On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates

We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter $\varepsilon>0$. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in $\varepsilon$. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due the particular structure of the regularized problems, classical techniques \`a la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in $\varepsilon$ in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.


Introduction and setting of the problem
Let us consider a bounded Lipschitz domain Ω ⊂ R , > 1, the boundary Ω of which is partitioned into two sets Γ andΓ. More precisely, Γ andΓ are non empty open sets for the topology induced on Ω from the topology on R , Ω = Γ ∪Γ and Γ ∩Γ = ∅ (see Fig. 1). The Cauchy problem we are interested in consists, for some data ( 0 , 1 ) ∈ 1/2 (Γ) × −1/2 (Γ), in finding ∈ 1 (Ω) such that  noise. Due to Holmgren's theorem, the Cauchy problem (1.1) has at most one solution. However it is ill-posed in the sense of Hadamard: existence may not hold for some data ( 0 , 1 ), as for example shown in [3]. A possibility to regularize problem (1.1) is to use the quasi-reversibility method, which goes back to [31] and was revisited in [27]. The original idea was to replace an ill-posed Boundary Value Problem such as (1.1) by a family, depending on a small parameter , of well-posed fourth-order BVPs. Much later, the first author introduced the notion of mixed formulation of quasi-reversibility for the Cauchy problem of the Laplace equation [4]. This notion was extended to general abstract linear ill-posed problems in [7]. The idea is to replace the ill-posed second-order BVP by a family, again depending on a small parameter , of second-order systems of two coupled BVPs: the advantage is that the order of the regularized problem is the same as the original one, which is interesting when it comes to the numerical resolution. The price to pay is the introduction of a second unknown function in addition to the principal unknown . Such mixed formulation of quasi-reversibility is the following: for > 0, find ( , ) ∈ 0 ×˜0 such that for all ( , ) ∈ 0 ×˜0, where 0 = { ∈ 1 (Ω), | Γ = 0 }, 0 = { ∈ 1 (Ω), | Γ = 0} and˜0 = { ∈ 1 (Ω), |Γ = 0}. In (1.2), the brackets stand for the duality pairing between −1/2 (Γ) and˜1 /2 (Γ). Here˜1 /2 (Γ) is the subspace formed by the functions in 1/2 (Γ) which, once extended by 0 on Ω, remain in 1/2 ( Ω). We observe that in view of Poincaré inequality, the standard norm of 1 (Ω) in the spaces 0 and˜0 is equivalent to the semi-norm ‖ · ‖ defined by ‖ · ‖ 2 = ∫︀ Ω |∇ · | 2 d . Let us denote (·, ·) the corresponding scalar product. We remark that the weak formulation (1.2) is equivalent to the strong problem where we observe that the two unknowns and are harmonic functions which are coupled at the boundary Ω. We have the following theorem.
If in addition we assume that ( 0 , 1 ) is such that problem (1.1) has a (unique) solution (the data are said to be compatible), then there exists a constant which depends only on the geometry such that and lim →0 ‖ − ‖ 1 (Ω) = 0.
To prove such theorem, we need the following lemma, which establishes an equivalent weak formulation to problem (1.1) and which is proved in [7]. Proof of Theorem 1.1. Let us begin with the first part of the theorem. There exists a continuous lifting operator 0 ↦ → from 1/2 (Γ) to 1 (Ω) such that | Γ = 0 . Let us defineˆ= − ∈ 0 . By replacing in (1.2), we obtain that (ˆ, ) ∈ 0 ×˜0 satisfies, for all ( , ) ∈ 0 ×˜0, the system The Cauchy-Schwarz inequality implies The equivalence of the norm ‖ · ‖ and the standard 1 (Ω) norm in spaces 0 and˜0, the continuity of the trace operator and the continuity of the lifting operator 0 ↦ → yield Using the Young's inequality to deal with the right hand side of the above inequality, the result follows. Let us prove the second part of the theorem. In the case when the Cauchy data ( 0 , 1 ) is associated with the solution , then satisfies the weak formulation (1.4). By subtracting (1.4) to the second equation of (1.2), we obtain that for all ∈˜0, ∫︁ (1.5) Remark 1.3. Let us mention that another type of mixed formulation of quasi-reversibility was introduced in [20], in which the additional unknown lies in div (Ω) instead of 1 (Ω). In addition, a notion of iterative formulation of quasi-reversibility was introduced and analyzed in [19]. We believe that the quasi-reversibility formulation (1.2) is the easiest one to handle to establish regularity results of the weak solutions.
The estimates of Theorem 1.1 involve 1 (Ω) norms of the regularized solution ( , ) in the case of a Lipschitz domain Ω and for the natural regularity of the Cauchy data ( 0 , 1 ), that is 1/2 (Γ)× −1/2 (Γ). These estimates were derived in two different cases: the data ( 0 , 1 ) are compatible or not. The main concern of this paper is to analyze, when the domain Ω and the Cauchy data ( 0 , 1 ) are more regular than Lipschitz and 1/2 (Γ) × −1/2 (Γ), respectively, the additional regularity of the solution ( , ), whether the data ( 0 , 1 ) are compatible or not. We also want to obtain estimates in the corresponding norms. In order to simplify the analysis, the additional regularity of the data ( 0 , 1 ) is formulated in the following way: we assume that ( 0 , 1 ) is such that there exists a function in 2 (Ω) with ( | Γ , | Γ ) = ( 0 , 1 ) and that we can define a continuous lifting operator ( 0 , 1 ) ↦ → . Denoting = ∆ ∈ 2 (Ω) and considering the new translated unknown − → , the initial Cauchy problem (1.1) can be transformed into a homogeneous one (however still ill-posed): for ∈ 2 (Ω), (1.6) We emphasize that this regularity assumption made on the data is not an assumption of regularity of the solution . It is simple to construct smooth data in the sense above such that the corresponding is only in 1 (Ω) and not in 2 (Ω). The mixed formulation of quasi-reversibility for problem (1.6) takes the following form: for > 0, find ( , ) ∈ 0 ×˜0 such that for all ( , ) ∈ 0 ×˜0, (1.7) Note that the strong equations corresponding to problem (1.7) are (1.8) The analog of Theorem 1.1, the proof of which is skipped, is the following.
Theorem 1.4. For all ∈ 2 (Ω) and > 0, the problem (1.7) has a unique solution ( , ) ∈ 0 ×˜0. There exists a constant which depends only on the geometry such that If in addition we assume that is such that problem (1.6) has a (unique) solution , then there exists a constant which depends only on the geometry such that The objective is now to study the regularity of the solution ( , ) to problem (1.7) and to complete the statements (1.9) and (1.10) of Theorem 1.4 by giving estimates in stronger norms. One objective, as will be seen in Section 6, is the following. In practice, one has to solve problem (1.7) in the presence of two approximations. Firstly, the data is altered by some noise of amplitude . Secondly, the problem (1.7) is discretized, for instance with the help of a Finite Element Method (FEM) based on a mesh of size ℎ. It is then desirable to estimate the error between the approximated solution and the exact solution as a function of , and ℎ. Such error estimate for the 1 (Ω) norm needs the solution to be in a Sobolev space (Ω), with > 1. It could be noted that in a recent contribution [13] (see also [9][10][11][12]), a discretized method was proposed in order to regularize the Cauchy problem (1.1) in the presence of noisy data without introducing a regularized problem such as (1.7) at the continuous level. In some sense, the method of [13] relies on a single asymptotic parameter, that is ℎ, instead of two in our method, that is and ℎ. However, we believe that from the theoretical point of view, the regularity of quasi-reversibility solutions is an interesting problem in itself. To our best knowledge, it has never been investigated up to now. The difficulty stems from the fact that we analyze the regularity of a problem involving a small parameter which degenerates when tends to 0. There are other contributions (see e.g. [15,16,18,26,35,36]) where regularity results or asymptotic expansions are obtained in situations where the limit problem has a different nature from the regularized one. For example in [18], the authors study a mixed Neumann-Robin problem where the small parameter is the inverse of the Robin coefficient. But while both the perturbed problem and the limit one are well-posed in [18], only the perturbed problem is well-posed in our case, the limit problem being ill-posed (in any framework). Our contribution is original in this sense. In the present work, we study the regularity of the solution of the regularized problem as tends to zero. We emphasize that computing an asymptotic expansion of the solution with respect to and proving error estimates (e.g. as in [24,33]) remains an open problem, the reason being that, due to the ill-posedness of the limit problem, no result of stability can be easily established.
Our paper is organized as follows. First we consider the simple case of a smooth domain in Section 2, where classical regularity results (see e.g. [8]) can be used. The case of the polygonal domain is introduced in Section 3, where we also analyze the regularity of the quasi-reversibility solution in corners delimited by two edges of Γ or two edges ofΓ. In this case, the regularity of functions and can be analyzed separately with the help of the classical regularity results of [22] in a polygon for the Laplace equation with Dirichlet or Neumann boundary conditions. In Section 5 we consider the more difficult case of a corner of mixed type, that is delimited by one edge of Γ and one edge ofΓ. This analysis relies on the Kondratiev approach [28], which is based on some properties of weighted Sobolev spaces which are recalled in Section 4. Section 6 is dedicated to the application of our regularity results to derive some error estimate between the exact solution and the quasi-reversibility solution in the presence of two perturbations: noisy data and discretization with the help of a FEM. We also try to illustrate our error estimate by presenting a numerical example. Two appendices containing technical results, which are used in Section 5, complete the paper. The main results of this article are Theorem 2.2 (uniform regularity estimates in smooth domains), Theorem 3.1 (uniform regularity estimates in 2D polygonal domains) and the final approximation analysis of Section 6.
Proposition 2.1. For ∈ 2 (Ω), the solution ( , ) ∈ 0 ×˜0 to the problem (1.7) is such that for all ∈ C ∞ 0 (Ω), and belong to 2 (Ω) and there exists a constant > 0 which depends only on the geometry such that ∀ ∈ (0, 1], If in addition is such that problem (1.6) has a solution , then where the norm ‖ · ‖ 1 (Δ,Ω) is defined by Proof. From the first equation of (1.8), we have that Clearly ∈ 2 (R ), which by using the Fourier transform implies that From (1.9) we obtain that If in addition is such that problem (1.6) has a (unique) solution , from (1.10) we obtain The estimates of are obtained following the same lines.
Let us now establish a global regularity estimate (up to the boundary) in the restricted case when Γ ∩Γ = ∅ (see Fig. 1

right).
Theorem 2.2. For ∈ 2 (Ω), the solution ( , ) ∈ 0 ×˜0 to the problem (1.7) is such that and belong to 2 (Ω) and there exists a constant > 0 which depends only on the geometry such that If in addition is such that problem (1.6) has a solution , then Proof. Given Γ ∩Γ = ∅, we may find two infinitely smooth functions and˜such that ( ,˜) = (1, 0) in a vicinity of Γ and ( ,˜) = (0, 1) in a vicinity ofΓ. We have from the first equation of (1.8), Since = 0 on Γ, from a standard regularity result for the Poisson equation with Dirichlet boundary condition we obtain and from (1.9) we have From a standard continuity result for the normal derivative and using that − = 0 on Γ, we obtain From the second equation of (1.8) we have Combining the two previous estimates with the fact that = 0 onΓ implies the regularity estimate √ ‖ ‖ 2 (Ω) ≤ ‖ ‖ 2 (Ω) .

Main result
From now on, Ω is a polygonal domain in dimension 2. Our motivation is indeed to obtain error estimates in the context of the discretization with the help of a classical FEM: due to the meshing procedure in two dimensions, in practice the computational domain is often a polygon. We use the same notations as in [22] to describe the geometry of such a polygon. Let us assume that Ω is the union of segments Γ , = 1, . . . , , where is an integer. Let us denote the vertex such that = Γ ∩Γ +1 , the angle between Γ and Γ +1 from the interior of Ω, the unit tangent oriented in the counter-clockwise sense and the outward normal to Ω. We assume that Γ andΓ are formed by a finite number of edges, namely and˜, respectively, with +˜= . Let us denote ℋ(Γ) the subset of functions ( 0 , 1 ) ∈ 2 (Γ) × 2 (Γ) such that ( , ) := ( 0 | Γ , 1 | Γ ) ∈ 3/2 (Γ ) × 1/2 (Γ ), = 1, . . . , , with the following compatibility conditions at : and the equivalence ≡ +1 at means that for small > 0 where ( ) denotes the point of Ω which, for small enough | | (say | | ≤ ), is at distance (counted algebraically) of along Ω. More precisely, ( ) ∈ Γ if < 0 and ( ) ∈ Γ +1 if > 0. It is proved in [22], that for ( 0 , 1 ) ∈ ℋ, there exists a function ∈ 2 (Ω) such that for each = 1, . . . , , ( | Γ , | Γ ) = ( , ) and even a continuous lifting ( 0 , 1 ) ↦ → from ℋ to 2 (Ω). We are hence again in the framework of Section 1, where the problem to solve is (1.6).
Clearly, the interior estimates given by Proposition 2.1 are true in the polygonal domain since they are independent of the regularity of the boundary. Let us now analyze the regularity up to the boundary. As done in [22], the estimates are obtained by using a partition of unity, which enables us to localize our analysis in three different types of corners (see Fig. 2): -regularity at a corner delimited by two edges which belong to Γ, called a corner of type Γ, -regularity at a corner delimited by two edges which belong toΓ, called a corner of typeΓ, -regularity at a corner delimited by one edge which belongs to Γ and one edge which belongs toΓ, called a corner of mixed type.
Let us denote by the set of such that is either a vertex of type Γ or a vertex of typeΓ and the set of such that is a corner of mixed type. We wish to prove the following theorem, which is obtained by gathering Propositions 2.1, 3.5, 3.6 and 5.12 hereafter.
}. For ∈ 2 (Ω) and > 0, the solution ( , ) ∈ 0 ×˜0 to the problem (1.7) is such that and belong to (Ω) and there exists a constant > 0 which depends only on the geometry such that If in addition we assume that is such that problem (1.6) has a (unique) solution , then

Regularity at a corner of type Γ
The regularity of solutions and near a corner delimited by two edges which belong to Γ can be analyzed separately. They will be obtained by directly applying the results of [22] for Dirichlet and Neumann Laplacian problems. Let us consider the vertex of a corner delimited by two edges Γ and Γ +1 which belong to Γ. Let us denote ( , ) the local polar coordinates with respect to the point and ∈ C ∞ (Ω) a radial function (depending only on ) such that = 1 for ≤ and = 0 for ≥ . We assume that is chosen such that = 0 in a vicinity of all edges Γ except for = or = + 1. In order to simplify notations, we skip the reference to index , denoting in particular = , Γ = Γ 0 and Γ +1 = Γ . Let us introduce the finite cone = Ω ∩ ( , ). The two following lemmata are proved in [22].
has a unique solution and there exists a unique constant ∈ R and a unique function ∈ 2 ( ) such that Moreover, there exists a constant > 0 such that In addition, if ≤ then = 0.
has a unique solution and there exists a unique constant ∈ R and a unique function ∈ 2 ( ) such that Moreover, there exists a constant > 0 such that (3.3) holds. In addition, if ≤ then = 0.
Proposition 3.5. Assume that is the vertex of a corner of type Γ. Let us consider < 1 + / if > and = 2 otherwise. For ∈ 2 (Ω), the solution ( , ) ∈ 0 ×˜0 to the problem (1.7) is such that and belong to (Ω) and there exists a constant > 0 which depends only on the geometry such that If in addition is such that problem (1.6) has a solution , then By using Lemma 3.3, we have that there exists a unique constant ∈ R and a unique function ∈ 2 ( ) such that and there exists a constant > 0 such that From (3.5), we deduce that we have From Theorem 1.4.5.3 of [21], the function ( , ) ↦ → ( ) / sin( / ) belongs to (Ω) for any < 1 + / . We conclude from (1.9) that there exists a constant > 0 which depends only on the geometry such that, for We remark from (1.8) that the function = − satisfies −∆ = in Ω and = 0 on Γ, which implies that satisfies problem (3.4) with By using Lemma 3.4, we have that there exists a unique constant ∈ R and a unique function and there exists a constant > 0 such that We infer that And we conclude from (1.9) that there exists a constant > 0 which depends only on the geometry such that so that = − satisfies the same estimate. The case when is such that there is a solution to (1.6) follows the same lines: it suffices to use estimate (1.10) instead of (1.9).

Regularity at a corner of typeΓ
We reuse the notations introduced in the last section.
Proposition 3.6. Assume that is the vertex of a corner of typeΓ. Let us consider < 1 + / if > and = 2 otherwise. For ∈ 2 (Ω), the solution ( , ) ∈ 0 ×˜0 to the problem (1.7) is such that and belong to (Ω) and there exists a constant > 0 which depends only on the geometry such that If in addition we assume that is such that problem (1.6) has a (unique) solution , then By using Lemma 3.3, we have that there exists a unique constant ∈ R and a unique function and there exists a constant > 0 such that We deduce the estimate And we conclude from (1.9) that there exists a constant > 0 which depends only on the geometry such that, We remark from (1.8) that the function = + satisfies −∆ = 0 in Ω and = 0 onΓ, which implies that By using Lemma 3.4, we have that there exists a unique constant ∈ R and a unique function and there exists a constant > 0 such that We infer that And we conclude from (1.9) that there is a constant > 0 which depends only on Ω such that so that = − satisfies the same estimate. The case when is such that there is a solution to (1.6) is similar.
Remark 3.7. We emphasize that the small parameter plays a different role in Proposition 3.5 and in Proposition 3.6. In Proposition 3.5, the exponent in is the same before and before because the corner is inside Γ and Γ is the support of the Cauchy data. In Proposition 3.6, the exponent in before is one more than the one before because the corner is insideΓ and data onΓ are unknown.
It remains to analyze the regularity of functions and at corners of mixed type and to derive corresponding estimates. As we will see, this is a much more difficult task. The main reason is that we do not know whether or not the eigenvectors of a certain symbol L defined on (0, ) (see (5.4)) form a Hilbert basis of 2 (0, )× 2 (0, ). To bypass this difficulty, we will apply the Kondratiev approach of the seminal article [28] (see also [29,30,32,34] for more recent presentations). We will follow strictly the methodology proposed in these works. However, we emphasize that in our study we have to keep track of the dependence in in all the estimates. This is the reason why we present the procedure in details. Let us mention that a somehow similar analysis has been conducted in a simpler situation in Annex of [14]. We start by presenting some preliminaries on weighted Sobolev spaces borrowed from [29].

Some preliminaries on weighted Sobolev spaces
Let us consider the strip = {( , ) ∈ R × (0, )} for > 0. For ∈ R and ∈ N, let us introduce the weighted Sobolev space We also denote˚( ) the closure of We recall the following properties of the Laplace transform.
(1) The Laplace transform is a linear and continuous map from C ∞ 0 (R) to the space of holomorphic functions in the complex plane. In addition, we have Hence, the Laplace transform (4.2) can be extended as an isomorphism The inverse Laplace transform is given by the formula By using the above properties, one can prove that for ∈ R and ∈ N, the norm (4.1) is equivalent to the norm given by Next, we introduce the infinite cone with ∈ (0, 2 ). For ∈ R and ∈ N, let us introduce the weighted Sobolev space ( ) as the closure of We also denote by˚( ) the closure of C ∞ 0 ( ) in ( ),˚, 0 ( ) the closure in ( ) of the set of functions in C ∞ 0 ( ) which vanish in a vicinity of 0 = ∩ { = 0},˚, ( ) the closure in ( ) of the set of functions in C ∞ 0 ( ) which vanish in a vicinity of = ∩ { = }. One can show that the norm of ( ) is equivalent to the norm The key point consists in the change of variable = ln , which transforms the cone = R * + × (0, ) into the strip = R × (0, ). In particular, if we introduce, for a function defined in , the function ℰ defined in by (ℰ )( , ) = ( , e ), since = (ℰ ), the norm (4.6) is equivalent to This shows that there exists an isomorphism between the spaces ( ) and − +1 ( ), or in other words, between ( ) and + −1 ( ). We point out that in [25], the weighted spaces were already used in the context of the regularization of the Cauchy problem (1.1).

The case of a corner of mixed type
The regularity of solutions and at a corner of mixed type can no longer be analyzed separately. We use the weighted Sobolev spaces introduced in the previous section. We first consider the quasi-reversibility problem in the strip . The strong equations corresponding to (1.7) in the strip are This operator is associated with the following problem in the strip : We have the following lemma.
Remark 5.2. We notice that the symbol L has complex eigenvalues and is not self-adjoint. This is a difference with the symbols which are involved when considering the Laplace equation with Dirichlet or Neumann boundary conditions.
Proof. We simply have By applying the Laplace transform to the problem (5.3) with respect to and by setting = with ∈ R, we obtain For fixed , this problem is equivalent to the weak formulation: find (̂︀ ,̂︀ ) ∈ 1 0,0 (0, ) × 1 0, (0, ), where By the Lax-Milgram Lemma, the weak formulation (5.7) is well-posed and there exists some constant > 0 (independent of and of ) such that Indeed, by setting =̂︀ and =̂︀ in (5.7), we obtain By using the Poincaré inequality and assuming that ≤ 1 we obtain that and where is independent of and . Now, given that which implies (5.8). Finally, we have for all = and which by integration on ℓ 0 and by definition of the norms ‖ · ‖ , (see (4.3)) implies This gives the estimate which proves that ℬ 0 is an isomorphism.
It remains to integrate the above estimate on ℓ − following the definition of the norm ‖ · ‖ ,0 given by (4.3).
In order to link the solutions of problem (5.26) obtained for different , we need to compute the adjoint of the symbol L defined in (5.4) and to specify its eigenvalues and eigenfunctions. Proof. For ( , ) ∈ (L ) and ( , ℎ) ∈ (L * ), we have by an integration by parts formula It is readily seen that all the boundary terms vanish due to the boundary conditions satisfied by ( , ) and ( , ℎ) at = 0 and = . This completes the proof.
The proof of Lemma 5.7 is the same as the proof of Lemma 5.1 and is therefore not given. Lastly, we will need a biorthogonality relationship between the eigenfunctions of L and that of L * .
Lemma 5.8. Assume that , ∈ Z and , = ± satisfy either + ̸ = −1 or + ̸ = 0. The eigenfunctions ( ± , ± ) of L and the eigenfunctions ( ± , ℎ ± ) of L * satisfy Proof. On the one hand, the assumption + ̸ = −1 or + ̸ = 0 is equivalent to ̸ = − . Let us first assume that ̸ = and = = +, which implies on the other hand that ̸ = . Skipping the sign +, we have Since 2 ̸ = 2 , this implies that for ̸ = and = = +, we have We clearly obtain the same result each time that ( , ) ̸ = ( , ). Let us now assume that = and = . We have ∫︁ In the next theorem, we compare two solutions of problem (5.3) associated with two different values of .
In view of the biorthogonality relationships of Lemma 5.8 and due to the fact that in case = − (that is + = −1 and + = 0) the first and third terms within the brackets above compensate one another as well as the second and fourth terms, we end up with On the other end, since 1 < 2 , the function 2 is more decreasing than 1 at +∞. And the situation is inverted at −∞. The same property holds for 2 and 1 . Since vanishes at −∞, we have that ( 2 , 2 ) ∈ (ℬ 1 ) ∩ (ℬ 2 ). Using an integration by parts in and the fact that − 2 < Re , we obtain that With the same argument, we obtain By combining (5.34)-(5.36), we get which completes the proof.
We end up with the main proposition of this section.
Note that this conclusion is very similar to the case of the Laplace equation with mixed Dirichlet-Neumann boundary conditions (see [22]).

Application to error estimates
In this last section, we use the regularity estimates for solutions of quasi-reversibility problem (1.7), in particular Theorem 3.1, to derive error estimates between the exact solution and the quasi-reversibility solution obtained in the presence of noisy data and with the help of a FEM.

Main analysis
Let us assume that Ω is a polygonal domain in two dimensions and that ∈ 1 (Ω) is the exact solution of problem (1.6) associated with the exact data ∈ 2 (Ω). In the context of inverse problems, usually is not available. Only an approximate data ∈ 2 (Ω) is available, with where can be viewed as the amplitude of noise. A natural idea is to solve problem (1.7) with instead of , and a practical way of proceeding is to discretize problem (1.7) with the help of a FEM. More precisely, we assume that Ω supports a triangular mesh which is regular in the sense of [17], the maximal diameter of each triangle being ℎ. Let us denote by 0,ℎ and˜0 ,ℎ the finite dimensional subspaces of 0 and˜0, respectively, formed by the continuous functions on Ω which are affine on each triangle and which vanish on the sides which belong to Γ andΓ, respectively. The discretized version of the mixed formulation of quasi-reversibility (1.7) is: for > 0, find ( ,ℎ , ,ℎ ) ∈ 0,ℎ ×˜0 ,ℎ such that for all ( ℎ , ℎ ) ∈ 0,ℎ ×˜0 ,ℎ , We denote ( ,ℎ , ,ℎ ) the solution to problem (6.2) which is associated with the noisy data instead of the exact data . In practice, the solution ,ℎ is the only approximate function of the exact solution which is accessible, this is why we are interested in the norm of the discrepancy ,ℎ − in the domain Ω. In this view, we write and estimate each term of this decomposition. The first term to estimate corresponds to the error due to the noisy data. Let us prove the following lemma.
The second term of (6.3) corresponds to the error due to discretization. Let us prove the following lemma, which is a consequence of Theorem 3.1.
There is a constant > 0 which depends only on the geometry and on such that where is given in the statement of Theorem 3.1.
Proof. The proof relies in particular on Céa's lemma. Since we need a uniform estimate with respect to , we detail the proof. For all ( ℎ , ℎ ) ∈ 0,ℎ ×˜0 ,ℎ , we have This implies that for all ( ℎ , ℎ ) ∈ 0,ℎ ×˜0 ,ℎ , But on the one hand, we have while on the other hand, there holds By using the classical interpolation error estimates in (Ω) for > 1 (see [22]), we know that there exists a constant > 0 which depends only on the geometry such that Theorem 3.1 in the case of exact data implies that there is a constant > 0 which depends on the geometry and on such that From the three above estimates, we get Eventually we end up with which completes the proof.
Estimating the third term in (6.3) is strongly related to the stability of the Cauchy problem for the Laplace equation, a topic which has a long history since the pioneering paper [23] (see e.g. [1-3, 5, 6, 38-40]). It is well-known that since such problem is exponentially ill-posed, the corresponding stability estimate is at best of logarithmic type (see e.g. [5]). To our best knowledge, an estimate of ( ) := ‖ − ‖ 1 (Ω) , which tends to 0 when tends to 0 in view of Theorem 1.4, is unknown. However, a logarithmic stability estimate for ‖ − ‖ 2 (Ω) can be derived from Theorem 1.9 in [1] and a Hölder stability estimate for ‖ − ‖ 1 ( ) can be derived from Propositions 2.2 and 2.3 in [5], where is a subdomain of Ω which excludes a vicinity ofΓ and a vicinity of corners.
By plugging these estimates in Theorem 1.9 of [1], we obtain the result.
Lemma 6.4. There exists a constant > 0 which depends only on the geometry and on and a constant ∈ (0, 1) which depends only on the geometry such that Proof. We start again from the system (6.6) satisfied by the function − in Ω. Let us consider some 0 ∈ Γ and a sufficiently small > 0 such that Γ 0 = Γ ∩ ( 0 , ) is the interior of a segment. We have that, by using a trace inequality, where 0 = Ω ∩ ( 0 , ). Then, by interpolation Now, the estimates of given by Theorems 1.4 and 2.2 provide We end up with Plugging the estimates in Propositions 2.3 (propagation of smallness from a subpart of the boundary to the interior of the domain) and 2.2 (interior propagation of smallness) in [5], we obtain the result.
Remark 6.5. Our analysis does not provide a uniform bound of ‖ − ‖ (Ω) with respect to for some > 1. Such uniform bound is required when trying to propagate smallness from the interior up to the boundary (see Prop. 2.4 in [5]). This is why a stability estimate for ‖ − ‖ 1 (Ω) can not be obtained from what precedes.
In conclusion, by gathering (6.3)-(6.5), we end up with the final estimate where is given in the statement of Theorem 3.1 and converges to 0 when tends to 0 at best with a logarithmic convergence rate in view of Lemma 6.3. An important application of the estimate (6.7) is that when → 0, we have to choose = ( ) and ℎ = ℎ( ) such that in order to obtain a good approximation of the exact solution from noisy data and by using our FEM.
Remark 6.6. Taking Lemma 6.4 into account, the estimate (6.7) is slightly improved in the truncated domain : where the exponent of ℎ is 1 because the domain excludes all the corners (we use a slight adaptation of Theorem 2.2).

Numerical illustrations
In this paragraph, we present the results of preliminary numerical experiments we conducted to illustrate certain features of the estimate (6.7). We set Note that the function satisfies = = 0 on Γ. As a consequence, is the solution of the Cauchy problem (1.6). Then for a given small > 0, we numerically approximate the solution of the mixed formulation of quasi-reversibility (1.7) using a P1 FEM. To proceed, we use the library FreeFem++ 3 . This gives us a numerical solution ,ℎ where ℎ corresponds to the mesh size. The mesh of the domain Ω is structured and composed of triangles that are all the same. We emphasize that in order to interpret the results more easily, that is to analyze the conjugate effects of and ℎ on the error, we take = .
In other words, we work with noiseless data.
In Figure 4, we have displayed the curve ‖ ,ℎ − ‖ 1 (Ω) as a function of ℎ for different values of . Given the geometry considered here, Theorem 3.1 ensures that we can take = 2 in (6.7). As a consequence, we have the theoretical estimate where the function is not known but at best logarithmic (see the discussion above). We observe that ‖ ,ℎ − ‖ 1 (Ω) is a function that decreases as ℎ tends to zero. However, such function seems linear for small values of and turns out to be a constant for large values of in the region where ℎ is small (see the left curve of Fig. 4). An attempt to explain such phenomenon is the following: for small values of , the first term in the right-hand side of (6.9) is much larger than the second one, so that the linearity with respect to ℎ is visible. This is confirmed, looking at the vertical scales indicated on the figure, by the fact that the maximal error is increasing when is decreasing. For large values of , the second term becomes dominant and does not depend on ℎ, which explains why a threshold is visible.
Such effect can be attenuated if we truncate the domain close to the boundary Ω ∖ Γ = (0, 1) × {1}, that is where the data are unknown. Indeed, as we can see on Figure 5, the numerical errors create some instability close to that part of the boundary.
In Figure 6, we set˜= (0, 1) × (0, 0.9) (interior domain) and we represent the curve ‖ ,ℎ − ‖ 1 (˜) as a function of ℎ for different values of . In that case, adapting a bit (6.8) (because Γ has some corners but angles are right angles), we obtain the theoretical estimate for some positive . In this situation, in agreement with (6.10), we observe that the linear behaviour with respect to ℎ as ℎ tends to zero appears quite clearly, because the first term in the right-hand side of (6.10) is not absorbed by the second one any more, for all values of that we consider. This may be due to the fact that the Hölder estimate is much smaller than the estimate ( ), which is at best logarithmic. Finally, in Figure 7 we show the curves ‖ ,ℎ − ‖ 1 (Ω) (left) and ‖ ,ℎ − ‖ 1 (˜) (right) with respect to log for a given ℎ, by using the same horizontal and vertical scales. In accordance with (6.9) and (6.10), we    observe that for a fixed ℎ, when decreases to zero, the errors firstly improve and secondly deteriorate. This is especially observable for the error in Ω (left picture). The goal of the appendix is to establish the following result.
To prove Proposition A.1, we need three lemmas. We first consider a simple situation when is purely imaginary.
We will say that ∈ C is an eigenvalue of J if Ker J ( ) ̸ = {0}. We have the following lemma. Proof. Lemma A.2 indicates that the result is true for any ∈ R. It follows from the analytic Fredholm theorem that J ( ) : (J ) −→ 2 (0, ) is an isomorphism if and only if is not an eigenvalue of J . It is straightforward that the eigenvalues of J are = ( /2 + )/ , ∈ Z, the corresponding eigenfunctions being given by ( ) = sin(( /2 + ) / ). The result follows.
We now consider a situation where is no longer purely imaginary.
Proof of Proposition A.1. Lemma A.4 implies that for all ∈ C such that Re = and |Im( )| ≥ , we have the estimate ‖ 2 ‖ 2 (0, ) + | | 2 ‖ ‖ 2 (0, ) ≤ ‖ ‖ 2 (0, ) , where > 0 is independent of , and depends only on . For ∈ [ − , + ], the symbol J ( ) is invertible according to Lemma A.3. The analytic Fredholm theorem guarantees that the inverse operator ↦ → J ( ) −1 is continuous outside of its poles. Since the segment [− − , − + ] is compact, we deduce that the above estimate remains true for all such that Re = with a constant which depends neither on nor Im .
Observing that e 2| | ≤ 4 cosh( ) 2 , we see that to establish (B.3), it is sufficient to show that there is some > 0 such that We will study the two factors on the left hand side of (B.4) proving that for > 0 small enough they are both smaller than one. Let us consider the first one. A direct computation gives We see that the first factor on the left hand side of (B.4) is smaller than one as soon as is positive on R. Since (0) = 1 > 0, it is sufficient to show that its discriminant is negative. We find ∆ = (2 sin( ) sinh( ) + cosh( )) 2 − 4(cos( ) 2 cosh( ) 2 + sin( ) 2 sin( ) 2 ) = (︁ ( 2 − 4 cos( ) 2 ) cosh( ) + 4 sin( ) sinh( ) )︁ cosh( ).
Observing that | sinh( )| < cosh( ), we can write Therefore, since cos( ) ̸ = 0 when / ∈ {( /2 + )/ , ∈ Z}, we see that we can find > 0 small enough (but independent of ) such that ∆ < 0. This shows that the first factor on the left hand side of (B.4) is smaller than one. A completely similar approach allows one to prove that the second factor is also smaller than one. As a consequence, (B.4) is satisfied for small enough and so is (B.3). for small enough. This is enough to conclude.