Global Lipschitz stability estimates for polygonal conductivity inclusions from boundary measurements
aa r X i v : . [ m a t h . A P ] F e b GLOBAL LIPSCHITZ STABILITY ESTIMATES FORPOLYGONAL CONDUCTIVITY INCLUSIONS FROMBOUNDARY MEASUREMENTS
ELENA BERETTA AND ELISA FRANCINI
Abstract.
We derive Lipschitz stability estimates for the Hausdorff distanceof polygonal conductivity inclusions in terms of the Dirichlet-to-Neumann map. Introduction
In this paper we establish Lipschitz stability estimates for a certain class ofdiscontinuous conductivities γ in terms of the Dirichlet-to-Neumann map.More precisely, we consider the following boundary value problem(1.1) (cid:26) div ((1 + ( k − χ P ) ∇ u ) = 0 in Ω ⊂ R ,u = φ on ∂ Ω , where φ ∈ H / ( ∂ Ω), P is a polygonal inclusion strictly contained in a planar,bounded domain Ω and k = 1 is a given, positive constant.Our goal is to determine the polygon P from the knowledge of the Dirichlet-to-Neumann map Λ γ : H / ( ∂ Ω) → H − / ( ∂ Ω)with Λ γ ( f ) := γ ∂u∂ν ∈ H − / ( ∂ Ω) . This class of conductivity inclusions is quite common in applications, like for ex-ample in geophysics exploration, where the medium (the earth) under inspectioncontains heterogeneities in the form of rough bounded subregions (for example sub-surface salt bodies) with different conductivity properties [18].Moreover, polygonal inclusions represent a class in which Lipschitz stable recon-struction from boundary data can be expected [7]. In fact, it is well known that thedetermination of an arbitrary (smooth) conductivity inclusion from the Dirichlet-to-Neumann map is exponentially ill-posed [12]. On the other hand, restrictingthe class of admissible inclusions to a compact subset of a finite dimensional spaceregularizes the inverse problem and allows to establish Lipschitz stability estimatesand stable reconstructions (see [4],[10], [1], [13]). In order to show our main resultwe follow a similar approach as the one in [7] and take advantage of a recent re-sult obtained by the authors in [9] where they prove Fr´echet differentiability of theDirichlet-to-Neumann map with respect to affine movements of vertices of polygonsand where they establish an explicit representation formula for the derivative.We would like to mention that our result relies on the knowledge of infinitely
Mathematics Subject Classification.
Key words and phrases. polygonal inclusions, conductivity equation, stability, inverseproblems. many measurements though one expects that finitely many measurements shouldbe enough to determine a polygonal inclusion. In fact, in [5] the authors show thatif the inclusion is a convex polyhedron, then one suitably assigned current at theboundary of the domain Ω and the corresponding measured boundary potential areenough to uniquely determine the inclusion (see also [17] for the unique determi-nation of an arbitrary polygon from two appropriately chosen pairs of boundarycurrents and potentials and also [14] where a convex polygon is uniquely deter-mined in the case of variable conductivities). Unfortunately in the aforementionedpapers, the choice of the current fields is quite special and the proof of uniquenessis not constructive. In fact, to our knowledge, no stability result for polygons fromfew boundary measurements has been derived except for the local stability resultobtained in [6]. On the other hand, in several applications, like the geophysicalone, many measurements are at disposal justifying the use of the full Dirichlet-to-Neumann map, [3].The paper is organized as follows: in Section 2 we state our main assumptions andthe main stability result. Section 3 is devoted to the proof of our main result andfinally, Section 4 is devoted to concluding remarks about the results and possibleextensions. 2.
Assumptions and main result
Let Ω ⊂ R be a bounded open set with diam (Ω) ≤ L . We denote either by x = ( x , x ) and by P a point in R . We assume that ∂ Ω is of Lipschitz class withconstants r and K > P in ∂ Ω there exists acoordinate system such that P = 0 andΩ ∩ ([ − r , r ] × [ − K r , K r ]) = { ( x , x ) : x ∈ [ − r , r ] , x > φ ( x ) } for a Lipschitz continuous function φ with Lipschitz norm smaller than K .We denote by dist ( · , · ) the euclidian distance between points or subsets in R .Later on we will also define the Haussdorff distance d H ( · , · ).Let A the set of closed, simply connected, simple polygons P ⊂
Ω such that:(2.1) P has at most N sides each one with length greater than d ;(2.2) ∂ P is of Lipschitz class with constants r and K , there exists a constant β ∈ (0 , π/
2] such that the angle β in each vertex of P satisfies the conditions(2.3) β ≤ β ≤ π − β and | β − π | ≥ β , and(2.4) dist ( P , ∂ Ω) ≥ d . Notice that we do not assume convexity of the polygon.Let us consider the problem (cid:26) div( γ ∇ u ) = 0 in Ω ,u = φ on ∂ Ω , where φ ∈ H / ( ∂ Ω) and(2.5) γ = 1 + ( k − χ P , LOBAL LIPSCHITZ STABILITY FOR POLYGONAL INCLUSIONS 3 for a given k > k = 1 and for P ∈ A . The constants k , r , K , L , d , N and β will be referred to as the a priori data .In the sequel we will introduce a number of constants depending only on the apriori data that we will always denote by C . The values of these constants mightdiffer from one line to the other.Let us consider the Dirichlet to Neumann mapΛ γ : H / ( ∂ Ω) → H − / ( ∂ Ω) φ → γ ∂u∂n | ∂ Ω , whose norm in the space of linear operators L ( H / ( ∂ Ω) , H − / ( ∂ Ω)) is defined by k Λ γ k ∗ = sup (cid:8) k Λ γ φ k H − / ( ∂ Ω) / k φ k H / ( ∂ Ω) : φ = 0 (cid:9) . Theorem 2.1.
Let P , P ∈ A and let γ = 1 + ( k − χ P and γ = 1 + ( k − χ P . There exist ε and C depending only on the a priori data such that, if k Λ γ − Λ γ k ∗ ≤ ε , then P and P have the same number N of vertices (cid:8) P j (cid:9) Nj =1 and (cid:8) P j (cid:9) Nj =1 re-spectively. Moreover, (2.6) d H (cid:0) ∂ P , ∂ P (cid:1) ≤ C k Λ γ − Λ γ k ∗ for every j = 1 , . . . , N. Remark 2.2.
Observe that our stability estimate is a global one. In fact, if k Λ γ − Λ γ k ∗ > ε , since the following trivial inequality holds d H (cid:0) ∂ P , ∂ P (cid:1) ≤ L, we have trivially (2.7) d H (cid:0) ∂ P , ∂ P (cid:1) ≤ L ≤ L k Λ − Λ k ∗ ε Therefore, in any case, by (2.6), (2.7) we obtain the global estimate d H (cid:0) ∂ P , ∂ P (cid:1) ≤ (cid:18) C + 2 Lε (cid:19) k Λ − Λ k ∗ . Proof of the main result
The proof of Theorem 2.1 follows partially the strategy used in [7] in the case ofthe Helmholtz equation.The first step of the proof is a rough stability estimate for k γ − γ k L (Ω) which isstated in Section 3.1 and which follows from a result by Clop, Faraco and Ruiz [11].Then, in section 3.2, we show a rough stability estimate for the Hausdorff distanceof the polygons. We also show that if k Λ γ − Λ γ k ∗ is small enough, then the twopolygons have the same number of vertices and that the distance from vertices of P and vertices of P is small. For this reason it is possible to define a coefficient γ t that goes smoothly from γ to γ and the corresponding Dirichlet to Neumannmap. We prove that the Dirichlet to Neumann map is differentiable (section 3.3),its derivative is continuous (section 3.4) and bounded from below (section 3.5).These results finally give the Lipschitz stability estimate of Theorem 2.1. E. BERETTA ET AL.
A logarithmic stability estimate.
As in [7], we can show that, thanks toLemma 2.2 in [16] there exists a constant Γ , depending only on the a priori data,such that, for i = 0 , k γ i k H s (Ω) ≤ Γ ∀ s ∈ (0 , / . Due to this regularity of the coefficients, we can apply Theorem 1.1 in [11] andobtain the following logarithmic stability estimate:
Proposition 3.1.
There exist α < / and C > , depending only on the a prioridata, such that (3.2) k γ − γ k L (Ω) ≤ C | log k Λ γ − Λ γ k ∗ | − α /C , if k Λ γ − Λ γ k ∗ < / . A logarithmic stability estimate on distance of vertices.
In this sectionwe want to show that, due to the assumptions on polygons in A , estimate (3.2)yields an estimate on the Hausdorff distance d H ( ∂ P , ∂ P ) and, as a consequence,on the distance of the vertices of the polygons.It is immediate to get from (3.2) that(3.3) (cid:12)(cid:12) P ∆ P (cid:12)(cid:12) ≤ C | k − | | log k Λ γ − Λ γ k ∗ | − α /C Now, we show that (3.3) implies an estimate on the Hausdorff distance of theboundaries of the polygons.Let us recall the definition of the Hausdorff distance between two sets A and B : d H ( A, B ) = max { sup x ∈ A inf y ∈ B dist ( x, y ) , sup y ∈ B inf x ∈ A dist ( y, x ) } The following result holds:
Lemma 3.2.
Given two polygons P and P in A , we have d H ( ∂ P , ∂ P ) ≤ C p |P ∆ P | where C depends only on the a priori data.Proof. Let d = d H ( ∂ P , ∂ P ). Assume d > x ∈ ∂ P such that dist ( x , ∂ P ) = d . Then, B d ( x ) ⊂ R \ ∂ P . There are two possibilities:(i) B d ( x ) ⊂ R \ P or(ii) B d ( x ) ⊂ P .In case (i), B d ( x ) ∩ P ⊂ P \ P . The definition of A implies that, if d ≤ d ,there is a constant C > (cid:12)(cid:12) B d ( x ) ∩ P (cid:12)(cid:12) ≥ d C . If d ≥ d we trivially have (cid:12)(cid:12) B d ( x ) ∩ P (cid:12)(cid:12) ≥ (cid:12)(cid:12) B d ( x ) ∩ P (cid:12)(cid:12) ≥ d C , hence, in any case, for f ( d ) = (cid:26) d /C if d < d d /C if d ≥ d we have f ( d ) ≤ (cid:12)(cid:12) B d ( x ) ∩ P (cid:12)(cid:12) ≤ (cid:12)(cid:12) P ∆ P (cid:12)(cid:12) . Now, if (cid:12)(cid:12) P ∆ P (cid:12)(cid:12) < d C , then f ( d ) = d C ≤ (cid:12)(cid:12) P ∆ P (cid:12)(cid:12) gives d ≤ C p |P ∆ P | . Onthe other hand, if (cid:12)(cid:12) P ∆ P (cid:12)(cid:12) ≥ d C we have d C ≤ L C ≤ L C (cid:12)(cid:12) P ∆ P (cid:12)(cid:12) d /C that gives d ≤ LCd p |P ∆ P | .In case (ii), B d ( x ) ⊂ P , hence B d ( x ) \ P ⊂ P \ P ⊂ P ∆ P . Proceeding as above we have f ( d ) ≤ (cid:12)(cid:12) B d ( x ) \ P (cid:12)(cid:12) ≤ (cid:12)(cid:12) P ∆ P (cid:12)(cid:12) and the same conclusion follows. (cid:3) Proposition 3.3.
Given the set of polygons A there exist δ and C depending onlyon the a priori data such that, if for some P , P ∈ A we have d H ( ∂ P , ∂ P ) ≤ δ , then P and P have the same number N of vertices { P i } Ni =1 and { P i } Ni =1 , respec-tively, that can be ordered in such a way that dist ( P i , P i ) ≤ Cd H ( ∂ P , ∂ P ) for every i = 1 , . . . , N. Proof.
Let us denote by δ = d H ( ∂ P , ∂ P ) . Assume P has N vertices and that P has M vertices. We now will show that forany vertex P i ∈ ∂ P there exists a vertex P j ∈ ∂ P such that dist ( P i , P j ) < Cδ .By assumption (2.1) this implies that N ≤ M . Interchanging the role of P and P we get that M ≤ N which implies that M = N .Let P be one of the vertices in ∂ P and let us consider the side l ′ of ∂ P that isclose to P . Let us set the coordinate system with origin in the midpoint of l ′ andlet ( ± l/ ,
0) be the endpoint of l ′ .By definition of the Hausdorff distance, P ∈ U δ = (cid:8) x ∈ R : dist ( x, l ′ ) ≤ δ (cid:9) .Now we want to show that, due to the assumptions on A , for sufficiently small δ there is a constant C such that the distance between P and one of the endpointsof l ′ is smaller than Cδ . The reason is that if P is too far from the endpoints,assumption (2.3) on P cannot be true.Let us choose δ small enough to have:(3.4) δ < K r (this guarantees that the δ -neighborhood of each side of P does not intersect the δ -neighborhood of a non adjacent side), and(3.5) δ < d sin β . E. BERETTA ET AL.
Notice that, by assumption (2.3) and by (3.4), the rectangle R = (cid:20) − l δ sin β , l − δ sin β (cid:21) × [ − δ, δ ]does not intersect the δ -neighborhood of any other side of P .Let us now show that P cannot be contained in a slightly smaller rectangle R ′ = (cid:20) − l λ, l − λ (cid:21) × [ − δ, δ ] , where λ = δ sin β .Let us assume by contradiction that P ∈ R ′ and consider the two sides of ∂ P with an endpoint at P . These sides have length greater than d , hence they intersect ∂B λ/ ( P ) in two points Q and Q in R (because λ/ < λ − δ sin β ).Since λ/ > δ the intersection ∂B λ/ ( P ) ∩ R is the union of two disjoint arcs.We estimate the angle of P at P in the two alternative cases:(i) Q and Q are on the same arc or(ii) Q and Q are on different arcs.In case (i), the angle at P is smaller than arcsin (cid:0) δλ (cid:1) (the angle is smaller thanarcsin (cid:16) δ − b ) λ (cid:17) +arcsin (cid:16) δ + b ) λ (cid:17) , where b is the y -coordinate of P , that is maximumfor b = ± δ ).In order for (2.3) to be true we should havearcsin (cid:18) δλ (cid:19) = arcsin (cid:18)
23 sin β (cid:19) ≤ β that is not possible for β ∈ (0 , π/ π at most by arcsin (cid:0) δλ (cid:1) , which is again toosmall for (2.3) to be true.Since neither of cases (1) and (2) can be true, it is not possibile that P ∈ R ′ ,hence, P ∈ U δ \ R ′ which implies that there is one of the endpoints of l ′ , let us callit P ′ such that dist ( P, P ′ ) ≤ δ s β . (cid:3) Proposition 3.4.
Under the same assumptions of Theorem 2.1, there exist positiveconstants ε , α and C > , depending only on the a priori data, such that, if ε := k Λ γ − Λ γ k ∗ < ε , then P and P have the same number N of vertices (cid:8) P j (cid:9) Nj =1 and (cid:8) P j (cid:9) Nj =1 re-spectively. Moreover, the vertices can be order so that (3.6) dist (cid:0) P j , P j (cid:1) ≤ ω ( ε ) for every j = 1 , . . . , N, where ω ( ε ) = C | log ε | − α /C .Proof. It follows by the combination of Proposition 3.1, Lemma 3.2 and Proposition3.3. (cid:3)
LOBAL LIPSCHITZ STABILITY FOR POLYGONAL INCLUSIONS 7
Definition and differentiability of the function F . Let us denote by { P ji } Ni =1 the vertices of polygon P j for j = 0 , dist ( P i , P i ) ≤ ω ( ε ) for i = 1 , . . . , N , for ω ( ε ) as in Proposition 3.4 and the segment P ij P ij +1 is a side of P i for i = 0 , j = 1 , . . . , N .Let us consider a deformation from P to P : for t ∈ [0 ,
1] let P ti = P i + tv i , where v i = P i − P i , for i = 1 , . . . , N and denote by P t the polygon with vertices P tj and sides P tj P tj +1 .Let γ t = 1 + ( k − χ P t and let Λ γ t be the corresponding DtoN map.As we proved in [9, Corollary 4.5] the DtoN map Λ γ t is differentiable with respectto t .The function F ( t, φ, ψ ) = < Λ γ t ( φ ) , ψ >, for φ, ψ ∈ H / ( ∂ Ω), is a differentiable function from [0 ,
1] to R and we can writeexplicitly its derivative.Let u t , v t ∈ H (Ω) be the solutions to (cid:26) div( γ t ∇ u t ) = 0 in Ω ,u t = φ on ∂ Ω , and (cid:26) div( γ t ∇ v t ) = 0 in Ω ,v t = ψ on ∂ Ω , and denote by u et and v et their the restrictions to Ω \ P t (and by u it and v it theirrestrictions to P t ).Let us fix an orthonormal system ( τ t , n t ) in such a way that n t represents al-most everywhere the outward unit normal to ∂ P t and the tangent unit vector τ t isoriented counterclockwise. Denote by M t a 2 × ∂ P t with eigenvalues 1 and 1 /k and corresponding eigenvectors τ t and n t .Let Φ vt be a map defined on ∂ P t , affine on each side of the polygon and suchthat Φ vt ( P ti ) = v i for i = 1 , . . . , N. Then, it was proved in [9, Corollary 2.2] that, for all t ∈ [0 , ddt F ( t, φ, ψ ) = ( k − Z ∂ P t M t ∇ u et ∇ v et (Φ vt · n t ) . Continuity at zero of the derivative of F .Lemma 3.5. There exist constants C and β , depending only on the a priori data,such that (3.7) (cid:12)(cid:12)(cid:12)(cid:12) ddt F ( t, φ, ψ ) − ddt F ( t, φ, ψ ) | t =0 (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k φ k H / ( ∂ Ω) k ψ k H / ( ∂ Ω) | v | β t β . Proof.
This result corresponds to Lemma 4.4 in [9]. The dependence on | v | isobtained by refining estimate (3.5) in [9, Proposition 3.4] to get k u t − u k H (Ω) ≤ C k φ k H / ( ∂ Ω) (cid:12)(cid:12) P t ∆ P (cid:12)(cid:12) θ ≤ C k φ k H / ( ∂ Ω) | v | θ t θ , and by noticing that | Φ vt | ≤ C | v | . (cid:3) E. BERETTA ET AL.
Bound from below for the derivative of F . In this section we want toobtain a bound from below for the derivative of F at t = 0. Proposition 3.6.
There exist a constant m > , depending only on the a prioridata, and a pair of functions ˜ φ and ˜ ψ in H / ( ∂ Ω) such that (3.8) (cid:12)(cid:12)(cid:12)(cid:12) ddt F ( t, ˜ φ, ˜ ψ ) | t =0 (cid:12)(cid:12)(cid:12)(cid:12) ≥ m | v |k ˜ φ k H / ( ∂ Ω) k ˜ ψ k H / ( ∂ Ω) . Proof.
Let us first normalize the length of vector v and introduce H ( φ, ψ ) = Z ∂ P M o ∇ u e ∇ v e ˜Φ v · n , where ˜Φ v = Φ v/ | v | . By linearity, we have that ddt F ( t, φ, ψ ) | t =0 = | v | H ( φ, ψ ).Let m = k H k ∗ = sup (cid:26) H ( φ,ψ ) k φ k H / ∂ Ω) k ψ k H / ∂ Ω) : φ, ψ = 0 (cid:27) be the operator normof H , so that(3.9) | H ( φ, ψ ) | ≤ m k φ k H / ( ∂ Ω) k ψ k H / ( ∂ Ω) for every φ, ψ ∈ H / ( ∂ Ω) . Let Σ be an open non empty subset of ∂ Ω and let us extend Ω to a open domainΩ = Ω ∪ D that has Lipschitz boundary with constants r / K and suchthat Σ is contained in Ω (see [2] for a detailed construction). Let us extend γ by1 in D (and still denote it by γ ).We denote by G ( x, y ) the Green function corresponding to the operator div( γ ∇· )and to the domain Ω . The Green function G ( x, y ) behaves like the fundamentalsolution of the Laplace equation Γ( x, y ) for points that are far from the polygon.For points close to the sides of the polygon but far from its vertices, the asymp-totic behaviour of the Green function has been described in [2, Theorem 4.2] or [8,Proposition 3.4]: Let y r = Q + rn ( y ), where Q is a point on ∂ P whose distancefrom the vertices of the polygons is greater than r / n ( y ) is the unit outernormal to ∂ P . Then, for small r ,(3.10) (cid:13)(cid:13)(cid:13)(cid:13) G ( · , y r ) − k + 1 Γ( · , y r ) (cid:13)(cid:13)(cid:13)(cid:13) H (Ω ) ≤ C, where C depends only on the a priori data.Let us take u = G ( · , y ) and v = G ( · , z ) for y, z ∈ K , where K is a compactsubset of D such that dist ( K, ∂ Ω) ≥ r / K contains a ball of radius r / u and v are both solutions to the equation div( γ ∇· ) = 0 in Ω.Define the function S ( y, z ) = Z ∂ P M ∇ G ( · , y ) ∇ G ( · , z )( ˜Φ v · n )that, for fixed z , solves div( γ ∇ S ( · , z )) = 0 in Ω \ P and, for fixed y it solvesdiv( γ ∇ S ( y, · )) = 0 in Ω \ P .For y, z ∈ K , S ( y, z ) = H ( u , v ), hence, by (3.9)(3.11) | S ( y, z ) | ≤ C m r for y, z ∈ K, where C depend on the a priori data. LOBAL LIPSCHITZ STABILITY FOR POLYGONAL INCLUSIONS 9
Moreover, by (3.10), there exist ρ and E depending only on the a priori datasuch that(3.12) | S ( y, z ) | ≤ E ( d y d z ) − / for every y, z ∈ Ω \ (cid:0) P ∪ Ni =1 B ρ ( P i ) (cid:1) , where d y = dist ( y, P ).Since S is small for y, z ∈ K (see (3.11) and consider m small), bounded for y, z ∈ Ω \ P far from the vertices of the polygon, and since it is harmonic in Ω \ P ,we can use a three balls inequality on a chain of balls in order to get a smallnessestimate close to the sides of the polygon.To be more specific, let l i be a side of P with endpoints P i and P i +1 . Let Q i be the midpoint of l i and let y r = Q i + rn i where n i is the unit outer normal to ∂ P at Q i and r ∈ (0 , K r ). Lemma 3.7.
There exist constants
C > , β , and r < r /C depending only onthe a priori data, such that, for r < r (3.13) | S ( y r , y r ) | ≤ C (cid:18) ε ε + E (cid:19) βτ r ( ε + E ) r − , where ε = m C r − and τ r = − r/r ) .Proof. For the proof of Lemma 3.7 see [8, Proposition 4.3] where the estimate of τ r is slightly more accurate. (cid:3) Now, we want to estimate | S ( y r , y r ) | from below. In order to accomplish this,let us take ρ = min { d / , r / } and write (cid:12)(cid:12)(cid:12)(cid:12) S ( y r , y r ) k − (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂ P ∩ B ρ ( Q i ) M ∇ G ( · , y r ) ∇ G ( · , y r )( ˜Φ v · n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.14) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂ P \ B ρ ( Q i ) M ∇ G ( · , y r ) ∇ G ( · , y r )( ˜Φ v · n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.15) := I − I . (3.16)The behaviour of the Green function (see [2]) gives immediately that, for r < ρ/ I ≤ C , for some C depending only on the a priori data.In order to estimate I , we add and subtract Γ( · , y r ) to G ( · , y r ), then by Younginequality, (3.10), and by the properties of M , we get(3.18) I ≥ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂ P ∩ B ρ ( Q i ) |∇ Γ( · , y r ) | ( ˜Φ v · n i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − C , where C and C depend only on the a priori data.By definition of ˜Φ v we have (cid:12)(cid:12)(cid:12) ˜Φ v ( x ) − ˜Φ v ( Q i ) (cid:12)(cid:12)(cid:12) ≤ C | x − Q i | , so, by adding and subtracting Φ v ( Q i ) into the integral of (3.18), we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂ P ∩ B ρ ( Q i ) |∇ Γ( · , y r ) | ( ˜Φ v · n i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ α Z ∂ P ∩ B ρ ( Q i ) |∇ Γ( · , y r ) | − C Z ∂ P \ B ρ ( Q i ) |∇ Γ( · , y r ) | | x − Q i | , where α = | ˜Φ v ( Q i ) · n i | . By straightforward calculations one can see that(3.19) Z ∂ P ∩ B ρ ( Q i ) |∇ Γ( · , y r ) | ≥ C r and(3.20) Z ∂ P \ B ρ ( Q i ) |∇ Γ( · , y r ) | | x − Q i | ≤ C | log( ρ/r ) | . By putting together (3.14), (3.17), (3.19) and (3.20), we get(3.21) | S ( y r , y r ) | ≥ C αr − C | log( ρ/r ) | − C . By comparing (3.13) and (3.21) we get(3.22) C α ≤ C (cid:18) ε ε + E (cid:19) βτ r ( ε + E ) + C r | log( ρ/r ) | + C r. By an easy calculation one can see that βτ r ≥ r /C , hence(3.23) C α ≤ C (cid:18) ε ε + E (cid:19) r /C ( ε + E ) + C √ r. By choosing r = (cid:12)(cid:12)(cid:12) log (cid:16) ε ε + E (cid:17)(cid:12)(cid:12)(cid:12) − / and recalling that ε = C m r − we have | ˜Φ v ( Q i ) · n i | = α ≤ ω ( m ) , where ω ( t ) is an increasing concave function such that lim t → + ω ( t ) = 0.This estimate can also be obtained for ˜Φ v ( y ) · n i for every y ∈ B ρ ( Q i ) ∩ l i . Since˜Φ v is linear on the bounded side l i , | ˜Φ v ( y ) · n i | ≤ ω ( m ) for every y ∈ l i , and, in particular(3.24) (cid:12)(cid:12)(cid:12)(cid:12) v i | v | · n i (cid:12)(cid:12)(cid:12)(cid:12) = | ˜Φ v ( P i ) · n i | ≤ ω ( m )Repeating the same argument on the adjacent side, l i +1 , containing P i we obtainin particular that(3.25) (cid:12)(cid:12)(cid:12)(cid:12) v i | v | · n i +10 (cid:12)(cid:12)(cid:12)(cid:12) = | ˜Φ v ( P i ) · n i +10 | ≤ ω ( m )Then, there exists a constant C > (cid:12)(cid:12)(cid:12)(cid:12) v i | v | (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cω ( m ) LOBAL LIPSCHITZ STABILITY FOR POLYGONAL INCLUSIONS 11 and since one can apply the same procedure on each side of the polygon we have (cid:12)(cid:12)(cid:12)(cid:12) v i | v | (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cω ( m ) for i = 1 , . . . , N that yields 1 ≤ N Cω ( m ) ⇒ m ≥ ω − (1 /CN ) . By definition of the operator norm of H , there exist ˜ φ and ˜ ψ in H / ( ∂ Ω) such that | H ( ˜ φ, ˜ ψ ) | ≥ m k ˜ φ k H / ( ∂ Ω) k ˜ ψ k H / ( ∂ Ω) and (3.8) is true for m = ω − (1 /CN )2 . (cid:3) Remark 3.8.
Note that the lower bound for the derivative of F in Proposition 3.6holds for functions ˜ φ and ˜ ψ with compact support on an open portion of ∂ Ω . Lipschitz stability estimate.
In this section we conclude the proof of The-orem 2.1. Let ˜ φ and ˜ ψ the functions the satisfy (3.8) in Proposition 3.6.By (3.8) and by (3.7) we have (cid:12)(cid:12)(cid:12) < (Λ γ − Λ γ ) ( ˜ φ ) , ˜ ψ > (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) F (1 , ˜ φ, ˜ ψ ) − F (0 , ˜ φ, ˜ ψ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ddt F ( t, ˜ φ, ˜ ψ ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≥ ddt F ( t, ˜ φ, ˜ ψ ) | t =0 − Z (cid:12)(cid:12)(cid:12)(cid:12) ddt F ( t, ˜ φ, ˜ ψ ) − ddt F ( t, ˜ φ, ˜ ψ ) | t =0 (cid:12)(cid:12)(cid:12)(cid:12) dt ≥ (cid:0) m − C | v | β (cid:1) | v |k ˜ φ k H / ( ∂ Ω) k ˜ ψ k H / ( ∂ Ω) , that implies(3.26) ε = k Λ γ − Λ γ k ∗ ≥ (cid:0) m − C | v | β (cid:1) | v | . From (3.6), since | v | ≤ N max j dist ( P j , P j ) it follows that there exists ε > k Λ γ − Λ γ k ∗ ≤ ε , then (cid:0) m − C | v | β (cid:1) ≥ m / | v | ≤ m k Λ γ − Λ γ k ∗ . Finally, since d H (cid:0) ∂ P , ∂ P (cid:1) ≤ C | v | the claim follows. (cid:3) Finally, as a byproduct of Theorem 2.1 and of Proposition 3.3 we have thefollowing
Corollary 3.9.
Let P , P ∈ A and let γ = 1 + ( k − χ P and γ = 1 + ( k − χ P . There exist ε and C depending only on the a priori data such that, if k Λ γ − Λ γ k ∗ ≤ ε , then P and P have the same number N of vertices (cid:8) P j (cid:9) Nj =1 and (cid:8) P j (cid:9) Nj =1 re-spectively. Moreover, the vertices can be ordered so that (3.27) dist (cid:0) P j , P j (cid:1) ≤ C k Λ γ − Λ γ k ∗ for every j = 1 , . . . , N. Final remarks and extensions
We have derived Lipschitz stability estimates for polygonal conductivity inclu-sions in terms of the Dirichlet-to-Neumann map using differentiability propertiesof the Dirichlet-to-Neuman map.The result extends also to the case where finitely many conductivity polygonalinclusions are contained in the domain Ω assuming that they are at controlled dis-tance one from the other and from the boundary of Ω.We expect that the same result holds also when having at disposal local data. Infact, as we observed at the end of Proposition 3.6, the lower bound for the deriv-ative of F is obtained using solutions with compact support in a open subset of ∂ Ω and a rough stability estimate of the Hausdorff distance of polygons in termsof the local Dirichlet-to-Neumann map could be easily derived following the ideascontained in [15].Finally, it is relevant for the geophysical application we have in mind to extend theresults of stability and reconstruction to the 3-D setting possibly considering aninhomogeneous and/or anisotropic medium. This case is not at all straightforwardsince differentiability properties of the Dirichlet-to-Neumann map in this case arenot known.
Acknowledgment
The paper was partially supported by GNAMPA - INdAM.
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Dipartimento di Matematica “Brioschi”, Politecnico di Milano & New York Univer-sity Abu Dhabi
E-mail address : [email protected] Dipartimento di Matematica e Informatica “U. Dini”, Universit`a di Firenze
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