Constantin B Ȃcut ¸ȃ

AND VICTOR NISTOR Dedicated to Hengguang Li
email: li@wayne.edu

Monique Dauge

Nistor

V Nistor

DIFFERENTIAL OPERATORS ON DOMAINS WITH CONICAL POINTS: PRECISE UNIFORM REGULARITY ESTIMATES

HAL is

C. Bacuta has been partially supported by NSF Grant DMS-1522454. H. Li has been partially supported by the National Science Foundation Grant DMS-1418853, by the Natural Science Foundation of China Grant 11628104, and by the Wayne State University Grants Plus Program. V. Nistor has been partially supported by ANR-14-CE25-0012-01 (SINGSTAR). Manuscripts available from http://iecl.univ-lorraine.fr/˜Victor.Nistor/ AMS 2000 Subject classification: Primary 35J25; Secondary 35R05, 65N30, 58J32, 52B70. Key words and phrases: Polyhedral domain, strongly elliptic equations, mixed boundary conditions, weighted Sobolev spaces, well-posedness, parametric family, Legendre approximation.

Introduction

We consider families of mixed boundary value problems on a bounded, domain Ω ⊂ R d with conical points (d ≥ 2). The associated differential operators belong to suitable families of strongly elliptic, second order differential operators with singular coefficients. In our method, it is necessary to consider certain singular coefficients even if one is interested only in the case of regular coefficients. Using appropriate weighted Sobolev spaces, we obtain concrete estimates on the norm and on the regularity of the solutions of our boundary value problems in terms of the norms of the coefficients of the operators and their coercivity constants. In addition, we provide weighted Sobolev space conditions on the coefficients that ensure an analytic dependence of the solution on the coefficients of the operators and on the forcing terms (the free term).

To better explain our results, it is useful to put them into perspective. A classical result in Partial Differential Equations states that a second order, strongly coercive, strongly elliptic partial differential operator P induces an isomorphism [START_REF] Adler | Graded mesh approximation in weighted Sobolev spaces and elliptic equations in 2[END_REF] P :

H m+1 (G) ∩ {u| ∂Ω = 0} → H m-1 (G) ,
for all m ∈ Z + := {0, 1, . . .}, provided that G is a smooth, bounded domain in some euclidean space. See, for example, [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF][START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF][START_REF] Hörmander | The analysis of linear partial differential operators[END_REF][START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF] and the references therein. This result has many applications and extensions. However, it does not extend directly to non-smooth domains. In fact, on non-smooth domains, the solution u of P u = F will have singularities, even if the right hand side F is smooth. See Kondratiev's fundamental 1967 paper [START_REF] Kondrat Ev | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF] for the case of domain with conical points and Dauge's comprehensive Lecture Notes [START_REF] Dauge | Elliptic boundary value problems on corner domains[END_REF] for the case of polyhedral domains. See [START_REF] Bacuta | Regularity estimates for elliptic boundary value problems in Besov spaces[END_REF][START_REF] Bacuta | Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains[END_REF][START_REF] Bacuta | Improving the rate of convergence of high-order finite elements on polyhedra. I. A priori estimates[END_REF][START_REF] Costabel | Analytic regularity for linear elliptic systems in polygons and polyhedra[END_REF][START_REF] He | The highest superconvergence of the tri-linear element for Schrödinger operator with singularity[END_REF][START_REF] Kozlov | Spectral problems associated with corner singularities of solutions to elliptic equations[END_REF][START_REF] Li | Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains[END_REF][START_REF] Li | Analysis of a modified Schrödinger operator in 2D: regularity, index, and FEM[END_REF][START_REF] Nazarov | Elliptic problems in domains with piecewise smooth boundaries[END_REF][START_REF] Rukavishnikov | On the isomorphic mapping of weighted spaces by an elliptic operator with degeneration on the domain boundary[END_REF] for a sample of related results. These theoretical results have been a critical ingredient in developing effective numerical methods approximating singular solutions. See for example [START_REF] Babuška | Direct and inverse error estimates for finite elements with mesh refinements[END_REF][START_REF] Bramble | Uniform convergence of the multigrid V -cycle for an anisotropic problem[END_REF]. In addition, we mention that estimates for equations on conical manifolds can also be obtained using the method of layer potentials (see, for example, [START_REF] Carvalho | Layer potentials C * -algebras of domains with conical points[END_REF][START_REF] Fabes | Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains[END_REF][START_REF] Kohr | Layer potential analysis for pseudodifferential matrix operators in Lipschitz domains on compact Riemannian manifolds: applications to pseudodifferential Brinkman operators[END_REF][START_REF] Mitrea | On the Besov regularity of conformal maps and layer potentials on nonsmooth domains[END_REF][START_REF] Qiao | Double layer potentials on polygons and pseudodifferential operators on Lie groupoids[END_REF] and references therein).

For polygonal domains (and, more generally, for domains with conical points), Kondratiev's results mentioned above extend the isomorphism in [START_REF] Adler | Graded mesh approximation in weighted Sobolev spaces and elliptic equations in 2[END_REF] to polygonal domains by replacing the usual Sobolev spaces H m (Ω) with the Kondratiev type Sobolev spaces. Let Ω be then a curvilinear polygonal domain (see Definition 3.1, in particular, the sides are not required to be straight), and r Ω > 0 be a smooth function on Ω that coincides with the distance to its vertices close to the vertices. We let

(2) K m a (Ω) := { u : Ω → C, r |α|-a Ω ∂ α u ∈ L 2 (Ω), |α| ≤ m } ,
where ∂ i := ∂ ∂xi , i = 1, . . . , d, and

∂ α := ∂ α1 1 ∂ α2 2 . . . ∂ α d d .
Kondratiev's results [START_REF] Kondrat Ev | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF] (see also [START_REF] Costabel | Analytic regularity for linear elliptic systems in polygons and polyhedra[END_REF][START_REF] Kozlov | Spectral problems associated with corner singularities of solutions to elliptic equations[END_REF]) give that the Laplacian ∆ := i≤d ∂ 2 i induces an isomorphism of weighted Sobolev spaces. More precisely, [START_REF] Ammann | Sobolev spaces on Lie manifolds and regularity for polyhedral domains[END_REF] ∆ :

K m+1 a+1 (Ω) ∩ {u| ∂Ω = 0} → K m-1 a-1 (Ω)
is a continuous bijection with continuous inverse for m ∈ Z + := {0, 1, . . .} and |a| < π/α M AX , where α M AX is the maximum angle of Ω. One can extend this result by interpolation to the usual range of values for m [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF]. A similar result holds also for more general strongly elliptic operators [START_REF] Kozlov | Spectral problems associated with corner singularities of solutions to elliptic equations[END_REF]. In [START_REF] Bacuta | Improving the rate of convergence of high-order finite elements on polyhedra. I. A priori estimates[END_REF], this result of Kondratiev was extended to three dimensional polyhedral domains and in [START_REF] Bacuta | Interface and mixed boundary value problems on n-dimensional polyhedral domains[END_REF] it was extended to general d-dimensional polyhedral domains using as a main ingredient a suitable generalization of Hardy's inequality. In three dimensions and higher, this type of results is not enough for numerical methods. Thus, in [START_REF] Bacuta | Improving the rate of convergence of high-order finite elements on polyhedra. II. Mesh refinements and interpolation[END_REF], an anisotropic regularity and well-posedness result was proved for three dimensional polyhedral domains, building on previous results of Babuška and Guo [START_REF] Guo | Regularity of the solutions for elliptic problems on nonsmooth domains in R 3 . I. Countably normed spaces on polyhedral domains[END_REF] and Buffa, Costabel, and Dauge [START_REF] Buffa | Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron[END_REF]. See also [START_REF] Costabel | Analytic regularity for linear elliptic systems in polygons and polyhedra[END_REF] for further references and for related results, including analytic regularity.

In this paper, we generalize Kondratiev's result by allowing families of operators, by allowing low-regularity coefficients, and by studying the quantitative and qualitative dependence of the solution on these coefficients. To state our main result, let us fix some notation. Let β := (a ij , b i , c) denote the coefficients of ( 4)

p β u := - d i,j=1 ∂ i (a ij ∂ j u) + d i=1 b i ∂ i u - d i=1 ∂ i (b d+i u) + cu ,
a second order differential operator in divergence form on our domain Ω ⊂ R d . Many concepts discussed in the paper make sense for any dimension d ≥ 1. Nevertheless, the main results we prove are for d = 2. Thus, we assume for the rest of this introduction that Ω is a two-dimensional curvilinear polygonal domain. The coefficients β of the operator p β are obtained using weighted W m,∞ -type space defined by

(5) W m,∞ (Ω) := { u : Ω → C | r |α| Ω ∂ α u ∈ L ∞ (Ω), |α| ≤ m }
, where r Ω is as in Equation (2) (that is, it is equal to the distance function to the conical points when close to those points). We fix for the rest of the introduction m ∈ Z + := {0, 1, . . .} and we assume that a ij , r Ω b i , r 2 Ω c ∈ W m,∞ (Ω). We let [START_REF] Guo | Regularity of the solutions for elliptic problems on nonsmooth domains in R 3 . I. Countably normed spaces on polyhedral domains[END_REF] β

Zm := max{ a ij W m,∞ (Ω) , r Ω b i W m,∞ (Ω) , r 2 Ω c W m,∞ (Ω) } ,
(notice the factors involving r Ω !), and for P = p β and V = H 1 0 (Ω), define

) := ρ(P ) := inf (P v, v) v 2 H 1 (Ω) , v ∈ V , v = 0 , (7) ρ(β
where (z) = z denotes the real part of z. Our main result for Dirichlet boundary conditions in two dimensions is as follows.

THEOREM 1.1. Let Ω ⊂ R 2 be a curvilinear polygonal domain, η 0 > 0, m ≥ 0, and N m = 2 m+2 -m -3 ≥ 0. Then there exist γ and C m with the following property.

For any β ∈ Z m and any |a| < η := min{η 0 , γ -1 β -1 Z0 ρ(β)}, the operator p β defined in Equation (4) induces an isomorphism

(8) p β : K m+1 a+1 (Ω) ∩ { u| ∂Ω = 0 } → K m-1 a-1 (Ω) such that p -1 β : K m-1 a-1 (Ω) → K m+1 a+1 (Ω) ∩ {u| ∂Ω = 0} depends analytically on the coefficients β := (a ij , b i , c) and has norm p -1 β ≤ C m (ρ(β) -γ|a| β Z0 ) -Nm-1 β Nm Zm .
The parameter η 0 has to role of ensuring that a belongs to a fixed bounded set, so we can bound a 2 with |a| in the estimates involving β(a) (see Theorem 4.4). The bounds γ 1 and C m depend only on m, (Ω, ∂ D Ω), and η 0 .

Since the solution u of the equation p β u = F , u = 0 on the boundary, is in

K m+1 a+1 (Ω) for F ∈ K m-1
a-1 (Ω), |a| < η, we obtain the usual applications to the Finite Element Method on straight polygonal domains for m ≥ 1 and a > 0 [START_REF] Adler | Graded mesh approximation in weighted Sobolev spaces and elliptic equations in 2[END_REF][START_REF] Bȃcut ¸ȃ | Improving the rate of convergence of 'high order finite elements' on polygons and domains with cusps[END_REF][START_REF] Li | Analysis of a modified Schrödinger operator in 2D: regularity, index, and FEM[END_REF].

Theorem 1.1 is a consequence of Theorem 4.4, which deals with the mixed boundary value problem

(9)      p β u = f in Ω u = 0 on ∂ D Ω ∂ β ν u = h on ∂ N Ω , where (∂ β ν v) := d i=1 ν i ( d j=1 a ij ∂ j v + b d+i v .
An exotic example to which Theorem 4.4 applies is that of the Schrödinger operator H := -∆ + cr -2 Ω on Ω with pure Neumann boundary conditions (and suitable positivity conditions on c), see Theorem 5.4.

The main novelties of Theorem 4.4 (and of the paper in general) are the following:

(i) The precise estimate on the norm of the inverse of P β seems to be new even in the smooth case. (ii) We deal with singular coefficients of a type that has not been systematically considered in the literature on non-smooth domains. Thus our coefficients have both singular parts at the corners of the form r -j Ω (j ≤ 2) and have limited regularity away from the corners. (iii) We provide a new method to obtain higher regularity in weighted Sobolev spaces using divided differences; a method that is, in fact, closer to the one used in the classical case of smooth domains. (iv) Our method of obtaining higher regularity for the solution also yields regularity for the dependence on the coefficients, more precisely, in this case, an analytic dependence on the coefficients and the free term (f in Problem ( 9)).

The paper is organized as follows. In Section 2, we introduce the notation and necessary preliminary results for our problem in the usual Sobolev spaces. In particular, an enhanced Lax-Milgram Lemma (Lemma 2.5) provides uniform estimates for the solution of our problem [START_REF] Bacuta | Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains[END_REF] and analytic dependence of this solution on the coefficients β. In Section 3, we first define curvilinear polygonal domains (Definition 3.1). We then provide several equivalent definitions of the weighted Sobolev spaces K m a (Ω) and the form of our differential operators. Then, in Section 4, using local coordinate transformations, we derive our main result, the analytic dependence of the solution on the coefficients in high-order weighted Sobolev spaces (Theorem 4.4). Finally, Section 5 contains some consequences of Theorem 4.4 and some extensions. In particular, we consider a framework for the pure Neumann problem with inverse square potentials at vertices. For notational simplicity, we do not consider systems, although many of the techniques below apply to this more general setting.

We thank Martin Costabel, Monique Dauge, Markus Hansen, Serge Nicaise, and Christoph Schwab for useful discussions. We dedicate this paper to Monique Dauge for her many contributions to Scientific Computing and Analysis on singular domains.

Coercivity in classical Sobolev spaces

In this section, we recall some needed results on coercive operators, on (uniformly) strongly elliptic operators, and on analytic functions defined on open subsets of Banach spaces.

2.1. Function spaces and boundary conditions. Throughout the paper, Ω ⊂ R d , d ≥ 1, denotes a connected, bounded domain. Further conditions on Ω will be imposed in the next section. As usual, H m (Ω) denotes the space of (equivalence classes of) functions on Ω with m derivatives in L 2 (Ω). When we write A ⊂ B, we allow also

A = B.
In what follows, ∂ D Ω is a suitable closed subset of the boundary ∂Ω, where we impose Dirichlet boundary conditions.

We shall rely heavily on the weak formulation of Problem [START_REF] Bacuta | Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains[END_REF]. Thus, let us recall that H -1 (Ω) is defined as the dual space of (10)

H 1 0 (Ω) := { u ∈ H 1 (Ω) | u| ∂Ω = 0 } ,
with pivot L 2 (Ω). We introduce homogeneous essential boundary conditions abstractly, by considering a subspace V ,

H 1 0 (Ω) ⊂ V ⊂ H 1 (Ω) , (11)
such that V has the norm induced either from H 1 (Ω) or from K 1 1 (Ω) and H 1 0 (Ω) is a closed subspace of V . In many applications, V is closed in H 1 (Ω), but this is not the case in our application to the Neumann problem with inverse square potentials at vertices (see Theorem 5.4). Let V * be the dual of V with pivot space L 2 (Ω). Therefore, by ( , ) we shall denote both the inner product (f, g) = Ω f (x)g(x) dx on L 2 (Ω), and by continuous extension, also the duality pairing between V * and V . Thus,

V * = H -1 (Ω) if V = H 1 0 (Ω)
; otherwise, V * will incorporate also non-homogeneous natural boundary conditions.

For Problem (9), we choose

(12) V = H 1 D (Ω) := { u ∈ H 1 (Ω) | u = 0 on ∂ D Ω } ,
and assume that the Neumann part of the boundary contains no adjacent edges.

2.2. The weak formulation. Recall from Equation (4) the differential operator

p β u := - d i,j=1 ∂ i a ij ∂ j u + d i=1 b i ∂ i u - d i=1 ∂ i (b d+i u) + cu, which is used in Problem (9)
, where a ij , b i , c : Ω → C denote measurable complex valued functions as in (4) and β denotes the coefficients (a ij , b i , c). We shall make suitable further assumptions on these coefficients below. Equation [START_REF] Bacuta | Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains[END_REF], makes sense as formulated only if u is regular enough (at least in H 3/2+ , to validate the Neumann derivatives at the boundary). In order to use the Lax-Milgram Lemma for the problem (9), we formulate our problem in a more general way that allows u ∈ V . To this end, let us introduce the Dirichlet form B β associated to [START_REF] Bacuta | Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains[END_REF], that is, the sesquilinear form

B β (u, v) := d i,j=1 a ij ∂ j u, ∂ i v + d i=1 b i ∂ i u, v + d i=1 b d+i u, ∂ i v + cu, v = Ω d i=1 d j=1 a ij (x)∂ j u(x) + b d+i (x)u(x) ∂ i v(x) + d i=1 b i (x)∂ i u(x) + c(x)u(x) v(x) dx ,
where dx denotes the volume element in the Lebesgue integral on Ω ⊂ R d .

Remark 2.1. Let F (v) = Ω f (x)v(x)dx+ ∂ N Ω h(x)v(x)dS.
Then the weak variational formulation of Equation ( 9) is:

Find u ∈ V , such that (13) B β (u, v) = F (v) , for all v ∈ V .
We then define P β : V → V * by ( 14)

(P β u, v) := B β (u, v) , for all u, v ∈ V .
Thus, the weak formulation of Equation ( 9) is equivalent to (15)

P β u = F ∈ V * .
We are interested in the dependence of u on F and on the coefficients β := (a ij , b i , c) of P β . We notice that if the Neumann part of the boundary ∂ N Ω is empty, then p β and P β can be identified, but this is not possible in general. In fact, we are looking for an analytic dependence of the solutions on the coefficients. For this reason, it is useful to consider complex Banach spaces and complex valued coefficients.

2.3.

Bounded forms and operators. For two Banach spaces X and Y , let L(X; Y ) denote the Banach space of continuous, linear maps T : X → Y endowed with the operator norm

(16) T L(X;Y ) := sup x =0 T x Y x X .
We write L(X) := L(X; X). Let us define Z to be the set of coefficients β = (a ij , b i , c) such that the form B β is defined and continuous on V × V , and we give Z the induced norm. Thus Z is given the induced topology from L(V ; V * ).

It will be convenient to use a slightly enhanced version of the well-known Lax-Milgram Lemma stressing the analytic dependence on the operator and on the data. We thus first review a few basic definitions and results on analytic functions [START_REF] Dineen | Complex analysis on infinite-dimensional spaces[END_REF].

Let X and Y be Banach spaces. In what follows, L i (Y ; X) will denote the space of continuous, multi-linear functions L : Y × Y × . . . × Y → X, where i denotes the number of copies of Y . The norm on the space L

i (Y ; X) is L Li(Y ;X) := sup yj ≤1 L(y 1 , y 2 , . . . , y i ) X .
Of course, L 1 (Y ; X) = L(Y ; X), isometrically. We shall need analytic functions defined on open subsets of a Banach space. Let U ⊂ Y and consider the spaces

C k (U ; X), k ∈ Z + ∪ {∞, ω} defined as follows. If k ∈ Z + ∪ {∞}, then C k (U ; X)
denotes the space of functions v : U → X with continuous (Fréchet) derivatives

D i v : U → L i (Y ; X), i ≤ k. Similarly, C k b (U ; X) ⊂ C k (U ; X), k ∈ Z + ∪ {∞}, denotes the subspace of those functions v ∈ C k (U ; X) for which the derivatives D i v, i ≤ k, are bounded on U . For each finite j, we let (17) v C j b (U ;X) := sup i≤j, y∈U D i y v Li(Y ;X)
denote the natural Banach space norm on

C j b (U ; X), with D i a v ∈ L i (Y ; X) denoting the value of D i v at a.
The space C ω (U ; X) consists of the functions f : U → X that have, for any a ∈ U , an expansion

f (x) = ∞ k=0 1 k! D k a f (x -a, x -a, . . . , x -a)
that is uniformly convergent for x in a small, non-empty open ball centered at a. We let

C ω b (U ; X) := C ω (U ; X) ∩ C ∞ b (U ; X). If k is not finite, that is, if k = ∞ or k = ω, we endow C k b (U ; X)
with the Fréchet topology defined by the family of seminorms

• C j b (U ;X) , j ≥ 1.
We shall use that multilinear functions are analytic. We shall need the following standard result.

LEMMA 2.2. Let Y 1 , Y 2 be Banach spaces. (i) The map L(Y 1 ; Y 2 ) × Y 1 (T, y) → T y ∈ Y 2 is analytic. (ii) The map T → T -1 ∈ L(Y 1 ) is analytic on the open set L inv (Y 1 ) of invertible operators in L(Y 1 ) := L(Y 1 ; Y 1 ).
Proof. In (i), the given map is bilinear, and hence analytic. To prove (ii), we simply write the Neumann series

(T -R) -1 = ∞ n=0 T -1 (RT -1
) n , which is uniformly and absolutely convergent for R T -1 ≤ 1 -, for any 1 ≥ > 0.

2.4. An enhanced Lax-Milgram Lemma. We now recall the classical Lax-Milgram Lemma, in the form that we will need.

Definition 2.3. Let H 1 0 (Ω) ⊂ V ⊂ H 1 (Ω).
A continuous operator P : V → V * is called strongly coercive on V (or simply strongly coercive when there is no danger of confusion) if

0 < ρ(P ) := inf v∈V {0} (P v, v) v 2 V .
We shall usually write ρ(β) = ρ(P β ), where ρ(P β ) is as defined in Equation [START_REF] Babuška | Direct and inverse error estimates for finite elements with mesh refinements[END_REF].

For P = P β , we thus have ρ(β) v 2 H 1 (Ω) = ρ(P β ) v 2 H 1 (Ω) ≤ B β (v, v)
, for all v ∈ V . We shall need the following simple observation: Remark 2.4. If P : V → V * is strongly coercive on V and P 1 : V → V * satisfies P 1 := P 1 L(V,V * ) < ρ(P ), then P + P 1 is also strongly coercive on V and ρ(P + P 1 ) ≥ ρ(P ) -P 1 . Indeed, [START_REF] Chkifa | Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs[END_REF] (P + P 1 )u, u ≥ (P u, u) -P 1 u 2 V ≥ (ρ(P ) -P 1 ) u 2 V , and hence the set L(V ; V * ) c of strongly coercive operators is open in L(V ; V * ).

Recall now the standard way of solving Equation ( 13) using the Lax-Milgram Lemma for strongly coercive operators. LEMMA 2.5 (Analytic Lax-Milgram Lemma). Assume that P : V → V * is strongly coercive. Then P is invertible and P -1 ≤ ρ(P ) -1 . Moreover, the map L(

V ; V * ) c × V * (P, F ) → P -1 F ∈ V is analytic. Consequently, Z ∩ L(V ; V * ) c × V * (β, F ) → (P β ) -1 F ∈ V
is analytic as well.

Proof. The first part is just the classical Lax-Milgram Lemma [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF][START_REF] Ciarlet | The Finite Element Method for Elliptic Problems, volume 4 of Studies in Mathematics and Its Applications[END_REF][START_REF] Monk | Finite element methods for Maxwell's equations[END_REF], which states that "coercivity implies invertibility" and gives the norm estimate. The second part follows from Lemma 2.2. Indeed, the map Φ :

L(V ; V * ) c × V * → V , Φ(β, F ) := (P β ) -1 F is the composition of the maps L(V ; V * ) c × V * × V * (β, F ) → (P β , F ) ∈ L inv (V, V * ) × V * , L inv (V ; V * ) × V * (P, F ) → (P -1 , F ) ∈ L(V * ; V ) × V * , and
L(V * ; V ) × V * (P -1 , F ) → P -1 F ∈ V .
The first of these three maps is well defined and linear by the classical Lax-Milgram Lemma. The other two maps are analytic by Lemma 2.2. Since the composition of analytic functions is analytic, the result follows.

Examples of strongly coercive operators are obtained using "uniformly strongly elliptic" operators, whose definition we recall next. Definition 2.6. Let β ∈ Z. The operator P β is called uniformly strongly elliptic if there exists γ > 0 such that [START_REF] Ciarlet | The Finite Element Method for Elliptic Problems, volume 4 of Studies in Mathematics and Its Applications[END_REF]

d ij=1 a ij (x)ξ i ξ j ≥ γ ξ 2 ,
for all ξ = (ξ i ) ∈ R d and all x ∈ Ω. Here • denote the standard euclidean norm on R d . The largest γ with the property in [START_REF] Ciarlet | The Finite Element Method for Elliptic Problems, volume 4 of Studies in Mathematics and Its Applications[END_REF] will be denoted γ use (β) or γ use (P β ).

Then, we have the following standard example.

Example 2.7. Let β ∈ Z, as in Definition 2.6. We shall regard a matrix X := [x ij ], (X) ij = x ij , as a linear operator acting on C d by the formula Xζ = ξ, where ξ i = j x ij ζ j . We consider the adjoint and positivity with respect to the usual inner product on C d . We thus have X ≥ 0 if, and only if (Xξ, ξ) = ij x ij ξ j ξ i ≥ 0 for all ξ ∈ C d . Also, recall that X * , the adjoint of the matrix X, has entries (X * ) ij = x ji . Then P β is uniformly strongly elliptic if, and only if, there exists γ > 0 such that the matrix a(x) := [a ij (x)] of highest order coefficients of P β satisfies ( 20)

a(x) + a(x) * ≥ 2γI d , for all x ∈ Ω ,
where I d denotes the unit matrix on C d . Assume also that b i = c = 0. Then,

2 (P β u, u) := 2 Ω d i,j=1 a ij (x)∂ j u(x)∂ i u(x) dx = 2 (a∇u, ∇u) = (a∇u, ∇u) + (∇u, a∇u) = (a + a * )∇u, ∇u ≥ 2γ ∇u 2 L 2 (Ω) , for u ∈ H 1 D (Ω). (Recall that H 1 D (Ω) was defined in Equation (12). In particular, v = 0 on ∂ D Ω if v ∈ H 1 D (Ω).)
Remark 2.8. We use the notation of the previous example. If, moreover, ∂ D Ω has positive measure, then there exists

c = c Ω,∂ D Ω > 0 such that Ω |∇v| 2 dx ≥ c v 2 H 1 (Ω)
for all v ∈ H 1 D (Ω), and hence P β is strongly coercive on V = H 1 D (Ω). If ∂ N Ω has not adjacent edges, then P β is also strongly coercive on

V = K 1 1 (Ω) ∩ H 1 D (Ω)
, with the norm induced from K 1 1 (Ω), in view of the Hardy inequality [START_REF] Bacuta | Improving the rate of convergence of high-order finite elements on polyhedra. I. A priori estimates[END_REF][START_REF] Kozlov | Spectral problems associated with corner singularities of solutions to elliptic equations[END_REF]. Moreover, we will have ρ(P β ) ≥ cγ, with c depending only on the domain Ω.

We then have the following result that is standard for non-weighted spaces. (See also [START_REF] Nistor | High-order Galerkin approximations for parametric second-order elliptic partial differential equations[END_REF].)

PROPOSTION 2.9. If β = (a ij , b i , c) ∈ Z is such that P β is strongly coercive on V , H 1 0 (Ω) ⊂ V ⊂ H 1 (Ω)
, with the norm induced from H 1 (Ω) or from K 1 1 (Ω), then P β is uniformly strongly elliptic, more precisely, the estimate (19) is satisfied for any γ ≤ ρ(β) := ρ(P β ). Moreover, P β : V → V * is a continuous bijection and (P β ) -1 F depends analytically on the coefficients β and on F ∈ V * .

Proof. The second part is an immediate consequence of the analytic Lax-Milgram Lemma. Let us concentrate then on the first part. Let us assume that the norm on V is the one induced from K 1 1 (Ω), the case of H 1 (Ω) being completely similar. Let us assume that P β is strongly coercive and let ξ = (ξ i ) ∈ R d . Also, let us choose an arbitrary smooth function φ with compact support D in Ω. We then define the function

ψ ∈ C ∞ c (Ω) ⊂ V by the formula ψ(x) := e ıtξ•x φ(x) ∈ C, where ı := √ -1 and ξ • x = d k=1 ξ k x k . Then ∂ j ψ(x) = ıtξ j e ıtξ•x φ(x) + e ıtξ•x ∂ j φ(x)
, and hence ıtξ j e ıtξ•x φ(x) is the dominant term in ∂ j ψ(x) as t → ∞. Taking into account all the indices j and computing the squares of the L 2 -norms, we obtain [START_REF] Costabel | Corner singularities and analytic regularity for linear elliptic systems. part i: Smooth domains[END_REF] lim

t→∞ t -2 ψ 2 K 1 1 (D) = d j=1 ξ 2 j D |φ(x)| 2 dx = ξ 2 D |φ(x)| 2 dx .
Similarly, the coefficients a ij of P β , are estimated using "oscillatory testing" [START_REF] Costabel | Analytic regularity for linear elliptic systems in polygons and polyhedra[END_REF]

lim t→∞ t -2 (P β ψ, ψ) = D d i,j=1 a ij (x, y)ξ i ξ j |φ(x)| 2 dx .
We then use Definition 2.3 for v = ψ and we pass to the limit as t → ∞. By coercivity and the definition of ρ(β) := ρ(P β ), we have that ρ(β) ψ 2 K 1 1 (D) ≤ (P β ψ, ψ). Dividing this inequality by t -2 and taking the limit as t → ∞, we obtain from Equations ( 21) and ( 22) that

ρ(β) ξ 2 D |φ(x)| 2 dx ≤ D ij a ij (x, y)ξ i ξ j |φ(x)| 2 dx .
Since φ is an arbitrary compactly supported smooth function on D, it follows that, for all

x ∈ D, ρ(β) ξ 2 ≤ ij a ij (x)ξ i ξ j .
Since ξ is arbitrary, we obtain Equation ( 19) with γ = ρ(P ).

An immediate corollary of Proposition 2.9 is COROLLARY 2.10. We have ρ(β) ≤ γ use (β).

In the following sections, this inequality will be used in the form γ -1 use

(P β ) = γ -1 use (β) ≤ ρ(β) -1 := ρ(P β ) -1 .

Polygonal domains, operators, and weighted Sobolev spaces

In this section, we introduce the domains for our boundary value problems, the weighted Sobolev spaces, and the differential operators that we shall use. We also provide equivalent definitions of the needed weighted Sobolev spaces and prove some intermediate results.

3.1. Polygonal domains and defining local coordinates. In this section, we let Ω be a curvilinear polygonal domain, although our method works without significant change for domains with conical points.

Let us describe in detail our domain Ω as a Dauge-type corner domain, with the purpose of fixing the notation and of introducing some useful local coordinate systems -called "defining coordinates" -that will be used in the proofs below. Let B j denote the open unit ball in R j . Thus B 0 := {0} is reduced to one point, B 1 = (-1, 1), and

B 2 = {(x, y) ⊂ R 2 , x 2 + y 2 < 1}. Definition 3.1. A curvilinear polygonal domain Ω ⊂ R 2 is
an open, bounded subset of R 2 with the property that, for every point p ∈ Ω, there exists j ∈ {0, 1, 2}, a neighborhood U p of p in R 2 , and a smooth map φ p : R 2 → R 2 that defines a diffeomorphism

φ p : U p → B j × B 2-j ⊂ R 2 , φ p (p) = 0, satisfying the following conditions: (i) If j = 2, then U p ⊂ Ω; (ii) If j = 1, then φ p (U p ∩ Ω) = B 1 × (0, 1) or φ p (U p ∩ Ω) = B 1 × (B 1 {0}); (iii) If j = 0, then φ p (U p ∩ Ω) = { (r cos θ, r sin θ), with r ∈ (0, 1), θ ∈ I p }
for some finite union I p of open intervals in S 1 . For p ∈ Ω, we let j p the largest j for which p satisfies one of the above properties.

These are (essentially) the corner domains in [START_REF] Dauge | Elliptic boundary value problems on corner domains[END_REF]. The definition above was generalized to arbitrary dimensions in [START_REF] Bacuta | Interface and mixed boundary value problems on n-dimensional polyhedral domains[END_REF]. See also [START_REF] Kozlov | Spectral problems associated with corner singularities of solutions to elliptic equations[END_REF][START_REF] Li | Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains[END_REF][START_REF] Mazzucato | Well-posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks[END_REF][START_REF] Nazarov | Elliptic problems in domains with piecewise smooth boundaries[END_REF]. The second case in (ii) corresponds to cracks in the domain. We continue with some remarks. Remark 3.2. We notice that in the two cases (i) and (iii) of Definition 3.1 (j = 2 and j = 0), the spaces φ p (U p ) = B j × B 2-j will be the same (up to a canonical diffeomorphism), but the spaces φ p (U p ∩ Ω) will not be diffeomorphic. Remark 3.3. Let Ω be a curvilinear polygonal domain and p ∈ Ω. Then p satisfies the conditions of the definition for exactly one value of j, except the case when p is on a smooth part of the boundary, when a choice of j = 1 or j = 0 is possible. This is the case exactly when j p = 1. If j = 0 is chosen, then I p is half a circle.

Remark 3.4. The set V g := {p ∈ Ω| j p = 0} is finite and is contained in the boundary of Ω. It is the set of geometric vertices.

Let us choose for each point p ∈ Ω a value j = i p that satisfies the conditions of the definition. If j p = 1, we choose i p = j p = 1, except possibly for finitely many points p ∈ Ω. These points will be called artificial vertices. The set of all vertices (geometric and artificial) is finite, which will be denoted by V, and will be fixed in what follows.

We assume that all points where the boundary conditions change are in V. We also fix the resulting polar coordinates r • φ p and θ • φ p on U p , for all p ∈ V.

Definition 3.5. The coordinate charts φ p : U p → B j ×B 2-j of Definition 3.1 that were chosen such that j = i p are called the defining coordinate charts of the curvilinear polygonal domain Ω. (Recall that j = i p = 0 if, and only if p ∈ V.) Remark 3.6. Artificial vertices are useful, for instance, in the case when we have a change in boundary conditions or if there are point singularities in the coefficients, see [START_REF] Li | Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains[END_REF][START_REF] Li | Analysis of a modified Schrödinger operator in 2D: regularity, index, and FEM[END_REF] and the references therein. The right framework is, of course, that of a stratified space [START_REF] Bacuta | Interface and mixed boundary value problems on n-dimensional polyhedral domains[END_REF], with j p denoting the dimension of the stratum to which p belongs, but we do not need this in the simple case at hand. Remark 3.7. It follows from Definition 3.1 that if Ω is a curvilinear polygonal domain, then the set ∂Ω V is the union of finitely many smooth, open curves e j : (-1, 1) → ∂Ω. The curves e j have as image the open edges of ∂Ω and we shall sometimes identify e j with its image. The curves e j are disjoint and have no self-intersections. The closure of the image of e j , for any j, is called a closed edge. Thus, the vertices are not contained in the open edges (but they are, of course, contained in the closed edges). Our assumption that all points where the boundary conditions change are in V implies that ∂ D Ω consists of a union of closed edges of Ω.

3.2.

Equivalent definitions of weighted spaces. In this section, we discuss some equivalent definitions of weighted Sobolev spaces. We adapt to our setting the results in [START_REF] Ammann | Sobolev spaces on Lie manifolds and regularity for polyhedral domains[END_REF], to which we refer for more details.

We shall fix, from now on, a finite set of defining coordinate charts

φ k = φ p k , for some p k ∈ Ω, 1 ≤ k ≤ N , so that U k := U p k , 1 ≤ k ≤ N ,
defines a finite covering of Ω. Thus, for p = p k such that j p = 0, the coordinates are (x, y) ∈ R 2 . Otherwise, these coordinates will be denoted by (r, θ) ∈ (0, 1) × S 1 . We may relabel these points such that p k is a vertex if, and only if, 1 ≤ k ≤ N 0 . We then have the following alternative definition of the weighted Sobolev spaces K m a (Ω). We denote [START_REF] Costabel | Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation[END_REF] X k u := ∂ x u and Y k u := ∂ y u , for N 0 < k ≤ N , in the coordinate system defined by φ k = φ p k = (x, y) ∈ R 2 that corresponds to one of the chosen points p k (recall that, for N 0 < k ≤ N , p k is not a vertex). If, however, p k is a vertex, then we let (24)

X k u := r∂ r u and Y k u := ∂ θ u , for 1 ≤ k ≤ N 0 ,
in the (polar) coordinate system defined by φ k = (r, θ) ∈ (0, 1) × S 1 . Note the appearance of r in front of ∂ r ! Recall that r Ω equals the distance to the vertices close to the vertices. Thus r Ω = r close to the vertex of a straight angle.

Remark 3.8. Assuming that the coefficients of p β are locally Lipschitz, we can express the differential operator r 2 Ω p β in any of the coordinate systems φ k : 23) and [START_REF] Dauge | Opérateur de Stokes dans des espaces de Sobolev à poids sur des domaines anguleux[END_REF].

U k → R 2 . That means that, for each 1 ≤ k ≤ N , we can find coefficients c, c 1 , c 2 , c 11 , c 12 , c 22 such that (25) p β u = (c 11 X 2 k + c 12 X k Y k + c 22 Y 2 k + c 1 X k + c 2 Y k + c)u on U k , with the vector fields X k and Y k introduced in Equations (
For each open subset U ⊂ Ω, let us denote

(26) u 2 K m a (U ) := |α|≤m r |α|-a Ω ∂ α u 2 L 2 (U ) . Thus, if U = Ω, u K m a (U ) = u K m a (Ω)
is simply the norm on K m a (Ω). Note that the weight r Ω is not intrinsic to the set U , but depends on Ω, which is nevertheless not indicated in the notation u 2 K m a (U ) , in order not to overburden it. We define the spaces W m,∞ (U ) similarly as in [START_REF] Antoine | Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential. Part II: Discretization and numerical results[END_REF] with the same weight r Ω . Let U k := U p k . PROPOSTION 3.9. Let u : Ω → C be a measurable function and U ⊂ Ω be an open subset. We have that u ∈ K m a (U ) if, and only if, r -a

Ω X i k Y j k u ∈ L 2 (U ∩ U k ), for all 1 ≤ k ≤ N and all i + j ≤ m (recall that U k = U p k ). Moreover, the K m a (U )-norm is equivalent to the norm |||u||| U := N k=1 i+j≤m r -a Ω X i k Y j k u L 2 (U ∩U k ) .
Proof. This follows right away from the definition of the K m a (U )-norm. Indeed, away from the vertices, both the ||| • ||| U -norm and the K m a -norm coincide with the usual H m -norm. On the other hand, near a vertex, or more generally on an angle Ξ := {(r, θ)| α < θ < β}, both norms are given by i+j≤m r -a (r∂ r ) i ∂ j θ u L 2 (Ξ) . For the K m a (U )-norm this is seen by writing ∂ x and ∂ y in polar coordinates, more precisely, from [START_REF] Fabes | Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains[END_REF] r∂ x = (cos θ)r∂ r -(sin θ)∂ θ and r∂ y = (sin θ)r∂ r + (cos θ)∂ θ .

See [START_REF] Ammann | Weighted Sobolev spaces and regularity for polyhedral domains[END_REF][START_REF] Li | Analysis of a modified Schrödinger operator in 2D: regularity, index, and FEM[END_REF] for more details.

We finally have the following corollary.

COROLLARY 3.10. The norm u K m+1 a (Ω) is equivalent to the norm

|||u||| := u K m a (Ω) + N k=1 X k u K m a (U k ) + Y k u K m a (U k ) .
Proof. In the definition of |||u||| , Proposition 3.9, with m replaced by m + 1, we collect all the terms with i + j ≤ m and notice that they give a norm equivalent to the norm for K m a . The rest of the terms will contain at least one differential X k or one differential Y k and thus are of the form r -a

Ω X i k Y j k Y k u L 2 (Ω) or r -a Ω X i k Y j k X k u L 2 (Ω) , i + j ≤ m, since the differential operators X k and Y k commute on U k .
3.3. The differential operators. We include in this subsection the definition of our differential operators and three needed intermediate results (lemmas).

We introduce now our set of coefficients. Recall the norm β Zm introduced in Equation ( 6) and let [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] Z

m := { β = (a ij , b i , c), β Zm < ∞ } .
Note that for example, the Schrödinger operator -∆ + r -2 is an operator of the form P β for suitable β ∈ Z m .

Below, we shall often use inequalities of the form A ≤ CB, where A and B are expressions involving u and β and C ∈ R. We shall say that C is an admissible bound if it does not depend on u and β, and then we shall write A ≤ c B.

∈ W m-1,∞ (U k ). Moreover, c W m-1,∞ (U k ) + c 1 W m-1,∞ (U k ) + . . . + c 22 W m-1,∞ (U k ) ≤ c β Zm . If p β is moreover uniformly strongly elliptic, then |c -1 22 | ≤ c γ -1 use (β) on U k . Proof.
We first notice that since m ≥ 1, we can convert our operator to a nondivergence form operator. Indeed, one can simply replace a term of the form ∂ i a∂ j u with a∂ i ∂ j u + (∂ i a)∂ j u, where u ∈ K m+1 a+1 (Ω) and r Ω ∂ i a ∈ W m-1,∞ (Ω). We deal similarly with the terms of the form ∂ i (b i u). This accounts for the loss of one derivative in the regularity of the coefficients of c, . . . , c 22 .

We need to show that the coefficients c, . . . , c 22 are in W m-1,∞ (Ω)(U k ) and that they have the indicated bounds. To this end, we consider the two possible cases: when U k contains no vertices of Ω (equivalently, if k > N 0 ) and the case when U k is centered at a vertex (equivalently, if k ≤ N 0 ).

If k > N 0 , then the coefficients c, . . . , c 22 can be expressed using the coordinate chart φ k = φ p k of Definition 3.1 and its derivatives linearly in terms of the coefficients β on the closure of U k . Since there is a finite number of such neighborhoods and φ k and its derivatives are bounded on the closure of U k , the bound for the coefficients c, . . . , c 22 in terms of β Zm on U k follows using a compactness argument. In particular, the bound |c -1

22 | ≤ c γ -1 use (β) follows from the uniform ellipticity of p β on U k .

If, on the other hand, k ≤ N 0 (that is, U k is centered at a vertex). Let us concentrate on the highest order terms, for simplicity. We then have, up to lower order terms (denoted l.o.t)

r 2 ∂ 2 x = (cos θ) 2 (r∂ r ) 2 -2(sin θ cos θ)r∂ r ∂ θ + (sin θ) 2 ∂ 2 θ + l.o.t. r 2 ∂ x ∂ y = (sin θ cos θ)(r∂ r ) 2 + (cos 2 θ -sin 2 θ)r∂ r ∂ θ + (sin θ cos θ)∂ 2 θ + l.o.t. r 2 ∂ 2 y = (sin θ) 2 (r∂ r ) 2 + 2(sin θ cos θ)r∂ r ∂ θ + (cos θ) 2 ∂ 2 θ + l.o.t.
The bound on the coefficients c, . . . , c 22 follows since sin θ and cos θ are in W m,∞ (U k ) for all m. This gives also that c 22 = a 11 cos 2 θ + 2a

12 cos θ sin θ + a 22 sin 2 θ ≥ γ use (β) for the coefficient c 22 of Y 2 k = ∂ 2 θ . (Thus |c -1 22 | ≤ γ -1 use (β) on U k , for k ≤ N 0 .
) For instance, for the Laplacian in polar coordinates, we have

r 2 Ω ∆ = (r∂ r ) 2 + ∂ 2 θ = X 2 k + Y 2 k
in the neighborhood U k of the vertex p k of a straight angle.

The following lemma will be used in the proof of Theorem 4.4 and explains some of the calculations there. LEMMA 3.12. For two functions b and c, we have Proof. This is a direct calculation. Indeed, the first two relations are based on the rule

(i) bc K m a (Ω) ≤ c b W m,∞ (Ω) c K m a (Ω) . (ii) bc W m,∞ (Ω) ≤ c b W m,∞ (Ω) c W m,∞ (Ω) , so W m,∞ (Ω) is an algebra. (iii) If b ∈ W m,∞ (Ω) and b -1 ∈ L ∞ (Ω), then b -1 ∈ W m,∞ (Ω) and b -1 W m,∞ (Ω) ≤ c C b -1 m+1 L ∞ (Ω) b m W m,∞ (
∂ α (bc) = β≤α α β ∂ β b ∂ α-β c. The last one is obtained from the relation ∂ α (b -1 ) = b -1-|α| Q, where Q = Q(b, ∂ 1 b, ∂ 2 b, . . . , ∂ α b) is a polynomial of degree |α| in all derivatives ∂ β b, with 0 ≤ β ≤ α, where each ∂ β u is considered of degree 1.

This relation is proved by induction on |α|.

For further reference, we shall need the following version of "Nirenberg's trick," (see, for instance, [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF][START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]). LEMMA 3.13. Let T : X → Y be a continuous, bijective operator between two Banach spaces X and Y . Let S X (t) and S Y (t) be two c 0 semi-groups of operators on X, respectively Y , with generators denoted by A X and, respectively, A Y . We assume that for any t > 0, there exists T t ∈ L(X; Y ) such that S Y (t)T = T t S X (t). Assume that t -1 (T t -T ) converges strongly as t → 0 to a bounded operator B. Then T maps bijectively the domain of A X to the domain of A Y and we have that

A X T -1 ξ = T -1 A Y ξ -BT -1 ξ , for all ξ in the domain of A Y . Consequently, A X T -1 ξ X ≤ T -1 A Y ξ Y + B T -1 ξ X .
Proof. We have that ξ ∈ X is in D(A X ), the domain of the generator A X of S X if, and only if, the limit A X ξ := lim t→0 t -1 S X (t) -1)ξ exists. The definition of T t gives

t -1 S Y (t) -1)T ξ = t -1 T t -T )S X (t)ξ + t -1 T S X (t) -1)ξ .
Since t -1 (T t -T )ζ → Bζ for all vectors ζ ∈ X and B : X → Y is bounded, we obtain that the limit lim t→0 t -1 S Y (t) -1)T ξ exists if, and only if, the limit lim t→0 t -1 S X (t) -1)ξ exists. This shows that T maps bijectively the domain of A X to the domain of A Y and that A Y T = B + T A X as unbounded operators with domain D(A X ). Multiplying by T -1 to the left and to the right gives the desired result.

One can use Lemma 3.13 as a regularity estimate.

Higher regularity in weighted Sobolev spaces

In this section, we prove our main result, Theorem 4.4. Theorem 1.1 is then an immediate consequence of this theorem and of Remark 4.3. Recall that r Ω : Ω → [0, ∞) denotes a continuous function, smooth and > 0 outside the vertices, such that r Ω is the distance to the vertices, close to the vertices. 4.1. The higher regularity problem. We now come back to the study of our mixed problem, as formulated in Equation [START_REF] Bacuta | Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains[END_REF]. We are interested in solutions with more regularity than the ones provided by the space V appearing in its weak formulation, Equation [START_REF] Bacuta | Improving the rate of convergence of high-order finite elements on polyhedra. II. Mesh refinements and interpolation[END_REF] or Equation [START_REF] Buffa | Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron[END_REF]. While for the weak formulation the classical Sobolev spaces suffice, the higher regularity is formulated in the framework of the weighted Sobolev spaces considered by Kondratiev [START_REF] Kondrat Ev | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF] and others, see also [START_REF] Costabel | Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation[END_REF][START_REF] Dauge | Opérateur de Stokes dans des espaces de Sobolev à poids sur des domaines anguleux[END_REF].

We assume from now on that V := {u ∈ K 1 1 (Ω), u = 0 on ∂ D Ω} and that it has the induced norm. We then introduce

V m (a) := K m+1 a+1 (Ω) ∩ {u| ∂ D Ω = 0} for m ∈ Z + = {0, 1, 2, . . .} and V - m (a) := K m-1 a-1 (Ω) ⊕ K m-1/2 a-1/2 (∂ N Ω) for m ∈ N = {1, 2, . . .}. In particular, V m (a) = K m+1 a+1 (Ω) ∩ r a Ω V . The spaces K m-1/2 a-1/2 (∂ N Ω), m ≥ 1
, are the spaces of traces of functions in K m a (Ω), in the sense that the restriction at the boundary defines a continuous, surjective map [START_REF] Ammann | Weighted Sobolev spaces and regularity for polyhedral domains[END_REF]. The space K m a (∂ N Ω) can be defined directly for m ∈ Z + in a manner completely analogous to the usual Kondratiev spaces. For non-integer regularity, they can be obtained by interpolation, [START_REF] Ammann | Sobolev spaces on Lie manifolds and regularity for polyhedral domains[END_REF][START_REF] Ammann | Weighted Sobolev spaces and regularity for polyhedral domains[END_REF].

K m a (Ω) → K m-1/2 a-1/2 (∂ N Ω)
We recall that for all m ∈ Z + and a ∈ R, the differentiation defines continuous maps ∂ j : K m a (Ω) → K m-1 a-1 (Ω). In the same way, the combination of the normal derivative at the boundary (∂

β ν v) := d i=1 ν i ( d j=1 a ij ∂ j v + b d+i v and restriction at the boundary define a continuous, surjective map ∂ β ν : K m a (Ω) → K m-3/2 a-3/2 (∂ N Ω), m ≥ 2.
LEMMA 4.1. We have continuous maps

P β (m, a) := (p β , ∂ β ν ) : V m (a) → V - m (a) , m ≥ 1 , P β (m, a)(u) = ij ∂ i (a ij ∂ j u) + i b i ∂ i u + cu, ij ν i a ij ∂ j u| ∂ N Ω .
Therefore the operators P β (m, a), m ∈ N, a ∈ R, are given by the same formula (but have different domains and ranges).

Remark 4.2. Let us assume for this remark that a = 0 and discuss this case in more detail. If ∂ N Ω contains no adjacent edges, then the Hardy inequality [START_REF] Bacuta | Improving the rate of convergence of high-order finite elements on polyhedra. I. A priori estimates[END_REF][START_REF] Kozlov | Spectral problems associated with corner singularities of solutions to elliptic equations[END_REF] shows that the natural inclusion ( 29)

K 1 1 (Ω) ∩ {u| ∂ D Ω = 0} → H 1 D (Ω) := H 1 (Ω) ∩ {u| ∂ D Ω = 0} is an isomorphism (that
is, it is continuous with continuous inverse). We thus consider V := V 0 (0) in general (for all ∂ N Ω, that is, even if it contains adjacent Neumann edges). For symmetry, we also let V - 0 (0) := V * and (30)

P β (0, 0) := P β : V 0 (0) = V → V - 0 (0)
:= V * , which is, of course, nothing but the operator studied before.

We then have

V m+1 (0) ⊂ V m (0) and V - m+1 (0) ⊂ V - m (0)
for all m ≥ 0 . This is trivially true for m > 0. For m = 0, in which case we need to construct the natural inclusion Φ :

V - m (0) → V - 0 (0), m ≥ 1. The map Φ associates to (f, h) ∈ V - m (0) := K m-1 -1 (Ω) ⊕ K m-1/2
-1/2 (∂ N Ω) the linear functional F := Φ(f, h) on V , F ∈ V * defined by the formula ( 31)

F (v) = Φ(f, h)(v) := Ω f v dx + ∂ N Ω h v dS ,
where dx is the volume element on Ω and dS is the surface element on ∂Ω. With this definition of the inclusion Φ : V - m (0) → V - 0 (0) := V * , we obtain that P β (m, 0) is the restriction of P β (0, 0) to V m (0). In other words, we have the commutative diagram [START_REF] Kohr | Layer potential analysis for pseudodifferential matrix operators in Lipschitz domains on compact Riemannian manifolds: applications to pseudodifferential Brinkman operators[END_REF] V m (0)

P β (m,0) -----→ V - m (0)     V 0 (0) := V P β (0,0):=P β --------→ V - 0 (
0) with the operators P β introduced in Lemma 4.1 and in Equation [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF].

See also Remark 2.1. We now return to the general case a ∈ R.

Remark 4.3. We then have

V m (a) = r a Ω V m (0) for m ≥ 0 and V - m (a) = r a Ω V - m (0) for m > 0 . We then let V - 0 (a) := r a Ω V - 0 (0) = r a Ω V * .
By symmetry, we obtain [START_REF] Kondrat Ev | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF] V m+1 (a) ⊂ V m (a) and V - m+1 (a) ⊂ V - m (a) for all m ≥ 0 , in general (for all a). In fact, the relation between the spaces above for different values of a allows us to reduce to the case a = 0 since, if β ∈ Z m , then there exists β(a) ∈ Z m such that (34) P β (m, a) = r a Ω P β(a) (m, 0)r -a Ω , m ≥ 1 . This can be seen from r a ∂ j (r -a u) = ∂ j u -r -1 (ax j r -1 )u and r -1 x j ∈ W m,∞ (Ω) for all m, which then gives [START_REF] Li | Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains[END_REF]

r a ∂ j ∂ k (r -a u) = ∂ j ∂ k u + r -1 φ∂ k u + ∂ j (r -1 ψu) + r -2 φψu = ∂ j ∂ k u + r -1 φ∂ k u + r -1 ψ∂ j + r -2 (r∂ j ψ + r -1 x j ψ + φψ)u ,
where φ := -ar -1 x j and ψ := -ar -1 x k . In particular, [START_REF] Li | Analysis of a modified Schrödinger operator in 2D: regularity, index, and FEM[END_REF] β(a) = β + aβ 1 + a 2 β 2 with β 1 , β 2 ∈ Z m depending linearly and continuously on β ∈ Z m . (This explains why it is crucial to consider coefficients in weighted spaces of the form W m,∞ (Ω) as well as in terms of the form ∂ i (b i u) in the definition of p β .) We use Equation ( 34) to define P β (0, a) for all a. Of course, P β (0, 0

) = P β : V → V * .
As mentioned in the introduction, an less common example for Theorems 1.1 and 4.4 is the Schrödinger operator H := -∆ + cr -2 Ω , c > 0, on Ω with pure Neumann boundary conditions, hence V := K 1 1 (Ω) in this example. See also Theorem 5.4. Our higher regularity problem is then to establish conditions for P β (m, a) to be an isomorphism, which is achieved in Theorem 4.4.

4.2.

Extension of Theorem 1.1 and its proof. For its proof, it will be convenient to extend the differential operators X k , Y k from U k to the whole domain Ω. We choose these extensions so that (i) If p k is a vertex, then all X j , Y j , j = k, vanish close to p k .

(ii) For all k, X k (regarded as a vector field) is tangent to all edges (if X k vanishes at a point on an edge, it is considered to be tangent to the edge at that point). Recall that ρ(P ) := inf v =0 (P v, v)/ v V , for any linear map P : V → V * , that ρ(β) := ρ(P β ), and that γ -1 use (β) ≤ ρ(β) -1 , by Corollary 2.10. Also, recall that β(a) is given by Equation (34) THEOREM 4.4. Let Ω ⊂ R 2 be a bounded, curvilinear polygonal domain and m ∈ Z + . There exist C m > 0 and N m ≥ 0 such that, if β = (a ij , b i , c) ∈ Z m and

P β : V → V * is strongly coercive, then P β (m, 0) : V m (0) → V - m (0) is invertible and P β (m, 0) -1 L(V - m ;Vm) ≤ C m ρ(β) -Nm-1 β Nm
Zm . Proof. Since the statement is for a = 0, we shall write V m (0) = V m and V - m (0) = V - m . We shall also denote (P β ) -1 m := (P β ) -1 L(V - m ;Vm) . For m = 0, we can just take C 0 = 1 and N 0 = 0 and then the result reduces to the Lax-Milgram Lemma (see Lemma 2.5). In general, we adapt to our setting the classical method based on finite differences (see for example [START_REF] Costabel | Corner singularities and analytic regularity for linear elliptic systems. part i: Smooth domains[END_REF][START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF][START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF]), which was used in similar settings in [START_REF] Bacuta | Improving the rate of convergence of high-order finite elements on polyhedra. I. A priori estimates[END_REF][START_REF] Bacuta | Improving the rate of convergence of high-order finite elements on polyhedra. II. Mesh refinements and interpolation[END_REF][START_REF] Chkifa | Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs[END_REF][START_REF] Li | Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric finite element method[END_REF][START_REF] Nistor | High-order Galerkin approximations for parametric second-order elliptic partial differential equations[END_REF] and many other papers. We thus give a summary of the argument. For simplicity, we drop Ω from the notation of the norms. In this proof, as throughout the paper, C is a parameter that is independent of β or F , and hence it depends only on Ω, ∂ N Ω, m, and the choice of the vector fields X k and Y k (and of their initial domains U k ), but not on F , u, β. We shall usually write

A ≤ c B instead of A ≤ CB, if C is such a bound.
Using the definition of R(β), the induction estimate of Equation [START_REF] Li | A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential[END_REF], and the relation β Zm P β (m -1, 0) -1 ≥ 1/C of Equation [START_REF] Li | Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric finite element method[END_REF], we obtain (41)

X k u K m 1 ≤ c R(β) 2Nm-1+1 ρ(β) F V - m .
We now turn to the study of the terms Y k u K m 1 , for which we need to use the strong ellipticity of P β (as in the classical methods [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF][START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF]) together with Lemmas 3.11 and 3.12. First of all, Lemma 3.11 provides us with the decomposition

c k Y 2 k u = r 2 Ω P β u -Q k u, where c k ∈ W m-1,∞ (U k ) and Q k is a sum of differential operators of the form Y k X k and X 2
k and lower order differential operators generated by X k and Y k with coefficients in W m-1,∞ (U k ). This gives using first the general form of the

• K m 1 (U k ) -norm (recall that X k and Y k commute on U k ) (42) Y k u K m 1 (U k ) ≤ c Y k u K m-1 1 + X k Y k u K m-1 1 (U k ) + Y 2 k u K m-1 1 (U k ) ≤ c u K m 1 + Y k X k u K m-1 1 (U k ) + Y 2 k u K m-1 1 (U k ) ≤ c u K m 1 + X k u K m 1 + c -1 k (r 2 Ω p β -Q k )u K m-1 1 (U k ) .
The first term in the last line of Equation ( 42) is estimated by the induction hypothesis in Equation ( 40). The second one is estimated in Equation [START_REF] Mitrea | On the Besov regularity of conformal maps and layer potentials on nonsmooth domains[END_REF]. To estimate the third term, we obtain directly from Lemma 3.11 the following

(1) each c k ∈ W m-1,∞ (U k ) is bounded in terms of β Zm , (2)
the coefficients of X 2 k , X k Y k , X k , and Y k and the free term of Q k (which is no longer in divergence form) are in W m-1,∞ (U k ) and are also bounded in terms of β Zm ,

(3) c -1 k L ∞ ≤ c γ -1 use (β) ≤ c ρ(β) -1
, by the uniform strong ellipticity of p β . Hence

(43) c -1 k W m-1,∞ (U k ) ≤ c c -1 k m L ∞ (U k ) c k m-1 W m-1,∞ (U k ) ≤ c ρ(β) -m β m-1 W m,∞ = ρ(β) -1 R(β) m-1 ,
where the first inequality is by Lemma 3.12(iii).

We have, successively

(44) r 2 Ω p β u K m-1 1 (U k ) ≤ c p β u K m-1 -1 (U k ) ≤ c p β u K m-1 -1 ≤ c F V - m-1 . Similarly, let n Q be the W m-1,∞ (U k ) norm of the coefficients of Q k , then n Q ≤ c β W m,∞ and hence (45) Q k u K m-1 1 (U k ) ≤ c n Q X 2 k u K m-1 1 (U k ) + Y k X k u K m-1 1 (U k ) + X k u K m-1 1 (U k ) + Y k u K m-1 1 (U k ) + u K m-1 1 (U k ) ≤ c β Zm X k u K m 1 + u K m 1 ≤ c (R(β) 2Nm-1+2 + R(β) Nm-1+1 ) F V - m-1 ≤ c R(β) 2Nm-1+2 F V - m-1
, where we have used also Equations ( 40) and [START_REF] Mitrea | On the Besov regularity of conformal maps and layer potentials on nonsmooth domains[END_REF]. Consequently,

(46) c -1 k (r 2 Ω p β -Q k )u K m-1 1 (U k ) ≤ c c -1 k W m-1,∞ r 2 Ω p β u -Q k u K m-1 1 (U k ) ≤ c R(β) m-1 ρ(β) 1 + R(β) 2Nm-1+2 ) F V - m-1 ≤ c R(β) 2Nm-1+m+1 ρ(β) F V - m-1
.

Substituting back into Equation ( 42) the estimates of Equations ( 40), [START_REF] Mitrea | On the Besov regularity of conformal maps and layer potentials on nonsmooth domains[END_REF], and [START_REF] Rukavishnikov | On the isomorphic mapping of weighted spaces by an elliptic operator with degeneration on the domain boundary[END_REF] for the respective terms, and then using Equation (37), we obtain

(47) Y k u K m 1 (U k ) ≤ c R(β) 2Nm-1+m+1 ρ(β) F V - m .
In a completely analogous manner, substituting back into Equation ( 39) the estimates of Equations ( 40), [START_REF] Mitrea | On the Besov regularity of conformal maps and layer potentials on nonsmooth domains[END_REF], and (47), we obtain

(48) u K m+1 1 ≤ c R(β) 2Nm-1+m+1 ρ(β) F V - m .
In all the statements above, saying v Z < ∞ for some Banach space Z means, implicitly, that v ∈ Z. We thus have that u ∈ K m+1 1 and that it satisfies the required estimate with N m = 2N m-1 + m + 1. The proof is complete.

We now record the obvious modification needed to deal with the additional parameter a. Remark 4.6. Remark 4.3 gives that there exists γ 1 such that ρ(β(a)) ≥ ρ(β)γ 1 |a| β Z0 , for a in a bounded interval. Moreover, an induction argument gives that N m = 2 m+2 -m -3 ≥ 0 in two dimensions. We ignore if this is true in higher dimensions as well.

See also [START_REF] Antoine | Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential. Part II: Discretization and numerical results[END_REF][START_REF] Ciarlet | The Fourier singular complement method for the Poisson problem[END_REF][START_REF] He | The highest superconvergence of the tri-linear element for Schrödinger operator with singularity[END_REF][START_REF] Karami | Singularities of stationary solutions to the Vlasov-Poisson system in a polygon[END_REF][START_REF] Kohr | Layer potential analysis for pseudodifferential matrix operators in Lipschitz domains on compact Riemannian manifolds: applications to pseudodifferential Brinkman operators[END_REF][START_REF] Li | A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential[END_REF] for extensions and related results.

Extensions and applications

We conclude with a few corollaries and extensions of our previous results. For simplicity, we formulate them only in the case a = 0, since Remark 4.3 allows us to reduce to the case a = 0. Throughout this section, we continue to assume that β = (a ij , b i , c) ∈ Z m and that Ω is a bounded, curvilinear polygonal domain with ∂ D Ω nonempty. 5.1. Corollaries of Theorem 4.4. Recall that L(V ; V * ) c denotes the set of strongly coercive operators and that we regard Z ⊂ L(V ; V * ) with the induced topology. In particular, L(V ; V * ) c ∩ Z denotes the set of coefficients that yield a strongly coercive operator.

(P β ) -1 F Vm ≤ C m β Nm Zm ρ(β) Nm+1 F V - m , (∀)β ∈ U, F ∈ V - m .
Proof. Recall that L(V ; V * ) c is open in L(V ; V * ) and that the map Z m → L(V ; V * ) c is continuous. Therefore U := L(V ; V * ) c ∩ Z m is open in Z m . Next we proceed as in Lemma 2.5 using that the map Φ : U × V - m → V m , Φ(β, F ) := (P β ) -1 F is the composition of the maps

U × V - m (β, F ) → (P β , F ) ∈ L inv (V m ; V - m ) × V - m , L inv (V m ; V - m ) × V - m (P, F ) → (P -1 , F ) ∈ L(V - m ; V m ) × V - m , and L(V - m ; V m ) × V - m (Q, F ) → QF ∈ V m .
The first of these three maps is well defined by Theorem 4.4. Since it is linear, it is also analytic. The other two maps are analytic by Lemma 2.2. Since the composition of analytic functions is analytic, the result follows.

The following result is useful in approximating solutions of parametric problems when one has uniform measures. (Note however that the estimates in Theorem 4.4 provide errors that are integrable with respect to lognormal measures.) We keep the notation in the last corollary. In particular, if the functions β(y) W m,∞ (Ω) and F (y) V - m are bounded and there exists c > 0 such that ρ(β(y)) > c, then (P β(y) ) -1 F (y) is a bounded analytic function.

Proof. The composition of two analytic functions is analytic. The first part is therefore an immediate consequence of the first part of Corollary 5.1. The second part follows also from Corollary 5.1.

The method used to obtain analytic dependence of the solution in terms of coefficients can be extended to other settings. m with W m-1 , and by using N m in the bounds for the norm. We thank Markus Hansen and Christoph Schwab for their input related to this remark. 5.2. General domains with conical points. The same argument as in the proof of Theorem 4.4 gives a proof of a similar result on general domains with conical points. In the neighborhood of a conical point, the domain is of the form Ω = {rx , 0 < r < 1, x ∈ ω}, where ω ⊂ S n-1 is a smooth domain on the unit sphere S n-1 . The main difference is that we will need to additionally straighten the boundary of ω.

Dirichlet and Neumann boundary conditions.

We conclude this paper by an application of Theorem 4.4 to estimates for Schrödinger operators. We note that the following result applies to arbitrary mixed boundary conditions (including pure Neumann).

LEMMA 3 . 11 .

 311 Let β = (a ij , b i , c) ∈ Z m , m ≥ 1,and let us express p β as in Remark 3.8. Then c, c 1 , c 2 , c 11 , c 12 , c 22

  Ω) . The parameter C in ≤ c depends only on m and Ω.

COROLLARY 4 . 5 .

 45 Let Ω ⊂ R 2 be a bounded, curvilinear polygonal domain, a ∈ R, m ∈ Z + . There exist C m > 0 and N m ≥ 0 such that, if β = (a ij , b i , c) ∈ Z m and P β(a) : V → V * is strongly coercive, then P β (m, a) : V m (a) → V - m (a) is invertible andP β (m, a) -1 L(V - m ;Vm) ≤ C m ρ(β(a)) -Nm-1 β(a) Nm Zm .Proof. Because β(a) depends analytically on β, in view of Remark 4.3 and of the relation in Equation[START_REF] Kozlov | Spectral problems associated with corner singularities of solutions to elliptic equations[END_REF], we can just substitute β(a) for β in Theorem 4.4.

COROLLARY 5 . 1 .

 51 Let U := L(V ; V * ) c ∩ Z m . Then U is an open subset of Z m and the map U × V - m (β, F ) → (P β ) -1 F ∈ V m is analytic. Moreover, there exist C m > 0 and N m ≥ 0 such that

COROLLARY 5 . 2 .

 52 Let Y be a Banach space and let U ⊂ Y be an open subset. Let F : U → V - m and β : U → L(V ; V * ) c ∩ W m,∞ (Ω) be analytic functions. Then U y → (P β(y) ) -1 F (y) ∈ V m is analytic and(P β(y) ) -1 F (y) Vm ≤ C m β(y) Nm Zm ρ(β(y)) Nm+1 F (y) V - m .

Remark 5 . 3 .

 53 Let us assume the following: (i) We are given continuously embedded Banach spacesW m+1 D ⊂ V ⊂ H 1 (Ω), W m-1 ⊂ V * ,and Z ⊂ Z m satisfying the following properties: (ii) For any β ∈ Z, the operatorP β defines continuous maps V → V * and W m+1 D → W m-1 . (iii) P β L(W m+1 D ; W m-1 ) ≤ c β Z and P β L(V ;V * ) ≤ c β Z .(iv) If β ∈ Z and P β : V → V * is strongly coercive, then the map (P β ) -1 :V * → V maps W m-1 to W m+1 Dcontinuously and there exists a continuous, increasing functionN m : R 2 + → R + such that (P β ) -1 L( W m-1 ;W m+1 D ) ≤ N m ρ(β) -1 , β Z .Then our previous results (in particular, Corollaries 5.1 and 5.2) extend to the new setting by replacing W m,∞ (Ω) with Z, V m with W m+1 D , V -

THEOREM 5 . 4 . 2 Ω

 542 Let P β u = -d ij=1 ∂ i (a ij ∂ j u) + c ru, c ≥ 0, be a strongly elliptic operator (so b i = 0). In case p ∈ V ⊂ ∂Ω is a vertex that belongs to two adjacent Neumann edges, we assume that c(p) > 0. Then P β is strongly coercive on V := {u ∈ K 1 1 (Ω), u = 0 on ∂ D Ω}. Moreover, (49)P β : K m+1 a+1 (Ω) ∩ {u| ∂ D Ω = 0} → K m-1 a-1 (Ω) ⊕ K m-1/2 a-1/2 (∂ N Ω) is an isomorphism and its inverse has norm(P β ) -1 ≤ C m ρ(β) -Nm-1 β Nm , |a| ≤ 1 ,with C m and N m as in Theorem 4.4 (hence independent of β and F ).

Let us notice that P β (m, 0) P β (m, 0) -1 ≥ P β (m, 0)P β (m, 0) -1 ≥ 1. Since P β (m, 0) m ≤ c β Zm , we have that β Zm (P β ) -1 m ≥ 1/C > 0 . When m = 0, we also have ρ(β) -1 ≥ (P β ) -1 =: (P β ) -1 0 , and hence ( 37)

0) is invertible and to obtain estimates on (P β ) -1 m := P β (m, 0) , we proceed by induction on m. As we have explained above, for m = 0, this has already been proved. We thus assume that P β (m -1, 0) is invertible and that it satisfies the required estimate, which we write as [START_REF] Li | A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential[END_REF] (

Let F ∈ V - m be arbitrary but fixed. We know by the induction hypothesis that

(recall that we drop Ω from the notation of our norms).

First of all, by Corollary 3.10, it is enough to estimate

and the first term on the right hand side is estimated by induction on m by ( 40)

Let us estimate now the remaining terms in the sum appearing on the right hand side of the inequality [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF]. Note that these terms are norms that are computed on smaller subsets U k ⊂ Ω. First, since X k is tangent to all edges of Ω, it integrates to a one parameter family of diffeomorphisms of Ω, and hence to strongly continuous one-parameter groups of continuous operators on X := V m-1 and Y := V - m-1 , due to the particular form of boundary conditions used to define these spaces. Let us denote by S X (t) : X → X and S Y (t) : Y → Y , t ∈ R, the operators defining these one-parameter groups of operators. We have that

and hence β ∈ Z m-1 by 3.11. Therefore B : X := V m-1 → Y := V - m-1 is bounded by Lemma 4.1. The assumptions of Lemma 3.13 are therefore satisfied. Moreover, B ≤ c β Zm-1 ≤ c β Zm , which allows us to conclude that

, which gives