,δ  )

Remark 3.2. Without loss of generality, we can assume that, in case x0 is interior of σ, the extremities of σ are not in B3 (x0 , δ  ), while in case x0 is an extremity of σ, we can assume that the second extremity of σ is not in B3 (x0 , δ  ). Remark 3.3. Observe that the problem is only on a neighbourhood B of σ ∩ (B3 (x0 , δ  ) \ B3 (x0 , δ)) as in B3 (x0 , δ), ∇η = 0 and the functions are in H 1 (B3 (x0 , δ  ) \ (B3 (x0 , δ) ∪ B)).

REGULARITY OF SOLUTIONS OF ELLIPTIC PROBLEMS WITH A CURVED FRACTURE

5

Proof. Step 1: Localisation. Let B and B  be open neighbourhoods of σ ∩ (B3 (x0 , δ  ) \ B3 (x0 , δ)) such that B ⊂ B  and consider the truncated function η1 such that η1 = 1 on B and supp(η1 ) ⊂ B  . Then we 1 (B3 (x0 , δ  ); σ), have, for all v ∈ H−β    u∇η · ∇v = η1 u∇η · ∇v + (1 − η1 )u∇η · ∇v. (3.2) B3 (x0 ,δ  )

B

B3 (x0 ,δ  )\(B3 (x0 ,δ)∪B)

Step 2: Regularization. As η1 u ∈ Hβ1 (B  ; σ), by making the change of variable induced by the −1  ; σ  = ϕ(B  ) diffeomorphism ϕ given by (3.1), we transform η1 u into η

∈ H 1 (B ˜ ) with B 1 u := (η1 u) ◦ ϕ β

 ⊂ B2 (0, 1 ) × (− 1 , 1 ) and and σ ˜ understood as the set {(0, 0, z) | z ∈ R}. Considering 1 such that B 1  , we have that η

˜ ). Moreover, the fact that ηˆ1 = η1 ◦ ϕ−1 = 0 outside of B 1 u ∈ Hβ (B2 (0, 1 ) × (− 1 , 1 ); σ 1 1 by Lemma 2.2, Hβ (B2 (0, 1 ) × (− 1 , 1 ); σ ˜ ) = Vβ (B2 (0, 1 ) × (− 1 , 1 ); σ ˜ ) and using that ηˆ1 = 0 outside 1 3  , we have η

of B ˜ ). By [26, p. 24], we deduce the existence of a sequence (ˆ un )n ⊂ D(R3 \ σ ˜) 1 u ∈ V (R ; σ β

1 3 such that u ˆn → η

˜ ). 1 u in Vβ (R ; σ

:= ϕ(B) and supp(η2 ) ⊂ B  and consider the sequence Let now η2 ∈ D(R3 ) such that η2 = 1 in B 

 m+2 by wn := (η2 u ˆn ) ◦ ϕ. This sequence satisfies (wn )n ⊂ C0 B \ σ and wn → η1 u in (wn )n defined  Hβ1 B  ; σ . Step 3: Conclusion. By Green’s formula, we have    η1 u∇η · ∇v = lim wn ∇η · ∇v = − lim (3.3) n→∞

B

n→∞

B

 B

v div(wn ∇η) = −

B

v div(η1 u∇η).

˚1 (B3 (x0 , δ  ) \ (B3 (x0 , δ) ∪ B))3 , applying Green’s formula we have On the other hand, as (1 − η1 )u∇η ∈ H   (1 − η1 )u∇η · ∇v = − v div((1 − η1 )u∇η). (3.4) B3 (x0 ,δ  )\(B3 (x0 ,δ)∪B)

B3 (x0 ,δ  )\(B3 (x0 ,δ)∪B)



The result follows from (3.2), (3.3), (3.4).

Proposition 3.4 (Localisation). Let σ be a one-dimensional submanifold of class C m+2 of R3 , x0 ∈ σ, ˚1 (O; σ) a solution of (1.1) and η ∈ D(R3 ) such that η = 1 in B3 (x0 , δ) and 0 < δ < δ  , β > 0, u ∈ H β ˚1 (B3 (x0 , δ  ); σ) is a weak solution of supp(η) ⊂ B3 (x0 , δ  ). Then u0 := ηu ∈ H β  B3 (x0 , δ  ), −Δu0 = q0 δσ + f0 , in (3.5) on ∂B3 (x0 , δ  ), u0 = 0, ˚1 (B3 (x0 , δ  ); σ) is the unique solution with q0 = ηq and f0 = −2∇η · ∇u − u Δη, in the sense that u0 ∈ H β of    ˚1 (B3 (x0 , δ  ); σ). ∇u0 · ∇v = q0 γσ v¯ + f0 v¯, ∀v ∈ H −β B3 (x0 ,δ  )

σ

B3 (x0 ,δ  )

Proof. By Lemma 3.1 and Leibnitz rule we have    u∇η · ∇v + η∇u · ∇v ∇u0 · ∇v = B3 (x0 ,δ  ) B3 (x0 ,δ  ) B3 (x 0 ,δ )  = − v div(u∇η) + ∇u · ∇(η v) − v ∇u · ∇η B3 (x0 ,δ  ) B3 (x0 ,δ  )  B3 (x0 ,δ )  = ∇u · ∇(η v) − v [2∇u · ∇η + u Δη] . O

B3 (x0 ,δ  )

We conclude using the fact that u is solution of (1.1), the uniqueness following from [15, Corollary 2.2].  Transformation to a problem with a straight fracture.

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S. ARICHE, C. DE COSTER AND S. NICAISE

Recall that, for all x0 ∈ σ, there exist δ  > 0, an open neighbourhood V  of 0 in R3 and a C m+2 −diffeoˆ satisfying ϕ(x0 ) = 0 and (3.1). Using the diffeomorphism ϕ, we morphism ϕ : B3 (x0 , δ  ) → V  : x → x transform the problem (3.5) into the weak problem  Lˆ u0 = qˆ0 δσˆ + fˆ0 , in V , (3.6) on ∂V  , u ˆ0 = 0, where • L is a second order operator with C m coefficients given by (3.7)

L=−

3  3  3  3 3   ∂ϕj ∂ϕk ∂ 2 ∂ 2 ϕj ∂ − ; ∂xi ∂xi ∂ x ˆj ∂ x ˆk i=1 j=1 ∂x2i ∂ x ˆj i=1 j=1 k=1

• qˆ0 (0, 0, zˆ) = (ηq) ◦ ϕ−1 (0, 0, zˆ);    3  3 3  ∂ϕj ∂ u ˆ ∂ϕk ∂ ηˆ • fˆ0 = −(Lˆ η )ˆ u−2 , where u ˆ = u ◦ ϕ−1 and ηˆ = η ◦ ϕ−1 ; ∂xi ∂ x ˆj ∂xi ∂ x ˆk i=1 j=1 k=1

• σ ˆ = {(0, 0, zˆ) | zˆ ∈ R} ∩ V  , in case x0 is interior to σ, σ ˆ = {(0, 0, zˆ) | zˆ ∈ R+ } ∩ V  , in case x0 is an extremity of σ; ˚1 (V  ; σ • the solution u ˆ0 = (ηu) ◦ ϕ−1 of (3.6) is in H ˆ ). β

Figure 2. Transformation in case x0 is an interior point of σ.

Figure 3. Transformation in case x0 is an extremity of σ. Hence, in order to study the local regularity of the solution u of (1.1), we study the regularity of the solution u ˆ0 of (3.6). Let δ˜ > 0 be small enough such that ˜ × (−δ, ˜ δ) ˜ ⊂ V  , in case σ ˆ = {(0, 0, zˆ) | zˆ ∈ R} ∩ V  , Wδ˜ := B2 (0, δ)  ˜ in case σ ˆ = {(0, 0, zˆ) | zˆ ∈ R+ } ∩ V  . Wδ˜ := B3 (0, δ) ⊂ V , Lemma 3.5. We have the following properties for the problem (3.6) ˆ ) and (i) u ˆ0 ∈ Vβ1 (Wδ˜; σ (3.8)

ˆ u0 Vβ1 (Wδ˜;ˆσ)  uHβ1 (O;σ) ;

REGULARITY OF SOLUTIONS OF ELLIPTIC PROBLEMS WITH A CURVED FRACTURE

7

(ii) fˆ0 ∈ L2β+1 (Wδ˜; σ ˆ ) and fˆ0 L2β+1 (Wδ˜;ˆσ) ≤ c˜uHβ1 (O;σ) ,

(3.9)

with c˜ = c˜(ϕC 2 (B3 (x0 ,δ )) , ηC 2 (B3 (x0 ,δ )) ); 2 (Wδ˜ \ σ ˆ ); (iii) u ˆ0 ∈ Hloc (iv) the operator L is uniformly elliptic on V  1. Proof. Step 1: Proof of (i). As u0 ∈ Hβ1 (B3 (x0 , δ  ); σ), we can easily prove by Lemma 2.2 or Lemma 2.3 that u ˆ0 ∈ Vβ1 (Wδ˜; σ ˆ ) with u0 Hβ1 (Wδ˜;ˆσ)  uHβ1 (B3 (x0 ,δ );σ)  uHβ1 (O;σ) . ˆ u0 Vβ1 (Wδ˜;ˆσ) ∼ ˆ Step 2: Proof of (ii). By definition of L and the regularity of ϕ and η, we obtain fˆ0 L2β+1 (Wδ˜;ˆσ)

≤ ≤ ≤

  3    ∂ϕ  ∂ϕ  (Lˆ η )ˆ u +2 u · ∇ˆ η   ∂xi · ∇ˆ  2 ∂xi Lβ+1 (Wδ˜;ˆ σ) i=1   c˜1 ˆ uL2β+1 (Wδ˜;ˆσ) uL2β+1 (Wδ˜;ˆσ) + ∇ˆ c˜2 uHβ1 (O;σ) , L2β+1 (Wδ˜;ˆ σ)

where c˜i = c˜(ϕC 2 (B3 (x0 ,δ )) , ηC 2 (B3 (x0 ,δ )) ) for i = 1, 2. Step 3: Proof of (iii). Let O be a neighborhood of σ ˆ , p ∈ Wδ˜ \ O and η1 be a truncation function ¯ˆ ). Observe that, for all x ∈ B3 (p, Rp ), we have such that supp(η1 ) ⊂ B3 (p, Rp ) with 0 < Rp < 12 d(p, σ d(x, σ ˆ ) ∼ 1. ˆ0 , then v is a solution of Let v := η1 u  Lv = η1 fˆ0 + g(η1 , u ˆ0 ), in B3 (p, Rp ), v = 0, on ∂B3 (p, Rp ), 3  ∂ϕj ∂ u ∂ϕk ∂η1 ˆ0 ∂xi ∂ x ˆj )( ∂xi ∂ x ˆk ). i=1 j=1 k=1 By Step 2, we have fˆ0 ∈ L2β+1 (V  ; σ ˆ ) and hence η1 fˆ0 ∈ L2 (B3 (p, Rp )). As the function g is defined by the derivative up to order 2 of η1 and the first derivative of u ˆ0 ∈ Vβ1 (V  ; σ ˆ ), we deduce that g ∈ L2 (B3 (p, Rp )). 2 ˆ0 ∈ Hloc (Wδ˜ \ σ ˆ ). By [9, Remark 4.2], we conclude that v ∈ H 2 (B3 (p, Rp )) and hence u

ˆ0 ) = −(Lη1 )ˆ u0 − 2 with g(η1 , u

3  3  (

Step 4: Proof of (iv). Let Lpp be the principal part of L, i.e. Lpp = −

3  3  3  ∂ϕj ∂ϕk ∂ 2 . ∂xi ∂xi ∂ x ˆj ∂ x ˆk i=1 j=1 k=1

This implies that Lpp (x, iξ) = −

3  3  3  ∂ϕj ∂ϕk (iξj )(iξk ) = ∇ϕ · ξ2 , ∂x ∂x i i i=1 j=1 k=1

∂ϕ

where ∇ϕ = ( ∂xij )1≤i,j≤3 . As ϕ is a C 1 −diffeomorphism and V  is compact, we have a constant C > 0 such that, for all x ∈ V  , Lpp (x, ξ) = ∇ϕ · ξ2 ≥ Cξ2 , which concludes the proof.



1by uniformly elliptic on V  , we mean that the principal part frozen at every point x ∈ V  satisfies L (x , iξ) ≥ γξ2 , pp 0 0 for all ξ ∈ Rn , ξ = 0,with a γ > 0 independent of x0 and ξ.

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S. ARICHE, C. DE COSTER AND S. NICAISE

4. Natural shift theorem in weighted Sobolev spaces in a dihedron According to the previous section, we are reduced to study problem (3.6). In order to prove the regularity of the solution near an interior point of σ, we need to prove a natural shift theorem in weighted Sobolev spaces for elliptic operators with variable coefficients defined in a dihedron (Theorem 4.4). This result extends [12, Theorem 5.1] where the result was proved for elliptic operators with constant coefficient and [13, Lemma 7.4.3] (see Lemma 5.1 below), where the corresponding result is given on truncated cones with vertex singularity. Before we start with two technical lemmas. Lemma 4.1. Let K be a cone in R2 and denote x = (x⊥ , x3 ) a point of K × R with x⊥ ∈ K et x3 ∈ R. Consider a second order elliptic operator with variable coefficients of class C m defined on K × R by  ∂γ aγ (x) γ . L(x, ∂x ) := ∂x |γ|≤2

For all positive number j ∈ N, every ν ∈ Z with |ν| < 2j+1 and > 0, let Lj,ν be the operator defined by

 ν

Lj,ν (ˆ x, ∂xˆ ) = 2−2j L 2−j (ˆ x + (0, 0, )), 2j ∂xˆ . 2 Define also Lpp the principal part of L frozen at (0, 0, 2 (4.1)

−j

2

ν

) and





2−j ν

2−j ν j  2−j ν ∂ γ ), ∂xˆ := 2−2j Lpp (0, 0, ), 2 ∂xˆ = ) γ. aγ (0, 0, Lj,ν pp (0, 0, 2 2 2 ∂x ˆ |γ|=2

Then, for all 0 < ˜ < <  , all m ∈ N and all u ∈ H m+2 (A ) with A := {ˆ x⊥ ∈ K |   

} × (− 2 , 2 ), we have     j,ν

  2−j ν

j,ν  ), ∂xˆ u x, ∂xˆ ) − Lpp (0, 0,  2−j um+2,A . (4.2)   L (ˆ 2 m,A

˜ 4

< |ˆ x⊥ | <

Remark 4.2. Observe that the constant in (4.2) does not depend on j ≥ 0 and on ν ∈ Z with |ν| < 2j+1 but depends on and  . 

 2−j ν x, ∂xˆ ) − Lj,ν u = L1 u + L2 u, where Proof. By definition, we have Lj,ν (ˆ ˆ pp (0, 0, 2 ), ∂x L1 u :=

 |γ|=2

ν

2−j ν

))) − aγ (0, 0, ), 2 2 

ν  ∂ γ u 2j(|γ|−2) aγ 2−j (ˆ x + (0, 0, )) . L2 u := 2 ∂x ˆγ

bj,ν x) γ (ˆ

∂γ u ∂x ˆγ

with bj,ν x) = aγ (2−j (ˆ x + (0, 0, γ (ˆ

|γ|≤1

Let us prove by iteration on , that, for all u ∈ H +2 (A ) with 0 ≤  ≤ m, we have     j,ν

  2−j ν

j,ν −j  ), ∂xˆ u x, ∂xˆ ) − Lpp (0, 0, (4.3)    2 u+2,A .  L (ˆ 2 ,A Step 1: Case  = 0. We have L1 u0,A ≤

 |γ|=2



sup

x ˆ∈A



 ∂γ u    |bj,ν (ˆ x )| γ  ∂x ˆγ 

0,A

≤ sup |bj,ν x)| u2,A . γ (ˆ x ˆ∈A

x⊥ ∈ K | |ˆ x⊥ | <  } × (−  ,  ). Observe that, for j ≥ 0, ν ∈ Z with Let us consider the set B  := {ˆ 2−j ν   ˆ ∈ A , we have 2−j (ˆ x + (0, 0, ν |ν| < 2j+1 and every x 2 )) ∈ B and (0, 0, 2 ) ∈ B . This implies by the finite-increments formula that    −j 2−j ν  2−j ν

j,ν  ) − (0, 0, )  2−j |ˆ x)| ≤ sup |∇aγ (x)| 2 x ˆ + (0, 0, x|  2−j , (4.4) |bγ (ˆ 2 2 x∈B  and hence

L1 u0,A  2−j u2,A .

REGULARITY OF SOLUTIONS OF ELLIPTIC PROBLEMS WITH A CURVED FRACTURE

9

On the other hand, again using the fact that, for all j ≥ 0, |ν| < 2j+1 and every x ˆ ∈ A , we have ν −j  x + (0, 0, 2 )) ∈ B which is bounded, we obtain 2 (ˆ     ν   ∂ γ u   2j(|γ|−2) sup aγ (2−j (ˆ x + (0, 0, ))   2−j u1,A . L2 u0,A ≤  ∂x 2 ˆγ 0,A x ˆ∈A |γ|≤1

This proves the first step. Step 2: Let 1 ≤  ≤ m and assume that (4.3) is true at level  − 1 and let us prove that for u ∈ H +2 (A ), (4.3) is true at level . By definition of the norm we have 

    j,ν 2−j ν u x, ∂xˆ ) − Lj,ν ), ∂  L (ˆ x ˆ pp (0, 0, 2 ,A   (4.5)

 

  −j   j,ν  j,ν 2 ν 2−j ν j,ν   L (ˆ x, ∂xˆ ) − Lj,ν + L (ˆ x , ∂ ), ∂ .  ˆ u x ˆ ) − Lpp (0, 0, x ˆ u pp (0, 0, 2 ), ∂x 2   −1,A

,A

The first term can be managed by the assumption. Hence, we only have to estimate       μ     ∂ (L2 u)   ∂ μ (L1 u)   j,ν

  2−j ν

j,ν       ), ∂xˆ u x, ∂xˆ ) − Lpp (0, 0,  + . (4.6)  ∂x  L (ˆ 2 ˆμ 0,A  ∂ x ˆμ 0,A ,A |μ|=

By Leibniz’s formula we have    μ      ∂ μ−χ (bj,ν )   χ+γ      ∂ (L1 u)   j,ν   ∂ μ+γ u  u   γ    bγ   ∂  sup + sup .    ∂x  ∂x ˆμ 0,A ˆμ+γ 0,A ˆμ−χ   ∂ x ˆχ+γ 0,A x ˆ∈A ˆ∈A  ∂ x χ<μ x |γ|=2

|γ|=2

Observe that, for |μ − χ| ≥ 1, we have     μ−χ  ∂ μ−χ (bj,ν )   2−j ν    γ −j|μ−χ|  ∂ −j )) . ˆ + (0, 0, =2   ∂xμ−χ aγ (2 x  ∂x ˆμ−χ  2 Hence, by the finite increments formula (4.4)       ∂ μ (L1 u)  −j    2 (4.7)  ∂x ˆμ 0,A |μ|=

|μ|+|γ|=+2

as above, we have  μ+γ   ∂ u   +  ∂x ˆμ+γ   0,A

|χ|+|γ|<+2

 χ+γ   ∂ u    2−j u+2,A .  ∂x ˆχ+γ 0,A

In the same way, using Leibnitz formula, we have  μ+γ   μ     ∂ (L2 u)   u ν   j(|γ|−2) ∂    aγ (2−j (ˆ )))  2 x + (0, 0, sup   ∂x    μ μ+γ ˆ 2 ∂x ˆ x ˆ∈A 0,A 0,A |γ|≤1   μ−χ   ∂  ν  −j  + sup  μ−χ aγ (2 (ˆ x + (0, 0, )))  ˆ 2 ˆ∈A ∂ x χ<μ x

 χ+γ   ∂ u   .  ∂x ˆχ+γ 0,A

ˆ ∈ A , Now observe that, for |γ| ≤ 1, we have 2j(|γ|−2) ≤ 2−j and for |μ − χ| ≥ 1 and x  μ−χ    μ−χ  ∂   ν  ν  −j −j|μ−χ|  ∂ −j  ))) = 2 ))) a (2 (ˆ x + (0, 0, a (2 (ˆ x + (0, 0, γ  ∂x    ∂xμ−χ γ ˆμ−χ 2 2  μ−χ  ∂ aγ   2−j  1. ≤ 2−j sup   μ−χ  ∂x B This implies that     ∂ μ (L2 u)    (4.8)  ∂x ˆμ  |μ|=

0,A

 2−j



 |μ|+|γ|≤+1

 μ+γ  ∂ u    ∂x  + μ+γ ˆ 0,A



|γ|+|χ|<+1

 χ+γ   ∂ u    2−j u+2,A .  ∂x ˆχ+γ 0,A

As by assumption we have −j (Lj,ν − Lj,ν pp )u−1,A  2 u+1,A ,

we deduce (4.3) at level  from (4.5), (4.6), (4.7) and (4.8).



10

S. ARICHE, C. DE COSTER AND S. NICAISE

Lemma 4.3. Let ˜ < <  , A and A be defined by



x⊥ | < } × (− , ), A := {ˆ x⊥ ∈ K | < |ˆ 4 2 2 (4.9)

˜

  A := {ˆ x⊥ | <  } × (− , ). x⊥ ∈ K | < |ˆ 4 2 2 Let m ≥ 2 and L be a uniformly elliptic operator with variable coefficients of class C m defined on A by  ∂γ L= aγ (ˆ x) γ . ∂x ˆ |γ|≤2

Let u ∈ H 1 (A ) such that Lu ∈ H m−2 (A ). Consider the sets (A )1≤≤m such that A ⊂ A−1 , A1 = A and Am = A and define the functions η ∈ C ∞ (R3 ) for  = 2, . . . , m, such that η = 1 on A and supp(η ) ⊂ A−1 . Assume that, for all 2 ≤  ≤ m, we have η u ∈ H  (A ) and satisfies η u,A  L(η u)−2,A + η u1,A .

(4.10) Then u ∈ H m (A) and satisfies

um,A  Lum−2,A + u1,A .

Proof. Let us prove by recurrence on  that, for all 2 ≤  ≤ m, u ∈ H  (A ) and (4.11)

u,A  Lu−2,A + u1,A .

Step 1:  = 2. By (4.10) for  = 2, Leibniz’s rule and the definition of η2 , we have

(4.12)

u2,A2 ≤ η2 u2,A  L(η2 u)0,A + η2 u1,A   ∂ μ u ∂ γ−μ η2 = η2 Lu + aγ (ˆ x) Cγμ μ 0,A + η2 u1,A ∂x ˆ ∂x ˆγ−μ μ<γ |γ|≤2   ∂μu  Lu0,A +  μ 0,A + η2 u1,A  Lu0,A + u1,A . ∂x ˆ μ<γ |γ|≤2

Step 2: Let 3 ≤  ≤ m and assume that u ∈ H −1 (A−1 ) and satisfies (4.11) for  − 1. Let us prove that u ∈ H  (A ) and that (4.11) is satisfies for . By (4.10) we have v = η u ∈ H  (A ) and (4.13)

u,A ≤ η u,A  L(η u)−2,A + η u1,A  L(η u)−2,A + u1,A ,

By definition of the norm in H −2 (A ) and of L and as supp(η ) ⊂ A−1 and A−1 ⊂ A , we have, by (4.11) for  − 1, L(η u)−2,A  Lu−2,A−1 + u−1,A−1  Lu−2,A + u1,A . 

This allows to conclude the proof by (4.13). Let us now give the natural shift theorem in weighted Sobolev spaces in a dihedron. Theorem 4.4. Let K be a cone in R2 , 0 < <  , W := (K ∩ B2 (0, )) × (− , )

and

W  := (K ∩ B2 (0,  )) × (−  ,  ).

Denote ∂W  := (∂K ∩ B2 (0,  )) × (−  ,  ). We define the edge e := {x = (0, x3 ) | −  < x3 <  } ⊂ W  . Let m ≥ 2 and L be a uniformly elliptic operator of the second order defined on W  with coefficients of class C m (W  ),  ∂γ aγ (x) γ . L= ∂x |γ|≤2

 m−2 2 (W  ; e) and u ∈ Hloc W \ e) ∩ Vβ1 (W  ; e) be the solution of Let β ∈ R, f ∈ Vβ+m−1  Lu = f, in W , (4.14) u = 0, on ∂W  . m (W ; e) and satisfies the estimate Then u ∈ Vβ+m−1 m uVβ+m−1 (W ;e)  f V m−2

β+m−1 (W

 ;e)

+ uVβ1 (W  ;e) .

REGULARITY OF SOLUTIONS OF ELLIPTIC PROBLEMS WITH A CURVED FRACTURE

11

Remark 4.5. We denote x = (x⊥ , x3 ) ∈ W  with x⊥ ∈ K and x3 ∈ R, hence d(x, e) = |x⊥ |. Proof. Let us consider the sets A and A defined by (4.9) as well as the sets (A )1≤≤m and the functions η ∈ C ∞ (R3 ) considered in Lemma 4.3. Let A˜ such that supp(η2 ) ⊂ A˜ ⊂ A˜ ⊂ A . As f ∈ H m−2 (A ) and ˜ and by assumption u ∈ H 2 (A ), we deduce by [11, Theorem 2.3.2 (i)] that u ∈ H m (A) 

(4.15) um,A˜ ≤ c Lum−2,A + u1,A . This proves the requested regularity of u far from e. The proof of the regularity of u near e is based on dyadic covering of W  using a combination of a homothety together with a translation in x3 : ν

ˆ → x = 2−j (ˆ x + (0, 0, )), hj,ν : x 2 j+1 with j ∈ N and ν ∈ Z such that |ν| < 2 . Observe that we have     W = hj,ν (A) and W  ⊃ hj,ν (A ). j∈N |ν|<2j+1

j∈N |ν|<2j+1

j,ν j,ν Let us denote by hj,ν (resp. f ◦ hj,ν ) defined on A and consider the ∗ u (resp. h∗ f ) the function u ◦ h j,ν  operator L (ˆ x, ∂xˆ ) defined for x ˆ = (ˆ x⊥ , x ˆ3 ) ∈ A by 

ν

x, ∂xˆ ) := 2−2j L 2−j (ˆ x + (0, 0, )), 2j ∂xˆ . Lj,ν (ˆ 2

Observe that hj,ν ∗ u is solution of  j,ν −2j j,ν L (ˆ x, ∂xˆ )(hj,ν h∗ f, ∗ u) = 2 (4.16) hj,ν ∗ u = 0,

in on

A , (∂K × R) ∩ A¯ .

j,ν m ˜ ˜ Define the function v := η hj,ν ∗ u. Observe that, as above, h∗ u ∈ H (A) and using supp(η ) ⊂ A, m  we have v ∈ H (A ). Hence, to apply Lemma 4.3, we prove that, for all 2 ≤  ≤ m, v satisfies (4.10).

 2−j ν Let Q an open subset of R3 of class C m with A ⊂ Q and consider the operator Lj,ν ˆ depp (0, 0, 2 ), ∂x −j

fined by (4.1), i.e. the principal part of Lj,ν (ˆ x, ∂xˆ ) frozen at (0, 0, 2 2 ν ). Observe that, as Lj,ν (ˆ x, ∂xˆ )v ∈

 2−j ν −2    j,ν −2 H (A ), v ∈ H (A ) and η = 0 on Q \ A−1 we have Lpp (0, 0, 2 ), ∂xˆ v ∈ H (Q). Hence by [11, Theorem 2.3.2(ii)], we deduce that v satisfies

  2−j ν

v ,Q ≤ c Lj,ν ), ∂xˆ v −2,Q + v 1,Q , pp (0, 0, 2 where c depends only on the ellipticity constant and no more on j and ν as we have a uniformly elliptic operator with constant coefficient (see [23, Theorem 2.3.1]). Hence, as v = 0 in Q \ A , by Lemma 4.1, we have

  2−j ν

v ,A ≤ c Lj,ν ), ∂xˆ v −2,A + v 1,A pp (0, 0, 2

  2−j ν

), ∂xˆ ) v −2,A + Lj,ν (ˆ x, ∂xˆ ) − Lj,ν ((0, 0, x, ∂xˆ )v m−2,A + v 1,A ≤ c  Lj,ν (ˆ pp 2

 x, ∂xˆ )v −2,A + v 1,A , ≤ c c2−j v ,A + Lj,ν (ˆ or else,

 x, ∂xˆ )v −2,A + v 1,A . (1 − cc 2−j )v ,A ≤ c Lj,ν (ˆ

For j ≥ j1 with j1 large enough such that cc 2−j1 ≤ 12 , we obtain

 v ,A ≤ 2c Lj,ν (ˆ x, ∂xˆ )v −2,A + v 1,A . By definition of v and using Lemma 4.3, we get

 j,ν (4.17) hj,ν ˜m 2−2j hj,ν ∗ um,A ≤ c ∗ f m−2,A + h∗ u1,A . Conclusion. Observe that (4.17) means   1   ∂γ u 2−j|γ| γ 2hj,ν (A) 2  ∂x |γ|≤m

|γ|≤m−2

2−j(2+|γ|)

 12   −j|γ| ∂ γ u 2  12 ∂γ f 2  + 2  . j,ν (A ) j,ν (A ) h h ∂xγ ∂xγ |γ|≤1

12

S. ARICHE, C. DE COSTER AND S. NICAISE

For all (j, ν), let us multiply each term by 2−j(β−1) and replace 2−j by r (as 2−j  r on hj,ν (A )). Hence we obtain   1    1   β−1+|γ| ∂ γ u 2  12 ∂γ u ∂γ f rβ−1+|γ| γ 2hj,ν (A) 2  rβ+1+|γ| γ 2hj,ν (A ) 2 + r  . j,ν (A ) h ∂x ∂x ∂xγ |γ|≤m

|γ|≤m−2

|γ|≤1

j+1

By summing up for j ≥ j1 and ν ∈ Z with |ν| < 2

, and using (4.15), we obtain the result.



5. Natural shift theorem in weighted Sobolev spaces on cones with singularities In order to study the regularity of the solution u of (1.1) near a boundary point of σ, we need a natural shift theorem in weighted Sobolev spaces on cones with corner and edge singularities (Theorem 5.3). Its proof is based on a dyadic covering of the domain and is inspired by [12, Proposition 6.1] (where the authors consider the constant coefficients case) and on [13, Lemma 7.4.3] (where the result is proved on cones with vertex singularity). To facilitate the reading we here recall [13, Lemma 7.4.3]: Lemma 5.1. Let K be a regular cone in Rn with vertex 0, 0 < R < R , KR := K ∩ BR

and

KR := K ∩ BR ,

where BR (resp. BR ) denotes the ball of center 0 and radius R (resp. R ). Denote ∂KR := ∂K ∩ BR . Let m ≥ 2 and L be a second order elliptic operator defined on KR with coefficients of class C m (KR ),  ∂γ L= aγ (x) γ . ∂x |γ|≤2

 1 2 Let α ∈ R, f ∈ Vαm−2 (KR ; 0) and u ∈ Hloc KR \ {0} ∩Vα−m+1 (KR ; 0) be the solution of  Lu = f, in KR , u = 0, on ∂KR . Then u ∈ Vαm (KR ; 0) and satisfies the estimate 1 uVαm (KR ;0)  f Vαm−2 (KR ;0) + uVα−m+1 (KR ;0) .

Proof. We only sketch the proof, for the details we refer to [13, Lemma 7.4.3]. • The proof of the requested regularity of u far from 0 can be deduced using [11, Theorem 2.3.2 (i)]. It remains to prove the regularity of u near 0, which is based on a dyadic covering of KR and KR , namely using the homothety hj : x ˆ → x = 2−j x ˆ, with j ∈ N, we cover KR (resp. KR ) with a sequence of domains of the form hj (A ) (resp. hj (A)) where A (resp. A) is a bounded domain of KR (resp. KR ) and such that the vertex 0 does not belong to A (resp. A). This transformation leads to a new problem with a new operator Lj and Dirichlet boundary conditions, but this time in a smooth domain A . • This transformed system is elliptic in A and using a cut-off function, we can apply shift theorem to obtain a priori estimate of the corner leading part Lpp of L at 0, which is an elliptic operator with constant coefficients. • We show that (Lj − Lpp )vm−2,A  2−j vm,A , for all v ∈ H m (A ) and m ≥ 2. • The two last steps allow to obtain an a priori estimate for the operator Lj in the standard Sobolev spaces with a constant independent of j. • The estimate in the weighted Sobolev spaces is deduced using the definition of these spaces and the fact that on hj (A ), the distance to 0 is equivalent to 2−j .  In order to solve (3.6) in case σ ˆ = {(0, 0, zˆ) | zˆ ∈ R+ } ∩ V  , we need first to prove a shift regularity  result between Ωce and Ωce where (5.1)

Ωce = {x ∈ V | r0 (x) < and ρce (x) < }, Ωce = {x ∈ V  | r0 (x) <  and ρce (x) <  },

REGULARITY OF SOLUTIONS OF ELLIPTIC PROBLEMS WITH A CURVED FRACTURE rσ (x) r0 (x)

with <  , r0 (x) = dist(x, 0), rσ (x) = dist(x, σ) and ρce (x) = in Figure 4.

13

the angular distance, as illustrated

Figure 4. The “corner/edge” neighborhood. The following Lemma is the equivalent to Lemma 4.3 in weighted Sobolev spaces. Lemma 5.2. Let ,  ,  ∈ R with  < <  and define the sets



< r0 < and ρce < }, < r0 <  and ρce <  } νˆ := {x ∈ V  | 4 4 and σ ˆ := ({0} × R+ ) ∩ νˆ . Let m ≥ 2 and L be a uniformly elliptic operator with variable coefficients of class C m defined on νˆ by  ∂γ L= aγ (x) γ . ∂x (5.2)

νˆ = {x ∈ V |

|γ|≤2

m−2 Let β ∈ R and u ∈ such that Lu ∈ Vβ+m−1 (ˆ ν; σ ˆ ). Consider the sets (ˆ ν )1≤≤m such that νˆ ⊂ νˆ−1 , νˆ1 = νˆ and νˆm = νˆ and define functions η ∈ C ∞ (R3 ) such that η = 1 on νˆ and supp(η ) ⊂ νˆ−1 .  (ˆ ν; σ ˆ ) and satisfies Assume that, for all 2 ≤  ≤ m, we have η u ∈ Vβ+−1

Vβ1 (ˆ ν; σ ˆ)

(5.3)

η uVβ+−1  (ˆ ν  ;ˆ σ )  L(η u)V −2

ν β+−1 (ˆ

 ;ˆ σ)

+ η uVβ1 (ˆν  ;ˆσ) .

m Then u ∈ Vβ+m−1 (ˆ ν; σ ˆ ) and satisfies m uVβ+m−1 (ˆ ν ;ˆ σ )  LuV m−2

ν β+m−1 (ˆ

 ;ˆ σ)

+ uVβ1 (ˆν  ;ˆσ) . 

Proof. The proof can be made exactly as in Lemma 4.3. Now we prove our natural shift theorem on cones with singularities.

Theorem 5.3. Let <  , Ωce and Ωce be defined by (5.1). Consider a uniformly elliptic operator L of the second order defined on Ωce with coefficients of class C m (Ωce ),  ∂γ aγ (x) γ . L= ∂x |γ|≤2

We define the set e := {x = (0, x3 ) | 0 < x3 <  }.

  m−2 2 Ωce \ e ∩ Vβ1 (Ωce ; e) such that Lu ∈ Vβ+m−1 (Ωce ; e). Then u ∈ Let m ≥ 2, β ∈ R, u ∈ Hloc m Vβ+m−1 (Ωce ; e) and satisfies m uVβ+m−1 (Ωce ;e)  LuV m−2

 β+m−1 (Ωce ;e)

+ uVβ1 (Ωce ;e) .

Proof. The proof is based on a locally finite dyadic covering of Ωce and Ωce . Let us introduce the reference domains νˆ and νˆ defined in (5.2), see Figure 5. For j ∈ N, let us define νj := 2−j νˆ and νj := 2−j νˆ , and observe that   νj and Ωce ⊃ νj . Ωce = j∈N

j∈N

14

S. ARICHE, C. DE COSTER AND S. NICAISE

Figure 5. The reference domains νˆ and νˆ . ˆ → x = 2−j x ˆ, which maps νˆ onto νj and νˆ onto νj . For all j ∈ N, consider the function hj : x j j Let us denote h∗ u (resp. h∗ f ) the function u◦hj (resp. f ◦hj ) defined on νˆ and by Lj (ˆ x, ∂xˆ ) the operator defined on νˆ by 

Lj (ˆ x, ∂xˆ ) := 2−2j L 2−j x ˆ, 2j ∂xˆ . Observe that hj∗ u is then solution of x, ∂xˆ )(hj∗ u) = 2−2j hj∗ f, Lj (ˆ

(5.4)

in νˆ .

m (ˆ ν ; e) and satisfies Let us prove first that the solution hj∗ u of (5.4) is in Vβ+m−1 j m x, ∂xˆ )(hj∗ u)V m−2 hj∗ uVβ+m−1 (ˆ ν ;e)  L (ˆ

ν β+m−1 (ˆ

 ;e)

+ hj∗ uVβ1 (ˆν  ;e) ,

with a constant that does not depend on j.

 In order to obtain the independence on j, as in Theorem 4.4, we work on Ljpp 0, ∂xˆ , the principal part of Lj (ˆ x, ∂xˆ ) frozen at 0, i.e., 

 ∂γ (5.5) Ljpp 0, ∂xˆ = aγ (0) γ . ∂x ˆ |γ|=2



Observe that now νˆ and νˆ are of the same form as W and W  of Theorem 4.4 with K = R2 \ {0}. Hence, in order to apply Theorem 4.4, we need to truncate hj∗ u. We work by recurrence on 1 ≤  ≤ m and consider the sets (ˆ ν )1≤≤m such that νˆ ⊂ νˆ−1 , νˆ1 = νˆ ∞ 3 and νˆm = νˆ and the functions η ∈ C (R ) such that η = 1 on νˆ and supp(η ) ⊂ νˆ−1 given in Lemma 5.2 and prove that, for all 1 ≤  ≤ m, (5.6)

∃j−1 > 0, ∀j ≥ j−1 ,

j hj∗ uVβ+−1 x, ∂xˆ )(hj∗ u)V −2  (ˆ ν ;e)  L (ˆ

ν−1 ;e) β+−1 (ˆ

+ hj∗ uVβ1 (ˆν−1 ;e) .

Define the functions v := η (hj∗ u). 2 (ˆ ν2 ; e) and satisfies Step 1 : Let us prove (5.6) for  = 2, i.e., hj∗ u ∈ Vβ+1 j j 2 0 x, ∂xˆ )(hj∗ u)Vβ+1 hj∗ uVβ+1 (ˆ ν2 ;e)  L (ˆ (ˆ ν1 ;e) + h∗ uVβ1 (ˆ ν1 ;e) .  γ To this end, we apply Theorem 4.4 to the operators Ljpp (ˆ x, ∂xˆ ) := aγ (2−j x ˆ) ∂∂xˆγ and Ljpp (0, ∂xˆ )

(5.7)

|γ|=2

between ν˜ and νˆ1 , where supp(v2 ) ⊂ ν˜ ⊂ ν˜ ⊂ νˆ1 , and to the function v2 . More precisely: 2 ˚1 (ˆ As the solution u is in Vβ1 (Ωce ; e), we obtain easily that v2 ∈ V ν1 \ e). Moreover, by β ν1 ; e) ∩ Hloc (ˆ 0 Leibniz formula, we have Lj (ˆ x, ∂xˆ )v2 ∈ Vβ+1 (ˆ ν1 ; e) and, as v2 ∈ Vβ1 (ˆ ν  ; e), we deduce that Ljpp (ˆ x, ∂xˆ )v2 ∈ 0  2 Vβ+1 (ˆ ν ; e). By Theorem 4.4 we obtain v2 ∈ Vβ+1 (˜ ν ; e) and again Theorem 4.4 applied with Ljpp (0, ∂xˆ ) yields (5.8)

j 2 0 v2 Vβ+1 ˆ )v2 Vβ+1 (˜ ν ;e)  Lpp (0, ∂x (ˆ ν1 ;e) + v2 Vβ1 (ˆ ν1 ;e) j j 0 0 ≤ Lj (ˆ x, ∂xˆ )v2 Vβ+1 + (L (ˆ x , ∂ x ˆ ) − Lpp (0, ∂x ˆ ))v2 Vβ+1 (ˆ ν1 ;e) (ˆ ν1 ;e) + v2 Vβ1 (ˆ ν1 ;e) .

As by definition we have   



j ∂ γ v2 ∂ γ v2 (aγ (2−j x ˆ) − aγ (0)) γ ˆ, ∂xˆ − Ljpp 0, ∂xˆ v2 = 2j(|γ|−2) aγ (2−j x ˆ) γ + L x ∂x ˆ ∂x ˆ |γ|≤1

|γ|=2

REGULARITY OF SOLUTIONS OF ELLIPTIC PROBLEMS WITH A CURVED FRACTURE

we obtain easily, as in Lemma 4.1,  j 

   L x ˆ, ∂xˆ − Ljpp 0, ∂xˆ v2 V 0

ν1 ;e) β+1 (ˆ

15

2 ≤ c2−j v2 Vβ+1 (˜ ν ;e) .

This implies by (5.8) that for j ≥ j1 with j1 large enough such that c2−j1 ≤ 12 , we get j 2 0 v2 Vβ+1 x, ∂xˆ )v2 Vβ+1 (˜ ν ;e)  L (ˆ (ˆ ν1 ;e) + v2 Vβ1 (ˆ ν1 ;e) .

We conclude by Lemma 5.2 that the estimate (5.7) is valid. Step 2 : Assume that (5.6) is true for  − 1 and prove it is true for . As before we apply Theorem 4.4 to the operator Ljpp (ˆ x, ∂xˆ ) (resp. Ljpp (0, ∂xˆ )) and v between ν˜ and νˆ−1 , with supp(v ) ⊂ ν˜ ⊂ ν˜ ⊂ νˆ−1 . Again it is enough to verify that −2 x, ∂xˆ )v ∈ Vβ+−1 (ˆ ν−1 ; e). Ljpp (ˆ m−2 (Ωce ; e), (5.6) for  − 1 and supp(η ) ⊂ νˆ−1 , we easily obtain that By Leibniz formula, f ∈ Vβ+m−1 m−2 −1 x, ∂xˆ )v ∈ Vβ+m−1 (ˆ ν−1 ; e). As by recurrence assumption hj∗ u ∈ Vβ+−2 (ˆ ν−1 ; e) and supp(η ) ⊂ νˆ−1 , Lj (ˆ −1 −2 j ν−1 ; e) and hence Lpp (ˆ x, ∂xˆ )v ∈ Vβ+−1 (ˆ ν−1 ; e) by (5.5). By Theorem 4.4, we we obtain v ∈ Vβ+−2 (ˆ  (˜ ν ; e) and satisfies obtain v ∈ Vβ+−1 j v Vβ+−1  ˆ )v V −2 (˜ ν ;e)  Lpp (0, ∂x

(5.9)

ν−1 ;e) β+−1 (ˆ

+ v Vβ1 (ˆν−1 ;e)

with a constant independent of j. As above, we have (5.10)

Ljpp (0, ∂xˆ )v V −2

ν−1 ;e) β+−1 (ˆ

≤ Lj (ˆ x, ∂xˆ )v V −2

ν−1 ;e) β+−1 (ˆ

+ (Lj (ˆ x, ∂xˆ ) − Ljpp (0, ∂xˆ ))v V −2

ν−1 ;e) β+−1 (ˆ

.

As supp(η ) ⊂ ν˜, we easily check that (Lj (ˆ x, ∂xˆ ) − Ljpp (0, ∂xˆ ))v V −2

(5.11)

ν−1 ;e) β+−1 (ˆ

≤ c˜2−j v Vβ+−1  (˜ ν ;e) ,

with c˜ > 0 independent of j. By (5.9), (5.10) and (5.11), we deduce that, for j ≥ j−1 with j−1 large enough such that c˜2−j−1 ≤ 12 , we have j v Vβ+−1 x, ∂xˆ )v V −2  (ˆ ν ;e)  L (ˆ

ν−1 ;e) β+−1 (ˆ

+ v Vβ1 (ˆν−1 ;e) .

Step 3 : Conclusion. As v = hj∗ u on νˆ and supp(η ) ⊂ νˆ−1 , by Lemma 5.2, we deduce that, for j ≥ ¯j = max{j1 , jm−1 }, j m hj∗ uVβ+m−1 x, ∂xˆ )(hj∗ u)V m−2 (ˆ ν ;e)  L (ˆ

(5.12)

ν β+m−1 (ˆ

 ;e)

+ hj∗ uVβ1 (ˆν  ;e) .

ˆ which maps νˆ onto νˆj (resp. νˆ onto νˆj ) and note that Consider the change of variable x ˆ → x = 2−j x j r(ˆ x) = 2 r(x). Hence, by (5.12), we obtain, for j ≥ ¯j, 2(β−1)j

 |γ|≤m

rβ−1+|γ|

 ∂γ u 0,νj  2(β−1)j γ ∂x



rβ+1+|γ|

|γ|≤m−2

  ∂ γ (Lu) ∂γ u 0,νj + rβ−1+|γ| γ 0,νj . γ ∂x ∂x |γ|≤1

Multiplying by 2(1−β)j and summing up on j ≥ ¯j, we obtain  m uVβ+m−1 (

j≥¯ j

For the remainder part Ωce \

 j≥¯ j

νj ;e)

 LuV m−2

β+m−1 (

 j≥¯ j

νj ;e)

+ uVβ1 ( 

j≥¯ j

νj ;e) .

νj , we use Theorem 4.4 which allows to conclude.



16

S. ARICHE, C. DE COSTER AND S. NICAISE

6. Regularity of the solution of problem (1.1) We are now ready to prove Theorem 1.1. Its proof is made in three steps: first the regularity near an interior point of σ (Theorem 6.1), then the regularity near a boundary point of σ (Theorem 6.2) and then the regularity far from σ (Theorem 6.4). Let us prove first a higher regularity of the solution of (1.1) near an interior point of σ. Theorem 6.1. Let m ≥ 1, O is a bounded domain of R3 with boundary of class C m and σ be a one˚1 (O; σ) the dimensional submanifold of class C m+2 with σ ⊂ O. Let q ∈ L2 (σ), β ∈ (0, 1) and u ∈ H β solution of (1.1) given by [15, Corollary 2.2]. Then, for every interior point x0 of σ, there exists δ > 0 m (B3 (x0 , δ); σ ∩ B3 (x0 , δ)) and satisfies such that u ∈ Vβ+m−1 m uVβ+m−1 (B3 (x0 ,δ);σ∩B3 (x0 ,δ))  q0,σ .

Proof. Let 0 < δ < δ  small enough such that the extremities of σ does not belong to B3 (x0 , δ  ) and there ˆ satisfying ϕ(x0 ) = 0 and (3.1). exists a C m+2 -diffeomorphism ϕ : B3 (x0 , δ  ) → V  : x → x For 1 ≤  ≤ m, let us consider three decreasing sequences (δ )1≤≤m , (δ˜,1 )1≤≤m , (δ˜,2 )1≤≤m such that δm = δ, δ1 = δ  and ϕ(B3 (x0 , δ+1 )) ⊂ B2 (0, δ˜,1 ) × (−δ˜,1 , δ˜,1 ) ⊂ B2 (0, δ˜,2 ) × (−δ˜,2 , δ˜,2 ) ⊂ ϕ(B3 (x0 , δ )). Consider the sequence of truncated functions (η )2≤≤m B3 (x0 , δ−1 ). Recall that, by Section 3, u ˆ = (η u) ◦ ϕ−1 is solution  Lˆ u = qˆ δσˆ + fˆ , u ˆ = 0,

such that η = 1 on B3 (x0 , δ ) and supp(η ) ⊂ of V , ∂V ,

in on

where q = η q, f = −2∇η ·∇u−uΔη , L, qˆ and fˆ are defined as in Section 3, σ ˆ = {(0, 0, zˆ) | zˆ ∈ R}∩V and V = ϕ(B3 (x0 , δ−1 )).  Let us prove by recurrence on 2 ≤  ≤ m that u ∈ Vβ+−1 (B3 (x0 , δ ); σ ∩ B3 (x0 , δ )) and satisfies uVβ+−1  (B3 (x0 ,δ );σ∩B3 (x0 ,δ ))  q0,σ .

(6.1)

2 Step 1:  = 2 : u ∈ Vβ+1 (B3 (x0 , δ2 ); σ ∩ B3 (x0 , δ2 )) and satisfies (6.1) for  = 2. By Lemma 3.5, all the assumptions of Theorem 4.4 are satisfied with K = R2 \ {0} (we do not need boundary conditions as measR2 ({0}) = 0) and W := (B2 (0, δ˜1,1 ) \ {0}) × (−δ˜1,1 , δ˜1,1 ) and W  := (B2 (0, δ˜1,2 ) \ {0}) × (−δ˜1,2 , δ˜1,2 ). 2 2 (W ; σ ˆ ) = Vβ+1 (B2 (0, δ˜1,1 ) × (−δ˜1,1 , δ˜1,1 ); σ ˆ ), we then conclude by Theorem 4.4 that u ˆ2 ∈ As Vβ+1 2 ˜ ˜ ˜ Vβ+1 (B2 (0, δ1,1 ) × (−δ1,1 , δ1,1 ); σ ˆ ) and satisfies

ˆ u2 V 2

˜

˜

˜

σ) β+1 (B2 (0,δ1,1 )×(−δ1,1 ,δ1,1 );ˆ

0  fˆ2 Vβ+1 u2 Vβ1 (W  ;ˆσ) . (W  ;ˆ σ ) + ˆ

By (3.8), (3.9) and (1.3), we obtain 2 2 u2 V 2 uVβ+1 (B3 (x0 ,δ2 );σ) = u2 Vβ+1 (B3 (x0 ,δ2 );σ)  ˆ

˜

˜

˜

σ) β+1 (B2 (0,δ1,1 )×(−δ1,1 ,δ1,1 );ˆ

 uHβ1 (O;σ)  q0,σ .

 Step 2 : Assume that u ∈ Vβ+−1 (B3 (x0 , δ ); σ ∩ B3 (x0 , δ )) and satisfies (6.1) for  and prove +1 that u ∈ Vβ+ (B3 (x0 , δ+1 ); σ ∩ B3 (x0 , δ+1 )) and satisfies (6.1) for  + 1. We apply Theorem 4.4 (with K = R2 \ {0}) with and W  = (B(0, δ˜,2 ) \ {0}) × (−δ˜,2 , δ˜,2 ). W = (B(0, δ˜,1 ) \ {0}) × (−δ˜,1 , δ˜,1 ) 2 As previously, u ˆ+1 ∈ Vβ1 (W  ; σ ˆ ) ∩ Hloc (W  \ σ ˆ ). In order to apply Theorem 4.4, we just have to prove −1  ˆ ˆ ˆ ). By Definition of f+1 , the embedding (2.1) and (6.1) for , we have that f+1 ∈ Vβ+ (W ; σ

fˆ+1 V −1 (W  ;ˆσ) β+

(6.2)

   

ˆ uV −1 (ϕ(B3 (x0 ,δ ));ˆσ) + ∇ˆ uV −1 (ϕ(B3 (x0 ,δ ));ˆσ) β+

β+

ˆ uVβ++1 uV −1 (ϕ(B3 (x0 ,δ ));ˆσ)  (ϕ(B3 (x0 ,δ ));ˆ σ ) + ∇ˆ β+

ˆ uVβ+−1 uV −1  (ϕ(B3 (x0 ,δ ));ˆ σ ) + ∇ˆ

σ) β+−1 (ϕ(B3 (x0 ,δ ));ˆ

uVβ+−1  (B3 (x0 ,δ );σ∩B3 (x0 ,δ ))  q0,σ .

REGULARITY OF SOLUTIONS OF ELLIPTIC PROBLEMS WITH A CURVED FRACTURE

17

+1 By Theorem 4.4 we obtain u ˆ+1 ∈ Vβ+ (W ; σ ˆ ) with

u+1 Vβ1 (W  ;ˆσ) . ˆ u+1 V +1 (W ;ˆσ)  fˆ+1 V −1 (W  ;ˆσ) + ˆ β+

β+

The result can be deduced from Lemma 3.5, (6.2) and uV +1 (B3 (x0 ,δ+1 );σ)  ˆ u+1 V +1 (W ;ˆσ) . β+

β+



As a second result consider the regularity of the solution of (1.1) near a boundary point of σ. Theorem 6.2. Let m ≥ 1, O is a bounded domain of R3 with boundary of class C m and σ be a one˚1 (O; σ) be the dimensional submanifold of class C m+2 with σ ⊂ O. Let q ∈ L2 (σ), β ∈ (0, 1) and u ∈ H β solution of (1.1) given by [15, Corollary 2.2]. Then, for a boundary point x0 of σ, there exists δ > 0 such m (B3 (x0 , δ); σ ∩ B3 (x0 , δ)) and satisfies that u ∈ Vβ+m−1 m uVβ+m−1 (B3 (x0 ,δ);σ∩B3 (x0 ,δ))  q0,σ .

Proof. Recall that in case x0 is a boundary point of σ, we need to solve (3.6). We will prove the result by iteration on 1 ≤  ≤ m. For 1 ≤  ≤ m, let us consider three decreasing sequences (δ )1≤≤m , (  )1≤≤m , (  )1≤≤m such that δm = δ, δ1 = δ  and ϕ(B3 (x0 , δ+1 )) ⊂ B3 (0,  ) ⊂ B3 (0,  ) ⊂ ϕ(B3 (x0 , δ )). Consider the sequence of truncated functions (η )2≤≤m with η = 1 on B3 (x0 , δ ) and supp(η ) ⊂ B3 (x0 , δ−1 ). Recall that, by Section 3, u ˆ = (η u) ◦ ϕ−1 is a weak solution of  V , Lˆ u = qˆ δσˆ + fˆ , in on ∂V , u ˆ = 0, ˆ = {(0, 0, zˆ) | zˆ ∈ R+ } where q = η q, f = −2∇η · ∇u − uΔη , L, qˆ and fˆ are defined as in Section 3, σ   ˆ and V = ϕ(B3 (x0 , δ−1 )). As meas(ˆ σ ) = 0, we have Lˆ u = f a.e. in V . Let us cover B3 (0,  ) by (6.3)

Ωc, = {x ∈ V | r0 (x) <  and ρce (x) > 2 }, Ωce, = {x ∈ V | r0 (x) <  and ρce (x) <  },

and B3 (0,  ) by (see Figure 6) 

Ωc, = {x ∈ V  | r0 (x) <  and ρce (x) > 2 }, Ωce, = {x ∈ V  | r0 (x) <  and ρce (x) <  }.

Figure 6. The covering of B3 (0,  ). In order to prove the expected regularity, we apply Theorem 5.3 on Ωce, and Lemma 5.1 on the domain

Ωc, .

  We will prove by iteration on 2 ≤  ≤ m that u ˆ ∈ Vβ+−1 (Ωce, ; σ ˆ ) (resp. u ˆ ∈ Vβ+−1 (Ωc, ; σ ˆ )) with

(6.4)

ˆ u Vβ+−1  (Ωce, ;ˆ σ )  q0,σ

(resp. ˆ u Vβ+−1  (Ωc, ;ˆ σ )  q0,σ ).

2 ˆ ) ∩ Hloc (B3 (0, 1 ) \ σ ˆ ) and By Lemma 3.5, we know that u ˆ2 ∈ Vβ1 (B3 (0, 1 ); σ

ˆ u2 Vβ1 (B3 (0,1 );ˆσ)  uHβ1 (O;σ)  q0,σ .

18

S. ARICHE, C. DE COSTER AND S. NICAISE

0 0 Step 1:  = 2. By Lemma 3.5, we know that fˆ2 ∈ Vβ+1 (Ωce,1 ; σ ˆ ) (resp. fˆ2 ∈ Vβ+1 (Ωc,1 ; σ ˆ )) and is such that ˆ 0 (Ω ;ˆσ)  q0,σ . 0 fˆ2 Vβ+1 (Ωce,1 ;ˆ σ )  q0,σ and f2 Vβ+1 c,1 2 By Theorem 5.3, we deduce that u ˆ2 ∈ Vβ+1 (Ωce,1 ; σ ˆ ) with 2 ˆ u2 Vβ+1 (Ωce,1 ;ˆ σ )  q0,σ .

2 In the same way, applying Lemma 5.1 we deduce that u ˆ2 ∈ Vβ+1 (Ωc,1 ; σ ˆ ) with 2 ˆ u2 Vβ+1 (Ωc,1 ;ˆ σ )  q0,σ .

2 This implies that u ˆ2 ∈ Vβ+1 (B3 (0, 1 ); σ ˆ ) with 2 ˆ u2 Vβ+1 (B3 (0,1 );ˆ σ )  q0,σ

2 and hence u ∈ Vβ+1 (B3 (x0 , δ2 ); σ) and satisfies 2 uVβ+1 (B3 (x0 ,δ2 );σ)  q0;σ .

−1 −1 Step 2: Assume that u ˆ−1 ∈ Vβ+−2 (Ωce,−1 ; σ ˆ ) (resp. u ˆ−1 ∈ Vβ+−2 (Ωc,−1 ; σ ˆ )) and (6.4) is true   ˆ ) (resp. u ˆ ∈ Vβ+−1 (Ωc, ; σ ˆ )) and (6.4) is true also for  − 1 and prove that u ˆ ∈ Vβ+−1 (Ωce, ; σ −2 (Ωce,−1 ; σ ˆ) for . As explained above, in order to prove that, we just need to prove that fˆ ∈ Vβ+−1 −2  (resp. fˆ ∈ Vβ+−1 (Ωc,−1 ; σ ˆ )) with

fˆ V −2

 σ) β+−1 (Ωce,−1 ;ˆ

 q0,σ

(resp. fˆ V −2

 σ) β+−1 (Ωc,−1 ;ˆ

 q0,σ ).

This can be easily deduced from the definition of fˆ as in the proof of Theorem 6.1. We conclude the proof of this step as in Step 1. This concludes the proof of the Theorem.



Remark 6.3. Clearly Theorems 6.1 and 6.2 remain valid if the fracture is polygonal, in the sense that it is continuous, piecewise C m+2 and for each “break” point, the same transformation (with the regularity ˆ , made of two straight half-lines C m+2 ) is used to straighten the two half parts of the fracture into σ having 0 as common extremity. It remains to prove the regularity of the solution far from σ. Observe that, in that case, the weighted norms are equivalent to the standard Sobolev norms. Hence we prove that the solution belongs to H m far from σ. Theorem 6.4. Let m ≥ 1, O be a bounded domain of R3 with boundary of class C m and σ be a one˚1 (O; σ) the dimensional submanifold of class C m+2 with σ ⊂ O. Let q ∈ L2 (σ), β ∈ (0, 1) and u ∈ H β solution of (1.1) given by [15, Corollary 2.2]. Define R = dist(∂Ω, σ) and choose 0 < < R, then u ∈ H m (O ) where {x ∈ O | dist(x, σ) > } ⊂ O ⊂ {x ∈ O | dist(x, σ) > 2 } is a bounded domain of R3 with boundary of class C m . Moreover, u satisfies um,O  q0,σ . Proof. Consider a decreasing sequence of sets (O )1≤≤m with Om = O , O1 ⊂ {x ∈ O | dist(x, ∂O) >  m 4 } and ∂O of class C . Consider also a sequence of truncated function (η )2≤≤m such that η = 1 in O and supp(η ) ⊂ O−1 . ˚1 (O ) is solution of As u ∈ Hβ1 (O; σ), we observe easily that w = η u ∈ H −1  Δw = uΔη + 2∇η · ∇u =: h , in O−1 , w = 0, on ∂O−1 , with h ∈ L2 (O−1 ), and satisfies w 1,O−1  uHβ1 (O;σ)  q0,σ . Let us prove by recurrence on 2 ≤  ≤ m that u ∈ H  (O ) with u,O  q0,σ .

REGULARITY OF SOLUTIONS OF ELLIPTIC PROBLEMS WITH A CURVED FRACTURE

19

Step 1:  = 2. As h2 ∈ L2 (O1 ), applying [9, Theorem 9.25], we obtain that w2 ∈ H 2 (O1 ) and hence, u ∈ H 2 (O2 ) with u2,O2  q0,σ . Step 2: Assume that the estimate is true for  − 1 and prove it is true for . As, by assumption u ∈ H −1 (O−1 ) and h = uΔη + 2∇η · ∇u, we have h ∈ H −2 (O−1 ) and, again by [9, Theorem 9.25], we conclude that w ∈ H  (O−1 ) and hence u ∈ H  (O ) with u,O ≤ w,O−1  h−2,O−1  q0,σ . 

This concludes the proof. In order to conclude, it remains to put every ingredients together.

Proof of Theorem 1.1. For all x0 ∈ σ, choose δx0 given by Theorem 6.1 in case x0 is interior to σ and by Theorem 6.2 in case x0 is a boundary point of σ. Hence, by compacity of σ, we have a finite sub-covering  {xj |1≤j≤n} B(xj , δxj ) of σ. Define = min{δxj | 1 ≤ j ≤ n}. By considering the set O of Theorem 6.4, we can conclude by Theorems 6.1, 6.2 and 6.4.  Remark 6.5. The proofs of Theorems 6.1 and 6.2 show that the regularity of the boundary does not ˚1 (O; σ) of (1.1) near the fracture. Hence if we assume that affect the regularity of the solution u ∈ H β the boundary of O is not smooth (a three-dimensional polyhedron), we can replace the standard shift theorem, i.e., [9, Theorem 9.25], by the corresponding regularity result [17, Theorem 17.13]. Remark 6.6. Owing to Remark 6.3, Theorem 1.1 remains valid if the fracture is polygonal. References [1] B. Ammann, A. D. Ionescu, and V. Nistor. Sobolev spaces on Lie manifolds and regularity for polyhedral domains. Doc. Math., 11:161–206 (electronic), 2006. [2] T. Apel, O. Benedix, D. Sirch, and B. Vexler. A priori mesh grading for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal., 49(3):992–1005, 2011. [3] R. Araya, E. Behrens, and R. Rodríguez. A posteriori error estimates for elliptic problems with Dirac delta source terms. Numer. Math., 105(2):193–216, 2006. [4] S. Ariche, C. De Coster, and S. Nicaise. Regularity of solutions of elliptic or parabolic problems with Dirac measures as data. SeMA Journal, 2016. DOI 10.1007/s40324-016-0077-x, to appear. [5] I. Babuška and V. Nistor. Interior numerical approximation of boundary value problems with a distributional data. Numer. Methods Partial Differential Equations, 22(1):79–113, 2006. [6] C. Bacuta, V. Nistor, and L. T. Zikatanov. Improving the rate of convergence of high-order finite elements on polyhedra. I. A priori estimates. Numer. Funct. Anal. Optim., 26(6):613–639, 2005. [7] M. Berger and B. Gostiaux. Géométrie différentielle: variétés, courbes et surfaces. Mathématiques. [Mathematics]. Presses Universitaires de France, Paris, second edition, 1992. [8] P. Bolley, M. Dauge, and J. Camus. Régularité Gevrey pour le problème de Dirichlet dans des domaines à singularités coniques. Comm. Partial Differential Equations, 10(4):391–431, 1985. [9] H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. [10] M. Costabel and M. Dauge. Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of nodal finite elements. Numer. Math., 93(2):239–277, 2002. [11] M. Costabel, M. Dauge, and S. Nicaise. Corner singularities and analytic regularity for linear elliptic systems.. Part I: Smooth domains. 211 pages, https://hal.archives-ouvertes.fr/hal-00453934/file/CoDaNi_Analytic_Part_I.pdf, Feb. 2010. [12] M. Costabel, M. Dauge, and S. Nicaise. Analytic regularity for linear elliptic systems in polygons and polyhedra. Math. Models Methods Appl. Sci., 22(8):1250015, 63, 2012. [13] M. Costabel, M. Dauge, and S. Nicaise. Corner singularities and analytic regularity for linear elliptic systems. Part II: Corner domains. in preparation. [14] M. Costabel, M. Dauge, and C. Schwab. Exponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains. Math. Models Methods Appl. Sci., 15(4):575–622, 2005. [15] C. D’Angelo. Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces: applications to one- and three-dimensional coupled problems. SIAM J. Numer. Anal., 50(1):194–215, 2012. [16] C. D’Angelo and A. Quarteroni. On the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems. Math. Models Methods Appl. Sci., 18(8):1481–1504, 2008. [17] M. Dauge. Elliptic boundary value problems on corner domains – smoothness and asymptotics of solutions. Lecture Notes in Mathematics, Vol. 1341. Springer, Berlin, 1988. [18] M. Dauge. Neumann and mixed problems on curvilinear polyhedra. Integral Equations Operator Theory, 15(2):227–261, 1992.

20

S. ARICHE, C. DE COSTER AND S. NICAISE

[19] P. Grisvard. Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, Vol. 24. Pitman, Boston–London–Melbourne, 1985. [20] V. A. Kondrat’ev. Boundary value problems for elliptic equations on domains with conical or angular points. Trudy Moskov. Mat. Obshch., 16:209–292, 1967. In Russian. [21] V. A. Kozlov, V. G. Maz ya, and J. Rossmann. Elliptic boundary value problems in domains with point singularities. Mathematical Surveys and Monographs, Vol. 52. American Mathematical Society, Providence, RI, 1997. [22] H. Li. A-priori analysis and the finite element method for a class of degenerate elliptic equations. Math. Comp., 78(266):713–737, 2009. [23] J.-L. Lions and E. Magenes. Problèmes aux limites non homogènes et applications. Vol. 1. Travaux et Recherches Mathématiques, No. 17. Dunod, Paris, 1968. [24] V. G. Maz’ya and B. Plamenevski˘ı. Lp -estimates of solutions of elliptic boundary value problems in domains with edges. Trudy Moskov. Mat. Obshch., 37:49–93, 1978. In Russian. Translated in Trans. Mosc. Math. Soc., 1:49–97, 1980. [25] V. G. Maz’ya and J. Roßmann. On the Agmon-Miranda maximum principle for solutions of elliptic equations in polyhedral and polygonal domains. Ann. Global Anal. Geom., 9:253–303, 1991. [26] V. G. Maz’ya and J. Roßmann. Elliptic equations in polyhedral domains. Mathematical Surveys and Monographs, Vol. 162. American Mathematical Society, Providence, RI, 2010. [27] A. L. Mazzucato and V. Nistor. Well-posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks. Arch. Ration. Mech. Anal., 195(1):25–73, 2010. [28] Y. Roitberg. Elliptic boundary value problems in the spaces of distributions, volume 384 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1996. Translated from the Russian by Peter Malyshev and Dmitry Malyshev. [29] M. Schechter. Negative norms and boundary problems. Ann. of Math. (2), 72:581–593, 1960. [30] R. Scott. Finite element convergence for singular data. Numer. Math., 21:317–327, 1973/74. *LABORATOIRE DE MATHÉMATIQUES PURES ET APPLIQUÉES, UNIVERSITÉ DE JIJEL, BP 98 OULED AISSA JIJEL 18000, ALGÉRIE, E-MAIL : SADJIA [email protected], **LABORATOIRE DE MATHÉMATIQUES ET SES APPLICATIONS DE VALENCIENNES, FR CNRS 2956, UNIVERSITÉ DE VALENCIENNES ET DU HAINAUT-CAMBRÉSIS, LE MONT HOUY, 59313 VALENCIENNES CEDEX 9, FRANCE, E-MAILS : [email protected], [email protected]