Unique continuation from the edge of a crack

In this work we develop an Almgren type monotonicity formula for a class of elliptic equations in a domain with a crack, in the presence of potentials satisfying either a negligibility condition with respect to the inverse-square weight or some suitable integrability properties. The study of the Almgren frequency function around a point on the edge of the crack, where the domain is highly non-smooth, requires the use of an approximation argument, based on the construction of a sequence of regular sets which approximate the cracked domain. Once a finite limit of the Almgren frequency is shown to exist, a blow-up analysis for scaled solutions allows us to prove asymptotic expansions and strong unique continuation from the edge of the crack.


Introduction and statement of the main results
This paper presents a monotonicity approach to the study of the asymptotic behavior and unique continuation from the edge of a crack for solutions to the following class of elliptic equations where Ω ⊂ R N +1 is a bounded open domain, Γ ⊂ R N is a closed cut, N ≥ 2, and the potential f satisfies either a negligibility condition with respect to the inverse-square weight, see assumptions (H1-1)-(H1-3), or some suitable integrability properties, see assumptions (H2-1)-(H2-5) below. We recall that the strong unique continuation property is said to hold for a certain class of equations if no solution, besides possibly the zero function, has a zero of infinite order. Unique continuation principles for solutions to second order elliptic equations have been largely studied in the literature since the pioneering contribution by Carleman [6], who derived unique continuation from some weighted a priori inequalities. Garofalo and Lin in [20] studied unique continuation for elliptic equations with variable coefficients introducing an approach based on the validity of doubling conditions, which in turn depend on the monotonicity property of the Almgren type frequency function, defined as the ratio of scaled local energy over mass of the solution near a fixed point, see [4].
Once a strong unique continuation property is established and infinite vanishing order for nontrivial solutions is excluded, the problem of estimating and possibly classifying all admissible vanishing rates naturally arises. For quantitative uniqueness and bounds for the maximal order of vanishing obtained by monotonicity methods we quote e.g. [23]; furthermore, a precise description of the asymptotic behavior together with a classification of possible vanishing orders of solutions was obtained for several class of problems in [15,16,17,18,19], by combining monotonicity methods with blow-up analysis for scaled solutions.
The problem of unique continuation from boundary points presents peculiar additional difficulties, as the derivation of monotonicity formulas is made more delicate by the interference with the geometry of the domain. Moreover the possible vanishing orders of solutions are affected by the regularity of the boundary; e.g. in [15] the asymptotic behavior at conical singularities of the boundary has been shown to depend of the opening of the vertex. We quote [2,3,15,24,29] for unique continuation from the boundary for elliptic equations under homogeneous Dirichlet conditions. We also refer to [28] for unique continuation and doubling properties at the boundary under zero Neumann conditions and to [11] for a strong unique continuation result from the vertex of a cone under non-homogeneous Neumann conditions. The aforementioned papers concerning unique continuation from the boundary require the domain to be at least of Dini type. With the aim of relaxing this kind of regularity assumptions, the present paper investigates unique continuation and classification of the possible vanishing orders of solutions at edge points of cracks breaking the domain, which are then highly irregular points of the boundary.
Elliptic problems in domains with cracks arise in elasticity theory, see e.g. [9,22,25]. The high non-smoothness of domains with slits produces strong singularities of solutions to elliptic problems at edges of cracks; the structure of such singularities has been widely studied in the literature, see e.g. [7,8,12] and references therein. In particular, asymptotic expansions of solutions at edges play a crucial role in crack problems, since the coefficients of such expansions are related to the so called stress intensity factor, see e.g. [9].
A further reason of interest in the study of problem (1.1) can be found in its relation with mixed Dirichlet/Neumann boundary value problems. Indeed, if we consider an elliptic equation with a homogeneous Dirichlet boundary condition on a flat portion of the boundary and a homogeneous Neumann condition on the complement, an even reflection through the flat boundary leads to an elliptic equation satisfied in the complement of the Dirichlet region, which then plays the role of a crack; the edge of the crack corresponds to the Dirichlet-Neumann junction of the original problem. In [14] unique continuation and asymptotic expansions of solutions for planar mixed boundary value problems at Dirichlet-Neumann junctions were obtained via monotonicity methods; the present paper is in part motivated by the aim of extending to higher dimensions the monotonicity formula obtained in [14] in the 2-dimensional case, together with its applications to unique continuation. For some regularity results for second-order elliptic problems with mixed Dirichlet-Neumann type boundary conditions we refer to [21,27] and references therein.
In the generalization of the Almgren type monotonicity formula of [14] to dimensions greater than 2, some new additional difficulties arise, besides the highly non-smoothness of the domain: the positive dimension of the edge, a stronger interference with the geometry of the domain, and some further technical issues, related e.g. to the lack of conformal transformations straightening the edge. In particular, the proof of the monotonicity formula is based on the differentiation of the Almgren quotient defined in (4.9), which in turn requires a Pohozaev type identity formally obtained by testing the equation with the function ∇u · x; however our domain with crack doesn't verify the exterior ball condition (which ensures L 2 -integrability of second order derivatives, see [1]) and ∇u · x could be not sufficiently regular to be an admissible test function.
In this article a new technique, based on an approximation argument, is developed to overcome the aforementioned difficulty: we construct first a sequence of domains which approximate Ω \ Γ, satisfying the exterior ball condition and being star-shaped with respect to the origin, and then a sequence of solutions of an approximating problem on such domains, converging to the solution of the original problem with crack. For the approximating problems enough regularity is available to establish a Pohozaev type identity, with some remainder terms due to interference with the boundary, whose sign can nevertheless be recognized thanks to star-shapeness conditions. Then, passing to the limit in Pohozaev identities for the approximating problems, we obtain inequality (3.11), which is enough to estimate from below the derivative of the Almgren quotient and to prove that such quotient has a finite limit at 0 (Lemma 4.7). Once a finite limit of the Almgren frequency is shown to exist, a blow-up analysis for scaled solutions allows us to prove strong unique continuation and asymptotics of solutions.
In order to state the main results of the present paper, we start by introducing our assumptions on the domain. For N ≥ 2, we consider the set Let us observe that assumption (1.2) is not a restriction but just a selection of our coordinate system and, from (1.2) and (1.3), it follows that for any x ∈ B R := {x ∈ R N −1 : |x | <R}, for someR > 0. This condition says that R N \ Γ is star-shaped with respect to the origin in a neighbourood of 0. It is satisfied for instance if the function g is concave in a neighborhood of the origin. We are interested in studying the following boundary value problem where BR = {x ∈ R N +1 : |x| <R}, for some function f : BR → R such that f is measurable and bounded in BR \ B δ for every δ ∈ (0,R). We consider two alternative sets of assumptions: we assume either that lim r→0 + ξ f (r) = 0, where the function ξ f is defined as Br |h|u 2 dx We also observe that condition (H2-1) is satisfied if f belongs to the Kato class K n+1 , see [13].
In order to give a weak formulation of problem (1.6), we introduce the space H 1 Γ (B R ) for every R > 0, defined as the closure in H 1 (B R ) of the subspace : u = 0 in a neighborhood of Γ}. We observe that actually , where τ Γ denotes the trace operator on Γ, as one can easily deduce from [5], taking into account that the capacity of ∂Γ in R N +1 is zero, since ∂Γ is contained in a 2-codimensional manifold.
Hence we say that u ∈ H 1 (BR) is a weak solution to (1.6) In the classification of the possible vanishing orders and blow-up profiles of solutions, the following eigenvalue problem on the unit N -dimensional sphere with a half-equator cut plays a crucial role.
We say that µ ∈ R is an eigenvalue of (1.7) is there exists an eigenfunction ψ ∈ H 1 . By classical spectral theory, (1.7) admits a diverging sequence of real eigenvalues with finite multiplicity {µ k } k≥1 ; moreover these eigenvalues are explicitly given by the formula be the multiplicity of the eigenvalue µ k and {Y k,m } m=1,2,...,M k be a L 2 (S N )-orthonormal basis of the eigenspace of (1.7) associated to µ k . (1.9) The main result of this paper provides an evaluation of the behavior at 0 of weak solutions u ∈ H 1 (BR) to the boundary value problem (1.6). Theorem 1.1. Let N ≥ 2 and u ∈ H 1 (BR) \ {0} be a non-trivial weak solution to (1.6), with f satisfying either assumptions (H1-1)-(H1-3) or (H2-1)-(H2-5). Then, there exist k 0 ∈ N, k 0 ≥ 1, and an eigenfunction of problem (1.7) associated with the eigenvalue µ k0 such that As a direct consequence of Theorem 1.1 and boundedness of eigenfunctions of (1.7) (see Appendix A), the following point-wise upper bound holds. Corollary 1.2. Under the same assumptions as in Theorem 1.1, let u ∈ H 1 (BR) be a non-trivial weak solution to (1.6). Then, there exists k 0 ∈ N, k 0 ≥ 1, such that A further relevant consequence of our asymptotic analysis is the following unique continuation principle, whose proof follows straightforwardly from Theorem 1.1.
Under the same assumptions as in Theorem 1.1, let u ∈ H 1 (BR) be a weak solution to (1.6) such that u(x) = O(|x| k ) as |x| → 0, for any k ∈ N. Then u ≡ 0 in BR. Theorem 6.7 will actually give a more precise description on the limit angular profile ψ: if M k0 ≥ 1 is the multiplicity of the eigenvalue µ k0 and {Y k0,i : 1 ≤ i ≤ M k0 } is as in (1.9), then the eigenfunction ψ in (1.10) can be written as where the coefficients β i are given by the integral Cauchy-type formula (6.40).
The paper is organized as follows. In Section 2 we construct a sequence of problems on smooth sets approximating the cracked domain, with corresponding solutions converging to the solution of problem (1.6). In Section 3 we derive a Pohozaev type identity for the approximating problems and consequently inequality (3.11), which is then used in Section 4 to prove the existence of the limit for the Almgren type quotient associated to problem (1.6). In Section 5 we perform a blow-up analysis and prove that scaled solutions converge in some suitable sense to a homogeneous limit profile, whose homogeneity order is related to the eigenvalues of problem (1.7) and whose angular component is shown to be as in (1.11) in Section 6, where an auxiliary equivalent problem with a straightened crack is constructed. Finally, in the appendix we derive the explicit formula (1.8) for the eigenvalues of problem (1.7).
Notation. We list below some notation used throughout the paper.
-For all r > 0, with radius r and center at 0.
-For all r > 0, B r denotes the open ball {x = (x , x N ) ∈ R N : |x| < r} in R N with radius r and center at 0. -dS denotes the volume element on the spheres ∂B r , r > 0.

Approximation problem
We first prove a coercivity type result for the quadratic form associated to problem (1.6) in small neighbourhoods of 0.
Now we are going to to construct suitable regular sets which are star-shaped with respect to the origin and which approximate our cracked domain. In order to do this, for any n ∈ N \ {0} let f n : R → R be defined as f n (t) = n|t| + 1 n e 2n 2 |t| n 2 |t|−2 , if |t| < 2/n 2 , n|t|, if |t| ≥ 2/n 2 , so that f n ∈ C 2 (R), f n (t) ≥ n|t| and f n increases for all t > 0 and decreases for all t < 0; furthermore f n (t) − t f n (t) ≥ 0 for every t ∈ R. (2.10) For all r > 0 we definẽ see Figure 1. Letγ r,n ⊂ ∂B r,n be the subset of B r defined as The setBr,n (b) Section ofBr,n Figure 1. Approximating domains andS r,n denote the set given by ∂B r,n \γ r,n . We note that, for any fixed r > 0, the setγ r,n is not empty andB r,n = B r provided n is sufficiently large.
Lemma 2.4. Let 0 < r ≤R. Then, for all n ∈ N \ {0}, the setB r,n is star-shaped with respect to the origin, i.e. x · ν ≥ 0 for a.e. x ∈ ∂B r,n , where ν is the outward unit normal vector.
Proof. Ifγ r,n is empty, thenB r,n = B r and the conclusion is obvious. Letγ r,n be not empty. The thesis is trivial if one considers a point x ∈ ∂B r,n \γ r,n . If x ∈γ r,n , then x = (x , g(x ) + f n (x N +1 ), x N +1 ) and the outward unit normal vector at this point is given by hence we have that From now on, we fix u ∈ H 1 (BR) \ {0}, a non-trivial weak solution to problem (1.6), with f satisfying either (H1-1)-(H1-3) or (H2-1)-(H2-5). Since u ∈ H 1 Γ (BR), there exists a sequence of functions g n ∈ C ∞ 0,Γ (BR) such that g n → u in H 1 (BR). We can choose the functions g n in such a way that <C n , if we chooseC as in (2.13). Then g n (x) = 0 in view of (2.12).
Now we construct a sequence of approximated solutions {u n } n∈N on the setsB r0,n . For each fixed n ∈ N, we claim that there exists a unique weak solution u n to the boundary value problem −∆u n = f u n inB r0,n , u n = g n on ∂B r0,n . (2.14) Letting v n := u n − g n , we have that u n weakly solves (2.14) if and only if v n ∈ H 1 (B r0,n ) is a weak solution to the homogeneous boundary value problem Lemma 2.6. Let r 0 be as in Lemma 2.1. Then, for all n ∈ N, problem (2.15) has one and only one weak solution v n ∈ H 1 0 (B r0,n ), whereB r0,n is defined in (2.11). Proof. Let us consider the bilinear form for every v, w ∈ H 1 0 (B r0,n ). Lemma 2.1 implies that a is coercive on H 1 0 (B r0,n ). Furthermore, from estimate (2.4) we easily deduce that a is continuous. The thesis then follows from the Lax-Milgram Theorem.
Proposition 2.7. Under the same assumptions of Lemma 2.6, there exists a positive constant Proof. Let us observe that f g n and −∆g n are bounded in H −1 (B r0 ) as a consequence of the boundedness of g n in H 1 (B r0 ): indeed, using (2.4), one has that, for any φ ∈ H 1 0 (B r0 ), for some c 1 > 0 independent on n and φ, and , for some c 3 > 0 independent on n. This completes the proof. Proposition 2.8. Under the same assumptions of Lemma 2.6, we have that u n u weakly in H 1 (B r0 ), where u n is extended trivially to zero in B r0 \B r0,n .
Proof. We observe that the trivial extension to zero of u n in B r0 \B r0,n belongs to H 1 (B r0 ) since the trace of u n onγ r0,n is null in view of Remark 2.5.
From Proposition 2.7 it follows that there existṽ ∈ H 1 0 (B r0 ) and a subsequence {v Arguing as in Remark 2.5, we can prove that φ ∈ H 1 0 (B r0,n k ) for all sufficiently large k. Hence, from (2.14) it follows that, for all sufficiently large k, where u n k is extended trivially to zero in B r0 \B r0,n k . Passing to the limit in (2.18), we obtain that . Furthermoreũ = u on ∂B r0 in the trace sense: indeed, due to compactness of the trace map γ : For every δ > 0 we have that Γ δ ∩ B r0 ⊂ B r0 \B r0,n provided n is sufficiently large. Hence, since u n is extended trivially to zero in B r0 \B r0,n , we have that, for every δ > 0, Testing equation (2.19) with U itself and using Lemma 2.1, we obtain that so that U = 0, hence u =ũ. By Urysohn's subsequence principle, we can conclude that u n u weakly in H 1 (B r0 ).
Our next aim is to prove strong convergence of the sequence {u n } n∈N to u in H 1 (B r0 ). Proposition 2.9. Under the same assumptions of Lemma 2.6, we have that as n → ∞. Thus, from Lemma 2.1, we deduce that v n H 1 0 (Br 0 ) → 0 as n → ∞, hence v n → 0 in H 1 (B r0 ). This yields that u n = g n + v n → u in H 1 (B r0 ).

Pohozaev Identity
In this section we derive a Pohozaev type identity for u n in which we will pass to the limit using Proposition 2.9. For every r ∈ (0, r 0 ) and v ∈ H 1 (B r ), we define Proof. Since u n solves (2.14) in the domainB r0,n , which satisfies the exterior ball condition, and f u n ∈ L 2 loc (B r0,n \ {0}), by elliptic regularity theory (see [1]) we have that u n ∈ H 2 (B r,n \ B δ ) for all r ∈ (0, r 0 ), n sufficiently large and all δ < r n , where r n is such that B rn ⊂B r,n . Since rn 0 ∂Br |∇u n | 2 + |f |u 2 n dS dr = Br n |∇u n | 2 + |f |u 2 n dx < +∞, there exists a sequence {δ k } k∈N ⊂ (0, r n ) such that lim k→∞ δ k = 0 and Testing (2.14) with x · ∇u n and integrating overB r,n \ B δ k , we obtain that Integration by parts allows us to rewrite the first term in (3.3) as where we used that x = rν onS r,n and that the tangential component of ∇u n onγ r,n equals zero, thus ∇u n = ∂un ∂ν ν onγ r,n . Furthermore, by direct calculations, the first term in (3.4) can be rewritten as Taking into account (3.3), (3.4) and (3.5), we obtain that Under assumptions (H1-1)-(H1-3), the Hardy inequality (2.5) implies that f u n (x · ∇u n ) ∈ L 1 (B r ) and hence On the other hand, if (H2-1)-(H2-5) hold, we can use the Divergence Theorem to obtain that Since, under assumptions (H2-1)-(H2-5), ∇f · x + (N + 1)f u 2 n ∈ L 1 (B r ), we can pass to the limit as k → ∞ in (3.8) taking into account also (3.2), thus obtaining that Letting k → +∞ in (3.6), by (3.2), (3.7), and (3.9), we obtain (3.1).
Combining Lemma 3.1 with the fact that the domainsB r,n (defined as in (2.11)) are star-shaped with respect to the origin, we deduce the following inequality.
Passing to the limit in (3.10) as n → ∞, a similar inequality can be derived for u.
Then, for a.e. r ∈ (0, r 0 ), we have that and Proof. In order to prove (3.11), we pass to the limit inside inequality (3.10). As regards the first term, it is sufficient to observe that for each fixed r ∈ (0, r 0 ), as a consequence of Proposition 2.9. In order to deal with the second term, we observe that, by strong H 1 -convergence of u n to u, (3.13) implies that F n → 0 in L 1 (0, r 0 ). Then there exists a subsequence F n k such that F n k (r) → 0 for a.e. r ∈ (0, r 0 ), hence S r,n k |∇u n k | 2 dS = ∂Br |∇u n k | 2 dS → ∂Br |∇u| 2 dS as k → ∞ for a.e. r ∈ (0, r 0 ). In a similar way, we obtain that It remains to prove the convergence of R(r, u n ). Under the set of assumptions (H1-1)-(H1-3), we first write (3.14) Hölder inequality, (2.5), and Proposition 2.9 imply that as n → ∞, for a.e. r ∈ (0, r 0 ), since ξ f (r) is finite a.e. as a consequence of assumption (H1-2). Hence, from (3.14) we deduce that under assumptions (H1-1)-(H1-3). To prove (3.15) under assumptions (H2-1)-(H2-5), we first use Proposition 2.9 and the Hölder inequality to observe that as n → ∞, for a.e. r ∈ (0, r 0 ), since η(r, ∇f · x) and η(r, f ) are finite a.e. as a consequence of assumptions (H2-4) and (H2-2) and {u n + u} n is bounded in H 1 (B r ) for every r ∈ (0, r 0 ). Furthermore, by the fact that f is bounded far from the origin and the compactness of the trace map from H 1 (B r ) to L 2 (∂B r ), it follows that for a.e. r ∈ (0, r 0 ). Hence, passing to the limit in R(r, u n ) we conclude the first part of the proof. Finally (3.12) follows by testing (2.14) with u n itself and passing to the limit arguing as above.

The Almgren type frequency function
Let u ∈ H 1 Γ (BR) be a non trivial solution to (1.6). For every r ∈ (0,R) we define and In the following lemma we compute the derivative of the function H. Proof. First we observe that Let φ ∈ C ∞ c (0,R). Since u, ∇u ∈ L 2 (BR), we obtain that where we set v(x) = φ(|x|). Thus we proved (4.3). Identity (4.4) follows from (4.3) and (3.12).
We now observe that the function H is strictly positive in a neighbourhood of 0. Proof. Assume by contradiction that there exists r 1 ∈ (0, r 0 ] such that H(r 1 ) = 0, so that the trace of u on ∂B r1 is null and hence u ∈ H 1 0 (B r1 \ Γ). Then, testing (1.6) with u, we obtain that Thus, from Lemma 2.1 and (4.6) it follows that which, together with Lemma 2.3, implies that u ≡ 0 in B r1 . From classical unique continuation principles for second order elliptic equations with locally bounded coefficients (see e.g. [31]), we can conclude that u = 0 a.e. in BR, a contradiction.
Let us now differentiate the function D and estimate from below its derivative. for a.e. r ∈ (0, r 0 ).
Lemma 4.6. Letting r 0 be as in Lemma 2.1 and N as in (4.9), there exists a positive constant C 2 > 0 such that N (r) ≤ C 2 (4.21) for all r ∈ (0, r 0 ).
Integrating over the interval (r, r σ ) and by continuity of H outside 0, we obtain (4.24) for some constant K 2 (σ) depending on σ.

The Blow-up Argument
In this section we develop a blow-up analysis for scaled solutions, with the aim of classifying their possible vanishing orders. The presence of the crack produces several additional difficulties with respect to the classical case, mainly relying in the persistence of the singularity even far from the origin, all along the edge. These difficulties are here overcome by means of estimates of boundary gradient integrals (Lemma 5.5) derived by some fine doubling properties, in the spirit of [19], where an analogous lack of regularity far from the origin was instead produced by many-particle and cylindrical potentials.
In the next lemma we prove a doubling type result.
It remains to prove part (i). From (5.29) and (5.28) it follows that H w (r) = r k0 . Therefore (5.27) and Lemma 4.1 applied to w imply that thus completing the proof.
In order to make more explicit the blow-up result proved above, we are going to describe the asymptotic behavior of H(r) as r → 0 + . Lemma 5.7. Let γ := lim r→0 + N (r) ∈ R be as in Lemma 4.7. The limit lim r→0 + r −2γ H(r) exists and it is finite.

Straightening the domain
In order to detect the sharp vanishing order of the function H and to give a more explicit blow-up result, in this section we construct an auxiliary equivalent problem by a diffeomorphic deformation of the domain, inspired by [15], see also [2] and [29]. The purpose of such deformation is to straighten the crack; the advantage of working in a domain with a straight crack will then rely in the possibility of separating radial and angular coordinates in the Fourier expansion of solutions (see (6.30)). Let u ∈ H 1 (BR) be a weak solution to (1.6). Then where where w λ is defined in (5.1) andv λ (x) := v(λx).
Hence, by continuity of H, which yields (6.10) in view of (6.5). From (6.1) and a change of variable it also follows that |∇w λ (y) Jac Φ(Φ −1 (λy))| 2 |det Jac Φ −1 (λy)|dy for all λ ∈ (0,r). The above identity, together with (6.3)-(6.5) and the boundedness in H 1 (B 1 ) of {w λ } established in Lemma 5.2, implies estimate (6.11). Lemma 6.3. Let v = u • Φ be as in (6.6) and let k 0 and γ be as in Lemma 5.6 (i). Then, for every sequence λ n → 0 + , there exist a subsequence {λ n k } k∈N and an eigenfunction ψ of problem (1.7) associated with the eigenvalue µ k0 such that ψ L 2 (S N ) = 1, the convergence (5.15) holds and v(λ n k ·) Proof. From Lemma 5.6, there exist a subsequence λ n k and an eigenfunction ψ of problem (1.7) associated with the eigenvalue µ k0 such that ψ L 2 (S N ) = 1 and (5.15) holds. From (5.15) it follows that, up to passing to a further subsequence, w λn k ∂B1 converges to ψ in L 2 (S N ) and almost everywhere on S N , where w λ is defined in (5.1). From Lemma 6.2 it follows that {v λ / H(λ)} λ is bounded in H 1 (B 1 ) and hence, in view of (6.10), there existsψ ∈ L 2 (S N ) such that, up to a further subsequence, v(λ n k ·) →ψ strongly in L 2 (S N ) and almost everywhere on S N . (6.12) To conclude it is enough to show thatψ = ψ. To this aim we observe that, for every ϕ ∈ C ∞ c (S N ), from (6.6), (6.10), and a change of variable it follows that In view of (6.4) and (6.5) we have that, for all θ ∈ S N , so that, by the Dominated Convergence Theorem, the right hand side of (6.13) converges to S N ψ(θ)ϕ(θ) dS. On the other hand (6.12) implies that the left hand side of (6.13) converges to S Nψ (θ)ϕ(θ) dS. Therefore, passing to the limit in (6.13), we obtain that thus implying that ψ =ψ.