Martin Traizet
email: martin.traizet@lmpt.univ-tours.fr

Exploring the space of embedded minimal surfaces of finite total curvature

we investigate, both numerically and mathematically, several questions about embedded minimal surfaces of finite total curvature in euclidean space. We also describe how a theoretical construction can be implemented numerically to procude pictures of such surfaces.

Introduction

This paper investigates the space of complete, embedded minimal surfaces of finite total curvature (FTC) in euclidean space. For a long time the only known examples were the plane and the catenoid. In 1982, C. Costa discovered a genus one example with 3 ends [START_REF] Costa | Example of a complete minimal immersion in R 3 of genus one and three embedded ends[END_REF]. D. Hoffman and W. Meeks proved that the Costa surface is embedded and constructed, for each genus ≥ 1, a one-parameter family of embedded FTC minimal surfaces with three ends [START_REF] Hoffman | Embedded minimal surfaces of finite topology[END_REF]. Since then, several general methods to construct examples have been proposed [START_REF] Weber | Teichmuller theory and handle addition for minimal surfaces[END_REF], [START_REF] Kapouleas | Complete embedded minimal surfaces of finite total curvature[END_REF].

In [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF], the author developed a construction which does not rely on symmetries as do the previous ones. The input data for this construction is a finite collection of points in the complex planes (the configuration), satisfying a set of algebraic equations (the balancing condition). The output is a family of embedded FTC minimal surfaces, whose geometry can be described quite explicitely from the configuration. Roughly speaking, the surface is made of planes with small catenoidal necks between them, and the configuration gives the position of the necks, see figure 1.

There are two aspects of this construction which can be explored numerically. The first one is the search for balanced configurations. Most of the interesting examples I have found were discovered numerically. Sometimes they can be fully understood mathematically. They provide numerical, or mathematical answers to several interesting questions about embedded FTC minimal surfaces.

The other aspect is pictures. How can we actually compute these minimal surfaces ? Theoretically, they are constructed using Weierstrass Representation. The Riemann surface is defined by opening nodes, an explicit algebraic construction. Then three holomorphic 1-forms φ 1 , φ 2 , φ 3 (the Weierstrass data) are defined abstractly by prescribing periods. The question is : how can we compute numerically these holomorphic 1-forms ? We provide a constructive answer to this problem when all parts of the noded Riemann surface have genus zero, and explain how this can be used to implement the construction in [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF].

Credits : the Maple software was used to carry all numerical computations described in this paper. The domains were triangulated using Mathlab. I would like to thank C. Georgelin for her help in using this package.

Balanced configurations

A configuration is a collection of points {p k,i : 1 ≤ k ≤ M, 1 ≤ i ≤ N k } in the complex plane, together with some positive real numbers c 1 , • • • , c M . These points are organized into layers : the points p k,1 , • • • , p k,N k form the k th layer. There are M layers, and N k points in the k th layer. We also say that the points p k,1 , • • • , p k,N k are the points at level k. The total number of points is N = N 1 + • • • + N M . The numbers c 1 , • • • , c M are called the neck-sizes. The type of the configuration is the sequence

N 1 , • • • , N M .
Given a configuration which is balanced and non-degenerate (we will explain these terms shortly), the output of Theorem 1 in [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF] is a one parameter family of FTC minimal surfaces {M s } 0<s<ε (ε small enough). These surfaces have genus N -M and have M + 1 embedded ends, asymptotic to half-catenoids or planes. They can be described geometrically as M + 1 horizontal planes with small catenoidal necks between them, see figure 1. There are M layers of necks and N k necks in the k th layer, whose positions are given by the points p k,1 , • • • , p k,N k . The necks in the k th layer have waist radius sc k . This geometric description only holds asymptotically when s → 0.

The planes are perturbed to have logarithmic growth at infinity : the surface has M + 1 catenoid type ends, whose logarithmic growths are as follows : let Q k = N k-1 c k-1 -N k c k (with the convention N 0 = N M +1 = 0). Then the logarithmic growth of the k th end is precisely sQ k for k = 1, • • • , M -1, and is asymptotically sQ k for k = M, M + 1 when s → 0. (When the logarithmic growth is zero, the end is asymptotic to a plane). The theorem also guarantees that provided

Q 1 ≤ • • • ≤ Q M -1 < Q M < Q M +1
, the surfaces M s are embedded (for s small enough). When this condition is satisfied, we say that the configuration is embedded (a rather clumsy, but convenient terminology).

Let me now explain the balancing condition. Let

F k,i = N k j=1 j =i 2c 2 k p k,i -p k,j - N k-1 j=1 c k c k-1 p k,i -p k-1,j - N k+1 j=1 c k c k+1 p k,i -p k+1,j
with the convention that N 0 = N M +1 = 0. Because of the analogy with 2D electrostatic forces, we call F k,i the force on p k,i . The point p k,i interacts with all other points in the same layer, and with the points in the layer directly below and above it. We require that the points in each layer are distinct, and are distinct from the points in the layer directly below and above it, so that F k,i is defined. (We say the configuration is non-singular).

The configuration is balanced provided all forces are zero :

∀k, ∀i, F k,i = 0.
These are N algebraic equations. These equations are not independent because one has k,i F k,i = 0. Moreover, a computation shows that

k,i p k,i F k,i = M k=1 n k (n k -1)c 2 k - M -1 k=1 n k n k+1 c k c k+1 . (1)
The right hand side only depends on the neck-sizes, and it must be equal to zero for balanced configurations to exist. Provided the neck-sizes satisfy this condition, we are left with N -2 equations to solve. The balancing condition is also invariant by translation and complex scaling of the points (transformations z → az + b). We may normalize the positions of two points, and are left with N -2 parameters.

We say the configuration is non-degenerate if the Jacobian matrix ∂F k,i /∂p ,j has complex rank N -2. This is the maximum rank it may have.

Basic example : Costa Hoffman Meeks

The simplest examples have M = 2 layers of necks, with N 1 = n ≥ 2 and N 2 = 1. The neck-sizes are c 1 = 1, c 2 = n -1. The configuration has dihedral symmetry of order n and is given by p 1,i = ω i and p 2,1 = 0, where ω = exp(2πi /n). It is non-degenerate, see details in [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF], proposition 1. The corresponding family of embedded FTC minimal surfaces is the Costa Hoffman Meeks family of genus n -1, or rather, the extreme part of this family. (I have no proof of this claim, except in the genus one case, by the classification in [START_REF] Costa | Uniqueness of minimal surfaces embedded in R 3 with total curvature 12π[END_REF]. All I really know is that the surfaces have the same symmetries and topology as the Costa Hoffman Meeks surfaces). These are also the only non-degenerate balanced configurations with 2 layers of necks, see [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF], proposition 1.

Dihedral configurations

In this section we investigate the following question :

What is the least genus for an embedded, FTC minimal surface with r ≥ 2 ends ?

N. Kapouleas has proven the existence of embedded, FTC minimal surfaces with any finite number of ends [START_REF] Kapouleas | Complete embedded minimal surfaces of finite total curvature[END_REF]. His examples are constructed by desingularization of a finite set of co-axial catenoids and horizontal planes. However, the genus of his examples is very large by construction (in fact, it seems hard to even estimate the genus). D. Hoffman and W. Meeks have conjectured that the answer to the above question is r -2. M. Weber and M. Wolf have constructed, for each r ≥ 4, a FTC minimal surface with genus r -2 and r ends [START_REF] Weber | Teichmuller theory and handle addition for minimal surfaces[END_REF]. However, they cannot mathematically prove that their examples are embedded, although numerical pictures seem to indicate that they are (at least for a few values of r !).

Our result lies in between : we can prove existence of embbeded FTC minimal surfaces with arbitrary number of ends, with an explicit formula for the genus, but the genus is not optimal.

The easiest way to compute balanced configurations with arbitrary number of layers M is to keep the symmetries of the Costa Hoffman Meeks configuration and increase the number of layers (see figure 3). Let M ≥ 2 be the number of layers. Take

N 1 = • • • = N M -1 = n and N M = 1,
where n ≥ 2 is some integer. We want the configuration to have dihedral symmetry of order n, so we set

p k,i = a k ω i , 1 ≤ k ≤ M -1, p M,1 = 0
where ω = exp(2πi /n) and a k is such that a n k ∈ R * . Equation (1) gives

M -1 k=1 n(n -1)c 2 k - M -2 k=1 n 2 c k c k+1 = nc M -1 c M . (2)
This determines c M as a function of the parameters c 1 , • • • , c M -1 . By symmetry p M,1 = 0 and p k,i F k,i is the same for all i. In fact, elementary computations give

p 1,i F 1,i = (n -1)c 2 1 -nc 1 c 2 a n 1 a n 1 -a n 2 (3)
and for 2 ≤ k ≤ M -2,

p k,i F k,i = (n -1)c 2 k -nc k-1 c k a n k a n k -a n k-1 -nc k c k+1 a n k a n k -a n k+1 . (4)
We can fix a 1 = 1, then these equations determine recursively

a 2 , • • • , a M -1 as functions of c 1 , • • • , c M -1 . The equation p M -1,i F M -1,i = 0 is then auto- matically satisfied since p k,i F k,i = 0. Alternately, we can choose the values of a 1 , • • • , a M -1
, and the equations determine the value of

c 1 , • • • , c M -1 .
For M = 2 we recover the Costa Hoffman Meeks configurations (see section 2.1). For M ≥ 3, these configurations yield minimal surfaces with r = M + 1 ends and genus g = (n -1)(M -1). In particular if n = 2, the genus is g = r -2, the critical case for the Hoffman Meeks conjecture. Unfortunately, it is not hard to check that when M ≥ 3 and n = 2, the configuration is never embedded, whatever the choices of the necksizes. I believe (this is pure speculation) that the surfaces we obtain in this case are in the same family as the examples constructed by M. Weber and M. Wolf in [START_REF] Weber | Teichmuller theory and handle addition for minimal surfaces[END_REF], but that these examples do not stay embedded all the way.

One question we have to answer is : are the above configurations nondegenerate ? It turns out that if M ≥ 3, they are not always non-degenerate, but the following is true : for generic values of the parameters c 1 , • • • , c M -1 , the configuration is non-degenerate. Here generic means : outside the zero set of a non-zero polynomial. Indeed, non-degeneracy can be written as a polynomial equation in c 1 , • • • , c M -1 . To prove the statement, it suffices to prove that this polynomial is not identically zero, so it suffices to exhibit one set of values of the parameters such that the configuration is non-degenerate, for each n and M . We give the details of this computation in appendix A.1. It is clear that for generic values of c 1 , • • • , c M -1 , the configuration is nonsingular, which in this case means that all a k are non-zero and a k = a k+1 .

It remains to choose the neck-sizes such that the configuration is embedded. Take

c 1 = • • • = c M -1 = 1. Equation (2) gives c M = n -M + 1. Then Q 1 = -n, Q 2 = • • • = Q M -1 = 0, Q M = M -1 and Q M +1 = n -M + 1. The condition Q M +1 > Q M gives n > 2(M -1)
, so we can take n = 2M -1. We obtain a family of embedded minimal surface, whose genus is 2(M -1) 2 and number of ends is M + 1. This gives This estimate is by no mean optimal since the genus grows quadratically with r, whereas the Hoffman Meeks conjecture asks for linear growth. It is possible to improve this estimate, but not very much. In fact, it is possible to prove that in general, for a configuration with M layers, if the total number of necks is less than M (M -1)/2, the configuration cannot be embedded, whatever the repartition of the necks and the neck-sizes. Hence one cannot construct minimal surfaces with r ends and genus less than (r -1)(r -2)/2 with this approach : quadratic growth of the genus cannot be avoided.

Asymmetrical configurations

In this section we investigate the following question :

What is the least genus for an embedded FTC minimal surface with no symmetries ? By a symmetry, I mean an ambient isometry preserving the surface (other than the identity). In [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF], an embedded, asymmetrical example of genus 45 with 5 ends was proven to exist, as well as examples of arbitrary high genus. Numerical search gives plenty of embedded, asymmetrical examples of much smaller genus. The smallest genus I have found so far is 6 (see figure 4). Also, this numerical investigation has uncovered a genus 7 example which is simple enough so that it can be proven mathematically to exist (see figure 6). In this section, we explain how one can compute configurations without the help of symmetries, a situation quite opposite to section 2.2. A proof for the genus 7 example of figure 6 is given in appendix A.2.

The only non-degenerate configurations with M = 2 layers are the Costa Hoffman Meeks configurations. The simplest next case is M = 3 with N 3 = 1, which already gives plenty of interesting examples. Let me explain how I compute examples in this particular case. The method generalizes to arbitrary number of layers and necks, but is especially successful in this case.

Note that the balancing condition is invariant by permutation of the points at each level, so we should not use p k,i as variables when computing configurations, else each configuration will be duplicated N 1 !N 2 ! times, so the list of configurations will be huge and in fact the system will be impossible to solve. So the right variables are the elementary symmetrical functions of the points at each level. Consider the polynomials

P k (z) = N k i=1 (z -p k,i ).
By translation, we may assume that a 3,1 = 0, so P 3 = z. Let us write the forces in terms of P 1 , P 2 . We have

P k P k (z) = i 1 z -p i if P k (z) = 0. P k P k (p k,i ) = j =i 2 p k,i -p k,j . F k,i = c 2 k P k P k -c k c k-1 P k-1 P k-1 -c k c k+1 P k+1 P k+1 evaluated at z = p k,i .
From this we get that the configuration is balanced if the polynomial

c 2 1 zP 1 P 2 + c 2 2 zP 2 P 1 -c 1 c 2 zP 1 P 2 -c 2 c 3 P 1 P 2 (5)
vanishes at the points

p 1,1 , • • • , p 1,N 1 and p 2,1 , • • • , p 2,N 2 .
Since these are N 1 + N 2 distinct points and this polynomial has degree ≤ N 1 + N 2 -1, it is identically zero. Writing that all coefficients of this polynomial are zero gives a system of N 1 + N 2 quadratic equations in the coefficients of P 1 , P 2 . The leading coefficient of ( 5) is

N 1 (N 1 -1)c 2 1 + N 2 (N 2 -1)c 2 2 -N 1 N 2 c 1 c 2 -N 2 c 2 c 3 (6)
so we recover equation [START_REF] Costa | Example of a complete minimal immersion in R 3 of genus one and three embedded ends[END_REF]. We are left with N 1 + N 2 -1 equations to solve. The unknowns are the N 1 + N 2 coefficients of P 1 and P 2 .

This system has a special form which makes it easier to solve. First of all, the balancing condition is invariant by complex scaling z → az, so we may normalize one coefficient of P 2 to be equal to 1. We may then use the first N 1 equations to express the coefficients of P 1 as functions of the coefficients of P 2 , solving a linear system of N 1 equations. By substitution in the last N 2 -1 equations, we obtain a system of algebraic equations in the N 2 -1 remaining coefficients of P 2 . This works fairly well if N 2 is small. If N 1 is small, we can exchange the roles of P 1 and P 2 . Once we have found a couple of polynomials P 1 , P 2 satisfying (5), we recover a configuration by computing their zeroes

p 1,1 , • • • , p 1,N 1 and p 2,1 , • • • , p 2,N 2 .
We still must check that the configuration is non-singular, in the sense that these points are distinct, so the forces are defined. In fact, (5) always has the trivial solution P k = z N k , which gives a useless configuration where all points are equal to 0. But we can make sure a priori that any other solution is non-singular as follows.

Note that the balancing condition does not make sense for a singular configuration, but equation ( 5) still does, and may be seen as a way to make sense of a balanced singular configuration. In fact, families of configuration typically become singular for some particular values of the neck-sizes, altough this can happen only for a finite number of values. To understand why, assume that z 0 is a zero of P k with multiplicity

m k ≤ N k , for k = 1, 2, 3 (if m 3 = 1 then z 0 = 0). Assume that m 1 + m 2 ≥ 2 or m 2 + m 3 ≥ 2, so the configuration is singular. Also assume that (m 1 , m 2 , m 3 ) = (N 1 , N 2 , N 3 ), else P k = z N k for k = 1,
2, 3 and we have the trivial solution to [START_REF] Kapouleas | Complete embedded minimal surfaces of finite total curvature[END_REF]. Writing

P k = λ k (z -z 0 ) m k +o((z -z 0 ) m k )
and replacing in [START_REF] Kapouleas | Complete embedded minimal surfaces of finite total curvature[END_REF], we obtain the equation

m 1 (m 1 -1)c 2 1 + m 2 (m 2 -1)c 2 2 -m 1 m 2 c 1 c 2 -m 2 m 3 c 2 c 3 = 0. ( 7
)
Eliminating c 3 from ( 6) and ( 7) we obtain a quadratic homogenous equation in the unknowns c 1 , c 2 , whose coefficients depend on (m 1 , m 2 , m 3 ). By in-spection, we find that the coefficients of this equation are not all zero, so normalizing by c 1 = 1, there are at most two possible values for c 2 . Since there are only a finite possible number of values for the triple (m 1 , m 2 , m 3 ), there are only a finite number of values of c 2 for which a singular configuration (besides the trivial solution P k = z N k ) can occur.

Figure 6 displays a configuration of type 5,4,1. This configuration is almost singular, it becomes singular when c 2 = 1. One can compute the configuration quite explicitely by hand when c 2 = 1, and conclude that it is asymmetrical. The details of this computation are given in appendix A.2.

This proves :

Theorem 2 There exists embedded, asymmetrical FTC minimal surfaces of genus 7 with 4 ends.

A minimal surface with a planar end of order 2

In this section we investigate the following question :

Can an embedded FTC minimal surface have a planar end of order two ?

We are talking about the order of the extended Gauss map at the puncture corresponding to the end. This order is always at least 2 for a planar end.

There are examples of periodic minimal surfaces with planar ends of order 2, like the Riemann minimal examples. For the previously known examples of FTC minimal surfaces with planar ends, the order of the Gauss map at the end was always at least 3. But this was in fact forced by the symmetries of these surfaces. In this section we exhibit an example with a planar end of order 2.

We consider a configuration of type 5, 4, 1. We may normalize c 1 = 1, and we choose c 2 = 5/4. Equation (1) gives c 3 = 11/4. The logarithmic growths of the ends are

Q 1 = -5, Q 2 = 0, Q 3 = 9/4, Q 4 = 11/4,
so the end at level 2 is planar. The configuration can be computed as in section 2.3.

The question is, how can we determine the order of the Gauss map at the planar end ? Theoretically, we have the following asymptotic for the Gauss map g s of M s in a neighborhood of the planar end (corresponding to z = ∞)

lim s→0 -2(sg s (z) -1 ) = N 1 i=1 c 1 z -p 1,i - N 2 i=1 c 2 z -p 2,i
.

The right hand side can be expanded as

(N 1 c 1 -N 2 c 2 )z -1 + (c 1 N 1 i=1 p 1,i -c 2 N 2 i=1 p 2,i )z -2 + o(z -2 ).
The first term vanishes because the end is planar. The Gauss map has order 2 if the second term is not zero. This condition is easy to check (note that we do not need to compute the points p k,i , the coefficients of the polynomials P k are enough).

The configuration can be computed using the methods of section 2.3. Unfortunately, I was not able to prove mathematically (meaning by hand) that this configuration is non-degenerate. (What I can prove is that the configuration is non-degenerate for generic values of the neck-sizes, but here the neck-sizes are fixed.)

The computation can easily be done with a formal computer, however. Moreover, the computation only involves rational numbers so can be carried using exact arithmetic. This gives a numerical proof that there exists embedded FTC minimal surfaces with a planar end of order 2. The details of this computation are given in appendix A.3.

Pictures

The goal of this section is to explain how, given a balanced configuration, one can compute numerically the corresponding family of minimal surfaces. This is illustrated in figure 7 in the case of the Costa Hoffman Meeks genus one family (recall that this family corresponds to a configuration of type 2,1, see section 2.1). The surface is decomposed into three pieces, one per end. Each piece is parametrized by a multi-circular domain, by which I mean the complex plane minus one or several round disks. The point at infinity corresponds to an end of the surface. (In practice we clip the ends, so each piece is parametrized by a big disk minus small disks.)

If we identify the circle marked A with the circle marked A , the circle marked B with the circle marked B and the circle marked C with the circle marked C , we obtain topologically a genus one surface with three ends. To see why this defines a Riemann surface, we must specify how we identify the circles together. This is in fact an instance of a standard construction called opening nodes which is fundamental to the construction in [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF], so let me explain this construction more closely. Consider three copies of the complex plane, labelled

C 1 , C 2 , C 3 . Take two distinct points a - 1,1 , a - 1,2 in C 1 , three distinct points a + 1,1 , a + 1,2 , a - 2,1 in C 2 and one point a + 2,1 in C 3 . Identify a - 1,1 ∼ a + 1,1 , a - 1,2 ∼ a + 1,2 and a - 2,1 ∼ a + 2,1
. This defines a (singular) Riemann surface with three nodes (or double points). The planes C 1 , C 2 , C 3 are called the parts of the noded Riemann surface.

To open the nodes, we consider three non-zero complex parameters t 1,1 , t 1,2 and t 2,1 . We remove the disk [START_REF] Costa | Example of a complete minimal immersion in R 3 of genus one and three embedded ends[END_REF][START_REF] Costa | Uniqueness of minimal surfaces embedded in R 3 with total curvature 12π[END_REF] and (2, 1). We identify the point z in the annulus [START_REF] Costa | Uniqueness of minimal surfaces embedded in R 3 with total curvature 12π[END_REF][START_REF] Costa | Example of a complete minimal immersion in R 3 of genus one and three embedded ends[END_REF]. This defines a Riemann surface because the change of coordinates z → z is holomorphic.

D(a - k,i , |t k,i |) in C k and the disk D(a + k,i , |t k,i | in C k+1 , for (k, i) equal to (1, 1),
|t k,i | < |z -a - k,i | < 1 in C k with the point z in the annulus |t k,i | < |z -a + k,i | < 1 in C k+1 such that (z -a - k,i )(z -a + k,i ) = t k,i , for (k, i) equal to (1, 1), (1, 2) and
Let me explain how this construction is related to figure 7. Note that the circle of center a - 1,1 in C 1 and the circle of center a + 1,1 in C 2 , both of radius |t 1,1 |, are identified : these are the circles labelled A and A on figure 7. A similar statement holds for the other pairs of circles. When t k,i goes to zero, the corresponding circles collapse into the node. In some sense, the Riemann surface converges to the noded Riemann surface. The argument of t k,i may be seen as a Dehn twist parameter.

The parameters in this construction are the six points a ± k,i and the three complex numbers t k,i , for a total of nine complex parameters. We can, however, normalize the value of six parameters by translation and complex scaling in each plane. Then we are left with three complex parameters, which is the dimension of the space of conformal structures on a torus with three punctures.

More generally, in the case of a configuration of type

N 1 , • • • , N M , we consider M + 1 copies of the complex plane, labelled C 1 , • • • , C M +1 . We take N k +N k-1 distinct points a - k,1 , • • • , a - k,N k and a + k-1,1 , • • • , a + k-1,N k-1 in C k (with the convention N 0 = N M +1 = 0). We identify a - k,i ∼ a + k,i for each possible couple (k, i) : this defines a noded Riemann surface with N = N 1 + • • • + N M nodes.
We open the nodes as explained above, introducing one parameter t k,i per node.

The minimal surface is parametrized by the Weierstrass Representation in the following form :

X(z) = Re z z 0 φ 1 , Re z z 0 φ 2 , Re z z 0 φ 3
where φ 1 , φ 2 and φ 3 are three holomorphic 1-forms on our Riemann surface, such that φ 2 1 + φ 2 2 + φ 2 3 = 0 (the conformality equation). The Period Problem asks that X(z) is well defined (independent of the integration path).

To find three holorphic 1-forms satisfying the conformality equation and the Period Problem, we follow the following strategy : we construct φ 1 , φ 2 , φ 3 by prescribing their periods around the necks. (These periods are imaginary, so part of the Period Problem is already solved by definition. Each imaginary part is a parameter). Then we adjust the parameters to solve φ 2 1 +φ 2 2 +φ 2 3 = 0 and the remaining Period Problem. This strategy has been used in [START_REF] Traizet | On the genus of triply periodic minimal surfaces[END_REF] to construct triply periodic minimal surfaces. We follow the construction in [START_REF] Traizet | On the genus of triply periodic minimal surfaces[END_REF], with a few modifications due to the fact that our minimal surfaces have ends. This is not the strategy used to construct the family of minimal surfaces in [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF], where we use the classical form of the Weierstrass Representation (with the Gauss map). The construction in [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF] seems more difficult to implement.

The main question we have to answer to implement numerically the construction in [START_REF] Traizet | On the genus of triply periodic minimal surfaces[END_REF] is : how can we compute numerically these holomorphic 1-forms ? For each one of them, we need some kind of formula, in each multi-circular domain. We answer this question in section 3.1. Each 1-form is represented by some kind of series, whose coefficients are determined by solving a linear system.

At this point we have a family of Riemann surfaces, and holomorphic 1-forms φ 1 , φ 2 , φ 3 , depending on a lot of parameters. We have a well defined immersion on each multi-circular domain, but there is no reason that the images fit together. For instance on figure 7, the image of the circles A and A differ by a translation, and so do the images of the circles B and B . But there is no reason that these translations are the same (as is the case on figure 7) : we need to adjust the parameters so that this is the case. This is the Period Problem.

We also have to adjust the parameters so that φ

2 1 + φ 2 2 + φ 2 3 = 0. Let ψ = φ 2 1 + φ 2 2 + φ 2 3
. This is a meromorphic quadratic differential with at most double poles at the ends. The space of such differentials has finite complex dimension 3N -M -1. The question is : how do we compute numbers from a quadratic differential ? Following the construction in [START_REF] Traizet | On the genus of triply periodic minimal surfaces[END_REF], we simply compute periods of ψ/dz along suitable circles in each multi-circular domain (where dz stands for the usual dz in each multi-circular domain, so is not globally defined on our Riemann surface) and check that we obtain 3N -M -1 independent equations. (Another possibility would be to divide ψ by a global 1-form, say φ 3 , and compute periods. It is much simpler numerically to compute periods of ψ/dz.)

The total number of (real) equations we have to solve is 9N -5M -3. This is a quite large number of equations : for example, for the asymmetrical example of figure 5, with N = 10 and M = 3, we have 72 equations to solve. How do we solve such a large nonlinear system ? Theoretically, in [START_REF] Traizet | On the genus of triply periodic minimal surfaces[END_REF], we solve these equations using the Implicit Function Theorem. The proof of the Implicit Function Theorem, which is based on the contraction mapping principle, gives a method to numerically compute the solution, namely : assume we have a system of n equations in n variables, depending on some parameter s, which we write as f s (x) = 0. Assume that when s = 0, we are given a solution x 0 , and we know that A = df 0 (x 0 ) is invertible. Then for s small, we can solve f s (x) = 0 by the following iteration scheme : define the sequence {x k } k∈N inductively by x 0 = x 0 and

x k+1 = x k -A -1 f s (x k ). ( 8
)
Then for s small, {x k } k∈N converges to a solution x of f s (x) = 0.

We apply this scheme to solve our equations. Without entering into too much details, at the point x 0 , the underlying Riemann surface is fully noded (all t k,i are zero : the necks collapse to double points), and the center of the circles are given by the points of the configuration. We can compute explicitely all equations at this point, see [START_REF] Traizet | On the genus of triply periodic minimal surfaces[END_REF] for more details. The equations boil down to the balancing condition. Moreover, we can compute explicitely the Jacobian matrix A (derivative of equations with respect to parameters) at the point x 0 . Provided the configuration is non-degenerate, the Jacobian matrix is invertible. We then invert numerically the Jacobian and apply the iteration scheme [START_REF] Weber | Teichmuller theory and handle addition for minimal surfaces[END_REF]. For small values of s it converges very quickly to a fixed point (several digits of precision are gained at each loop).

When we increase the value of s it doesn't converge anymore. Here is the idea to push the parameter s further : start from a previously computed solution x s , increase s by a small amount, and apply again the above iteration scheme with x 0 = x s and A equal to the Jacobian at x s . The basic problem is that as soon as s is not zero, the nodes open (the t k,i are non-zero) so we cannot compute explicitely the Jacobian matrix. However, we can compute a good approximation of the Jacobian by pretending that the Riemann surface is still noded. Because the radii remain quite small (even though the other parameters move quite a lot), this gives us an approximate Jacobian which we can use instead of A.

Finally, to plot the surface, we need to integrate φ 1 , φ 2 , φ 3 . Because these 1-forms are represented as series, their integrals are readily computed : no numerical integration is required.

The rest of the section is organized as follows : in section 3.1 we explain how we compute numerically a holomorphic 1-form defined by prescribing periods in the case of opening nodes. In section 3.2, we give more details about the construction of the family of minimal surfaces.

Opening nodes : a model case

In this section we explain how to compute holomorphic 1-forms on Riemann surfaces defined by opening nodes. For the simplicity of notations, we first consider the case where the noded Riemann surface has only one part. Then we explain how one can generalize the construction to other cases.

Consider 2N distinct points a - 1 , • • • , a - N , a + 1 , • • • , a +
N in the complex plane We assume for convenience that the disks of radius one centered at these points are disjoint. Identify for each i = 1, • • • , N the point a - i with the point a + i . This defines a noded Riemann surface Σ 0 with N nodes, which we call

a 1 , • • • , a N . To open the nodes, consider N complex numbers t 1 , • • • , t N such that 0 < |t i | < 1. Remove the 2N disks D(a ± i , |t i |). Identify the annulus |t i | < |z-a - i | < 1 with the annulus |t i | < |z -a + i | < 1 under the identification rule (z -a - i )(z -a + i ) = t i .
This defines an open Riemann surface Σ t , where t = (t 1 , • • • , t N ). We compactify Σ t by adding the point at infinity, and we still denote by Σ t the compactification. The genus of Σ t is N .

It is well known that the space of holomorphic 1-forms on Σ t has complex dimension N , and that a holomorphic 1-form is uniquely defined by prescribing its periods on the circles around the points a + 1 , • • • , a + N . Let ω be the holomorphic 1-form on Σ t defined by prescribing

C(a + i ,1) ω = 2πi c i , i = 1, • • • , N
where c 1 , • • • , c N are given complex numbers. The question is : how can we actually compute ω ?

We are aiming for a formula of the form

ω = ± N i=1 ∞ n=1 A ± i,n (z -a ± i ) n dz (9)
where the complex numbers A ± i,n are such that A ± i,n t -n i is bounded, so that the series converges on the domain |z -a ± i | > |t i |. (In the above formula, the sum on ± means that we add two terms, on for + and one for -). It is not hard to see that ω admits such a representation, using a Laurent series in the annuli |t i | < |z -a ± i | < 1 and the fact that a holomorphic 1-form on the Riemann sphere C ∪ {∞} must be identically zero.

The residues A ± i,1 are determined by the prescribed periods : since the circle C(a + i , 1) is homologous in Σ t to the circle C(a - i , 1) with the opposite orientation, we must have A ± i,1 = ±c i . (Note that this implies that the sum of all A ± i,1 is zero, so ω is holomorphic at infinity.) We want ω to be well defined on Σ t , namely invariant under the identification rule used to define Σ t . This should uniquely determine all coefficients A ± i,n . Let

ϕ i (z) = a - i + t i z -a + i so ϕ i maps the annulus |t i | < |z -a + i | < 1 to the annulus |t i | < |z -a - i | < 1 and Σ t is defined by identifying z with ϕ i (z). The fact that ω is well defined on Σ t is equivalent to ϕ * i ω = ω on the annulus |t i | < |(z -a + i )| < 1, for all i = 1, • • • , N . This is equivalent to ∀m ∈ Z, C(a + i ,1) (z -a + i ) m ϕ * i ω = C(a + i ,1) (z -a + i ) m ω (10)
By a change of variable,

C(a + i ,1) (z -a + i ) m ϕ * i ω = - C(a - i ,1) t i z -a - i m ω.
So (10) may be rewritten as

∀m ≥ 1, A ± i,m+1 = - t m i 2πi C(a ∓ i ,1) ω (z -a ∓ i ) m . (11)
(Notation : the sign ∓ on the right side is opposite to the sign ± on the left side). This is an infinite dimensional linear system in the unknowns A ± i,n , n ≥ 2. Let us introduce the following notations

ω 0 = ± N i=1 ±c i z -a ± i dz, A = (A ± i,n : i ≤ N, n ≥ 2), α(A) = ± N i=1 ∞ n=2 A ± i,n (z -a ± i ) n dz, F ± i,m (α) = - t m-1 i 2πi C(a ∓ i ,1) α (z -a ∓ i ) m-1 , F (α) = (F ± i,m (α) : i ≤ N, m ≥ 2)
. Then (11) may be rewritten as A = F (ω 0 + α(A)). We solve this fixed point problem using the standard iteration scheme : define the sequence {A k } k∈N by induction by A 0 = 0 and A k+1 = F (ω 0 + α(A k )). To see that {A k } k∈N converges to a fixed point, we introduce the following Banach norms :

||A|| = ± N i=1 ∞ n=2 |A ± i,n | ||α|| ∞ = ± N i=1 sup z∈C(a ± i ,1)

|α(z)|

Then straightforward estimates give

||α(A)|| ∞ ≤ ||A|| ||F (α)|| ≤ 2N i=1 |t i | 1 -|t i | ||α|| ∞
Hence ||F (ω 0 )|| < ∞ and provided all t i are small enough, A → F (α(A)) is a contracting linear operator. It follows, by the standard fixed point theorem, that the sequence {A k } k converges to a solution A of A = F (ω 0 + α(A)). Let ω = ω 0 + α(A). From (11), we have the estimate

|A ± i,m | ≤ |t i | m-1 ||ω|| ∞ (12) Hence each series n A ± i,n (z -a ± i ) -n converges for |z -a ± i | > |t i |, so ω is the desired one form.
What we have achieved is a constructive proof of the existence and uniqueness of ω on Σ t , provided all t i are small enough. The above method generalizes easily to the case of meromorphic differentials with prescribed principal part at the poles, provided the poles are outside the disks D(a ± i , 1) : simply add the principal parts to ω 0 . The method also generalizes to the case of several Riemann spheres connected by nodes (see section 3.2), the notations are just a little more cumbersome. On the other hand, it is essential to the above argument that all the parts of the noded Riemann surface have genus zero so we can represent ω as a series.

There are several reasons why a constructive proof is interesting. First of all, this allows us to compute numerically ω, which is our interest in this paper. The estimate (12) says that the coefficients A ± i,n decay rapidly with n, provided all t i remain small. So it is legitimate to truncate the series to some finite order. Typically for the examples we will consider, |t i | is of order 0.01 and we truncate the series to the order n = 10.

The method is also interesting from the theoretical point of view because it generalizes to the case of infinitely many Riemann spheres connected by nodes. In this case, by opening the nodes we obtain non compact Riemann surfaces of infinite genus. The standard "abstract algebraic geometry" machinery does not seem to apply to this setup. This might be useful to construct, for instance, non-periodic minimal surfaces of infinite topology.

The integral in (11) can be explicitely computed using the following formula

Res b 1 (z -a) n (z -b) m = n + m -2 m -1 (-1) m-1 (b -a) n+m-1 .

How we compute the family of minimal surfaces

In this section we give more details about how we compute numerically the family of minimal surfaces corresponding to a given configuration. As was already said, we follow very closely the construction in [START_REF] Traizet | On the genus of triply periodic minimal surfaces[END_REF], with a few modifications to to the fact that our surfaces have catenoidal ends. In particular, the notations and normalizations are as in this paper. Giving all the details of the construction would essentially amount to write a proof of the main theorem in [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF], which is not our point here, so we will be quite allusive.

Given a configuration of type N 1 , • • • , N M , a family of Riemann surfaces is constructed by opening nodes, as explained in the beginning of section 3. The parameters in this construction are the points a ± k,i and the complex numbers t k,i used to open the nodes. We compactify the Riemann surface by adding the points at infinity ∞ 1 , • • • , ∞ M +1 . Let Σ t be the resulting compact Riemann surface.

Next we define three meromorphic 1-forms φ 1 , φ 2 , φ 3 on Σ t , with poles at

∞ 1 , • • • , ∞ M +1
, by prescribing periods on the circles C(a + k,i , 1) as explained in section 3.1. The principal parts at the poles are forced by the fact that we want horizontal catenoidal (or planar) ends : φ 1 and φ 2 need double poles, with no residue, and φ 3 needs a simple pole (or no pole in the planar case) As explained in section 3.1, we represent φ ν as

φ ν,k = λ ν,k dz + N k i=1 ∞ n=1 A - ν,k,i,n (z -a - k,i ) n dz + N k-1 i=1 ∞ n=1 A + ν,k-1,i,n (z -a + k-1,i ) n dz. (13)
Here φ ν,k denotes φ ν in C k . The first term takes care of the double pole at ∞ k . We require that

λ 1,k = 1, λ 2,k = (-1) k+1 i , λ 3,k = 0.
The residues are determined by the period conditions :

A ± ν,k,i,1 = ±c ν,k,i
where c ν,k,i are prescribed real numbers.

As explained in section 3.1, the coefficients A ν,k,i,n for n ≥ 2 may be computed by solving a linear system by iteration. Adapted to the case at hand, this gives the following recipe :

A ± ν,k,i,m+1 ← -δ m,1 λ ν,k t k,i + (-t k,i ) m ∞ n=1 n + m -2 m -1    N k j=1 j =i A ∓ ν,k,j,n (a ∓ k,i -a ∓ k,j ) n+m-1 + N k∓1 j=1 A ± ν,k∓1,j,n (a ∓ k,i -a ± k∓1,j ) n+m-1    .
(δ i,j denotes the Kronecker symbol). Namely, we compute the right hand side for all ν, k, i and for all m ≥ 1, we replace all A ν,k,i,m+1 by the values we have found, and we iterate this process until each A ν,k,i,m+1 is equal to the right hand side at the desired order of accuracy. As was already said, we also truncate the series to some reasonnable order, depending on how small the parameters t k,i are.

The meromorphic 1-forms can be explicitely integrated :

X ν,k (z) = Re φ ν,k = Re(λ ν,k z) + N k i=1 A ν,k,i,1 log |z -a - k,i | + Re ∞ n=2 A - ν,k,i,n (1 -n)(z -a - k,i ) n-1 + N k-1 i=1 A ν,k-1,i,1 log |z -a + k-1,i | + Re ∞ n=2 A + ν,k-1,i,n (1 -n)(z -a + k-1,i ) n-1 .
The Period Problem can be written as

X ν,k (a - k,i + t k,i ) -X ν,k+1 (a + k,i + t k,i ) is independent of i. ( 14
) Let ψ = φ 2 1 + φ 2 2 + φ 2 3
. Following [START_REF] Traizet | On the genus of triply periodic minimal surfaces[END_REF], we solve the following equations :

C(a - k,i ,1) (z -a - k,i ) ψ dz = 0, 1 ≤ k ≤ M, 1 ≤ i ≤ N k (15) C(a + k,i ,1)
ψ dz = 0, 1 ≤ k ≤ M, 2 ≤ i ≤ N k (16) C(a - k,i ,1)
ψ dz = 0, 1 ≤ k ≤ M, 1 + 2δ k,1 ≤ i ≤ N k (17) Im 2 i=1 a - 1,i C(a - 1,i ,1)
ψ dz = 0 (18)
Provided the Period Problem is solved (namely all periods of φ 1 , φ 2 and φ 3 are pure imaginary), ψ also automatically satisfies the following equation

M +1 k=1 (-1) k N k i=1 C(a - k,i ,1)
z ψ dz + N k-1 i=1 C(a + k-1,i ,1) z ψ dz ∈ i R. ( 19
)
This mysterious relation is a consequence of Riemann's bilinear relation.

For completeness we give a proof of this equation in appendix A. [START_REF] Hoffman | Embedded minimal surfaces of finite topology[END_REF]. By the same argument as in [START_REF] Traizet | On the genus of triply periodic minimal surfaces[END_REF], one can prove that provided t is small enough, these 6N -2M -2 real equations are linearly independent, so solving this system guarantees that ψ = 0. For completeness, we provide a proof of this statement in appendix A.5. Of course this gives no guarantee that the equations are independant for a given t > 0, so after solving the equations we check that ψ indeed vanishes by expending the series representing it.

A Appendix

A.1 Dihedral configurations are non-degenerate

In this section we prove that the dihedral configurations of section 2.2 are non-degenerate for generic values of the parameters c 1 , • • • , c M -1 . As explained in this section, it suffices to find one set of values of the parameters such that the configuration is non-degenerate. We take a k = t 2 k for 1 ≤ k ≤ M -1. We shall prove that for t > 0 small enough, the configuration is non-degenerate.

The idea is the following : scale the configuration so that the points at level k are the nth roots of unity. Then the points at level < k go to infinity and the points at level > k go to 0 when t → 0, and in the limit we get a Costa Hoffman Meeks configuration. So for small t, the configuration may be seen as several Costa Hoffman Meeks configuration imbricated into each other. Non-degeneracy thus boils down to the fact that the Costa Hoffman Meeks configuration is non-degenerate.

We may normalize c 1 = 1. Then c 2 , • • • , c M -1 are determined inductively by (3), ( 4) and C M is determined by [START_REF] Costa | Uniqueness of minimal surfaces embedded in R 3 with total curvature 12π[END_REF]. Their limits when t → 0 are

lim c k = n -1 n k-1 for 1 ≤ k ≤ M -1 and lim c M = (n -1) M -1 n M -2 .
Tedious computations give the following limits for the partial derivatives of the forces :

lim t 2 k+1 ∂F k,i ∂p k,i = lim c 2 k   n -1 ω 2i -2 j =i 1 (ω i -ω j ) 2   . lim t 2 k+1 ∂F k,i ∂p k,j = lim c 2 k 2 (ω i -ω j ) 2 if j = i. lim t 2 k+1 ∂F k,i ∂p k-1,i = 0. lim t 2 k+1 ∂F k,i ∂p k+1,i = - c k c k+1 ω 2i . lim t (4-n)2 k-1 i ω i ∂F k,i ∂p k,j = lim c k c k-1 n 2 ω j with the convention c 0 = 0. lim t (4-n)2 k-1 i ω i ∂F k,i ∂p k-1,j = 0. lim t (4-n)2 k-1 i ω i ∂F k,i ∂p k+1,j = -δ 2,n n 2 c k c k+1 ω j .
Let A be the n × n complex matrix defined by

A i,i = n -1 ω 2i -2 j =i 1 (ω i -ω j ) 2 , A i,j = 2 (ω i -ω j ) 2 if j = i.
This is the Jacobian matrix of the Costa Hoffman Meeks configuration, with the last row and column removed. Since the Costa Hoffman Meeks configuration is non-degenerate, A has rank n -1 and any minor of size n -1 of A is invertible. Let B be the n × n matrix defined by B i,j = A i,j for i < n and B n,j = ω -j . Then B has rank n. Indeed, the operation C n ← ω j C j on the columns of B gives the column

C n = (0, • • • , 0, n).
Returning to the dihedral configuration, put the variables in lexicographic order :

p 1,1 , • • • , p 1,n , • • • , p M -1,1 , • • • , p M -1,n , p M,1
. Consider the Jacobian matrix, remove the first line, the first column, the last line and the last column. Let L k,i denote the row corresponding to F k,i . Perform the following row operations :

L k,1 ← t (4-n)2 k-1 n 2 c k c k-1 i ω i L k,i for k ≥ 2, L k,i ← t 2 k+1 c 2 k L k,i for k ≥ 2, i ≥ 2.
In particular P 1 and P 2 share three roots, so the configuration is singular. Let us prove that the configuration is asymmetrical if x 0 is not real. Let ϕ : C → C be a symmetry of the configuration (other than the identity), so ϕ fixes the set of roots of P k , for each k = 1, 2, 3. Then ϕ(0) = 0, and ϕ fixes the sum of the roots of P 1 , which is equal to 1. Hence ϕ(z) = z. This implies that the coefficients of P 1 are real, hence x 0 is real, a contradiction. So the configuration is asymmetrical.

For arbitrary values of c 2 , we can solve the equations in the same way, except that the computations cannot be explicitely done by hand, so we content ourself with the form of the solutions. In the following, the notation (•) means a linear function of its arguments whose coefficients are rational functions of c 2 . We normalize b 3 = 1 and write b 2 = x as before. Then (5) has the form

(a 4 , 1)z 7 + (a 3 , a 4 , x)z 6 + (a 2 , a 3 , a 4 x)z 5 + (a 1 , a 2 , a 3 x, b 0 )z 4 + (a 0 , a 1 , a 2 x, a 4 b 0 )z 3 + (a 0 , a 1 x, a 3 b 0 )z 2 + (a 0 x, a 2 b 0 )z = 0
Let E i be the coefficient of z i in this equation. Solving E 7 , E 6 and E 5 gives

a 4 = p 0 (x), a 3 = p 1 (x), a 2 = p 1 (x).
where the notation p r (x) denotes a polynomial of degree at most r in the variable x whose coefficients are rational functions of c 2 . Equations E 4 , E 2 and E 2 then give a linear system in the unknowns a 0 , a 1 and a 2 . The determinant ∆ of this system has the form ∆ = p 1 (x).

The Cramer formula gives

a 0 = p 3 (x) ∆ , a 1 = p 3 (x) ∆ , b 0 = p 3 (x) ∆ .
Multiplying E 1 by Delta and replacing the above values gives an equation of the form P (x) = 0, where P is a polynomial of degree at most 4 whose coefficients are rational functions of c 2 . When c 2 = 1, we have seen that that no division by zero occurs in this computation, and P has four simple roots. Therefore, this remains true for generic values of c 2 (namely, except for a finite number of values). Also for c 2 = 1, P has two complex roots. This remains true by continuity for c 2 close to 1.

We have seen in section 2.3 that for generic values of c 2 , the configuration is nonsingular. Let us now prove that it is non-degenerate, provided it is non-singular and x 0 is a simple root of P . Fix the values of c 1 , c 2 and c 3 . Let p k,i (t) be a deformation of the configuration with p k,i (0) = p k,i , where t is a real parameter. Assume that F k,i (t) = o(t). We must prove that up to complex scaling and translation, p k,i (t) = p k,i (0) + o(t). Normalize translation by p 3,1 (t) = 0. Define as before the polynomials P k,t = (zp k,i (t)). Equation [START_REF] Kapouleas | Complete embedded minimal surfaces of finite total curvature[END_REF] (Recall that the coefficients of P only depend c 2 , which is fixed, so P does not depend on t.) Since x 0 is a simple root of P , this implies that x t = x 0 + o(t). Then we have a i (t) = a i (0) + o(t) and b i (t) = b i (0) + o(t). Hence p k,i (t) = p k,i (0) + o(t), so the configuration is non-degenerate.

We conclude that if c 2 > 1 is close enough to 1, the configuration is non-singular, non-degenerate, asymmetric and embedded.

A.3 An example with a planar end of order 2

In this section, we give a computer proof that there exists embedded FTC minimal surfaces with a planar end of order 2. We continue with the example of type 5, 4, 1 of the previous section and take the value c 2 = 5/4, which gives a planar end at level 2. Here are the results : The discriminant of P is 137262067070756943236221183727217566386581834266359030391280500736000000 which is non-zero, so P has four simple roots. Each of them gives a non-degenerate configuration.

a 4 = 5
To prove that each configuration is non-singular, we use the argument described at the end of section 2.3. The only values of (m 1 , m 2 , m 3 ) which satisfy [START_REF] Traizet | On the genus of triply periodic minimal surfaces[END_REF], with m 1 + m 2 ≥ 2 or m 2 + m 3 ≥ 2, are (1, 4, 1) and (5, 4, 1). Both of them give P 2 = z 4 , which is not the case because b 3 = 2. So the configurations are non-singular. Finally, we have holomorphic quadratic differential must be replaced by that of a regular quadratic differential : a regular quadratic differential ψ is holomorphic outside the nodes (with at most double poles at ∞ 1 , • • • , ∞ M +1 ), and has at most double poles at each side a - k,i and a + k,i of each node, with the same residue. (The residue of a q-differential at a pole p is the coefficient of ζ -q in the expansion of ψ in term of a local coordinate ζ such that ζ(p) = 0. This is independent of the choice of the local coordinate). The space of regular quadratic differentials on Σ t depends holomorphically on t, including at t = 0.

Let ψ be a regular quadratic differential on Σ 0 satisfying equations (15) to (19). Since ψ has at most double poles at 15) imply that ψ/dz has at most simple poles at all a + k,i , a - k,i . Equations ( 16) and (17) imply that the only possible poles of ψ/dz are at a - ψ dz = 0.

∞ 1 , • • • , ∞ M +1 , ψ/dz is holomorphic at ∞ 1 , • • • , ∞ k . Equation (
These two equations imply that ψ/dz has no residue in C 1 , so ψ = 0.

Figure 1 :

 1 Figure 1: A configuration of type 3,4,1 and a genus 5 embedded minimal surface in the corresponding family. The stars, the circles and the dot represent the points at level 1, 2 and 3 respectively. The neck-sizes are c 1 = 7/4, c 2 = 1 and c 3 2.34.

•

Figure 2 :

 2 Figure 2: A genus one Costa Hoffman Meeks surface, as computed from the configuration on the left.

Figure 3 :

 3 Figure 3: A configuration of type 3,3,1 with dihedral symmetry, and a genus 4 embedded minimal surface in the corresponding family. The necksizes are c 1 = 1.35, c 2 = 1 and c 3 = 1.595.

Figure 4 :Theorem 1

 41 Figure 4: An asymmetrical configuration of type 3,4,2 (embedded, genus 6). The neck-sizes are c 1 = 7/10, c 2 = 1 and c 3 2.85.

Figure 5 :

 5 Figure 5: A genus 7 embedded, asymmetrical minimal surface (top view), computed from a configuration of type 4,5,1. The neck-sizes are c 1 = 3/2, c 2 = 1 and c 3 = 17/5.

Figure 6 :

 6 Figure 6: An asymmetrical configuration of type 5,4,1. The neck-sizes are c 1 = 1, c 2 = 1.01 and c 3 2.98. There is a cluster of 4 points on the left and a cluster of 3 points on the right. This configuration is almost singular.

Figure 7 :

 7 Figure 7: Triangulation of the three multi-circular domains generated with Mathlab (left) and their image in space (right), in the case of the Costa Hoffman Meeks genus one surface. The three pieces on the right match perfectly to give the whole surface.

P

  (x) = -1524209068800 x 2 -285169111920 x+2183134638080 x 3 +180302283315+1490178539520 x 4 .

c 1 i p 1 ,i -c 2 i p 2 ,i = -c 1 a 4 + c 2 b 3

 112243 map has order 2 at the planar end.

  gives that c 2 1 zP 1,t P 2,t + c 2 2 zP 1,t P 2,t -c 1 c 2 zP 1,t P 2,t -c 2 c 3 P 1,t P 2,t = o(t) at the points p 1,1 (t), • • • , p 1,N1 (t), p 2,1 (t), • • • , p 2,N2 (t). By linear algebra, the coefficients of the above polynomial are all o(t). If we call a 0 (t), • • • , a 4 (t) and b 0 (t), • • • , b 3 (t) the coefficients of P 1,t and P 2,t , normalize scaling by b 3 (t) = 2 and write b 2 (t) = x t , we obtain, by the above computation, the equation P (x t ) = o(t), where P is the same polynomial.

  1,1 , a - 1,2 , and a + k,1 for k = 1, • • • , M . Since ψ/dz has at most one simple pole in each C k ∪ {∞ k }, k ≥ 2, we conclude that ψ = 0 in each C k , k ≥ 2. Equations (18) and (19) imply that

	a -1,1 Res a -1,1	ψ dz	+ a -1,2 Res a -1,2	ψ dz	= 0.
	The Residue Theorem in C 1 gives
	Res a -1,1	ψ dz	+ Res a -1,2

By the above formulae, one obtains a matrix which converges when t → 0 to a matrix which has upper triangular block form, with M square blocks on the diagonal. The first block has size n -1 and is an invertible minor of A. The other M -1 blocks are equal to B so are invertible. Hence this limit matrix is invertible. It follows that the dihedral configuration is non-degenerate for t small enough.

A.2 An asymmetrical configuration of type 5, 4, 1.

In this section we give a proof that there exists a family of embedded asymmetrical configuration of type 5, 4, 1, namely we take N 1 = 5, N 2 = 4, N 3 = 1. We may normalize c 1 = 1. c 3 is determined in function of the free parameter c 2 by [START_REF] Traizet | An embedded minimal surface with no symmetries[END_REF]. By a straightforward computation, the configuration is embedded provided 1 < c2 < 5 -√ 5 2

1.381.

We first study the case c 2 = 1. Equation ( 6) gives c 3 = 3. Write

We assume that a 0 = 0, as the case a 0 = 0 only gives very symmetric configurations. We also assume that b 3 = 0, and take b 3 = 2 by scaling. Equation ( 5) with z = 0 gives b 1 = 0. Expanding equation ( 5) we find

Let E i be the coefficient of z i in this equation. Write x = b 2 . Equations E 7 to E 2 in this order determine all coefficients as functions of x by solving only linear equations. We find, in this order, a 4 = 1,

Reporting these values in E 1 gives the equation

This polynomial has four simple roots, two of which are complex. Let x 0 be one of them. Using Euclid's algorithm we find that

and P 1 , P 2 factor as

A.4 Proof of equation (19)

By the Residue theorem, we have

Let g be the genus of Σ and

We apply Riemann's bilinear relation ( [START_REF] Griffiths | Principles of Algebraic Geometry[END_REF], page 241) to the pair of meromorphic 1-forms (φ 1 , φ 2 ) (these are meromorphic differentials of the second kind)

By assumption, the period problem is solved, so all periods of φ 1 and φ 2 are imaginary.

Hence the left side is real. To compute the residues at ∞ k , we write, in a neighborhood of ∞ k ,

where µ 1,k , µ 2,k are some complex numbers and Q k is real. This gives

Equations ( 20), ( 21) and ( 22) prove (19).

A.5 Proof that the equations are independent

Let ψ be any meromorphic quadratic differential on Σ with at most double poles at ∞ 1 , • • • , ∞ M +1 . In this section we prove that equations (15) to (19) imply that ψ = 0, provided t is small enough. The idea is to prove that this is true if t = 0, and conclude by continuity. When t = 0, Σ t is a noded Riemann surface. In this case, the notion of


Marie-Françoise Bidaut-Véron

Rouba Borghol
email: borghol@univ-tours.fr

Laurent Véron
email: veronl@lmpt.univ-tours.fr

Boundary Harnack inequality and a priori estimates of singular solutions of quasilinear elliptic equations

Introduction

Let Ω be a domain is R N , N ≥ 2, d a locally bounded and measurable function defined in Ω and p a real number larger than 1. This article deals with the study of positive solutions of -div |Du| p-2 Dud(x)u p-1 = 0 in Ω (

which admit an isolated singularity on the boundary of Ω. It is known since the starting pioneering work of Serrin [START_REF] Serrin | Local behaviour of solutions of quasilinear equations[END_REF] that one of the main goals for studying the regularity of solutions of quasilinear equations consists in obtaining Harnack inequalities. The simplest form of this inequality is the following: Assume B 2r ⊂⊂ Ω and d ∈ L ∞ (B 2r ), then there exists a constant C = C(N, p, r d L ∞ (B2r )

1/p
) ≥ 1 such that any nonnegative solution u of (1.1 ) in B 2r satisfies u(x) ≤ Cu(y) ∀(x, y) ∈ B r × B r .

(1.2)

Actually this inequality is valid for a much wider class of operators in divergence form with a power-type growth. Among the important consequences of this inequality are the Hölder continuity of the weak solutions of (1.1 ) and the two-side estimate of solutions admitting an isolated singularity. Among more sophisticated consequences are the obtention of local upper estimates of solutions of the same equation near a singular point. This program has been carried out by Gidas and Spruck for equation

-∆u = u q (1.3)
in the case N ≥ 2 and 1 < q < (N + 2)/(N -2) [START_REF] Gidas | Local and global behaviour of positive solutions of nonlinear elliptic equations[END_REF], and recently by Serrin and Zhou [START_REF] Serrin | Isolated singularities of solutions of quasilinear equations[END_REF] for equation -div |Du| p-2 Du = u q (1.4) in the case N ≥ p and p -1 < q < N p/(Np) -1. A third type of applications of Harnack inequality linked to the notion of isotropy leads to the description of positive isolated singularities of solutions. This was carried out by Véron [START_REF] Véron | Singular solutions of some nonlinear elliptic equations[END_REF] for -∆u + u q = 0 (1.5) 1 in the case 1 < q < N/(N -2), and by Friedman and Véron [START_REF] Friedman | Singular solutions of some quasilinear elliptic equations[END_REF] for -div |Du| p-2 Du + u q = 0 (1.6) when p -1 < q < N (p -1)/(Np). When the singularity of u is not an internal point but a boundary point, the situation is more complicated and the mere inequality (1.2 ) with only one function has no meaning. Boundary Harnack inequalities which deals with two nonnegative solutions of (1.1 ) vanishing on a part of the boundary asserts that the two solutions must vanish at the same rate. For linear second order elliptic equations they are used for studying the properties of the harmonic measure [START_REF] Cafarelli | Boundary behavior of nonnegative solutions of elliptic operators in divergence form[END_REF] (see also [START_REF] Bauman | Positive solutions of elliptic equations in nondivergence form and their adjoints[END_REF]). For p-harmonic function in a ball, a sketch of construction is given by Manfredi and Weitsman [START_REF] Manfredi | On the Fatou Theorem for p-Harmonic Functions[END_REF] in order to obtain Fatou type results. In this article we consider singular solutions of (1.1 ) with a singular potential type reaction term. The first result we prove is the following: Assume ∂Ω is C 2 and d is measurable, locally bounded in Ω \ {a} for some a ∈ ∂Ω and satisfies

|d(x)| ≤ C 0 |x -a| -p
a. e. in B R (a) ∩ Ω (1.7)

for some C 0 , R > 0 and p > 1. Then there exists a positive constant C depending also on N , p and C 0 such that if u ∈ C 1 (Ω \ {a}) is a nonnegative solution of (1.1 ) vanishing on ∂Ω \ {a}, there holds

u(y) Cρ(y) ≤ u(x) ρ(x) ≤ Cu(y) ρ(y) ∀(x, y) ∈ Ω × Ω s.t. |x| = |y| . (1.8)
where ρ(.) is the distance function to ∂Ω. Another form of this estimate, usually called boundary Harnack inequality, asserts that if u 1 and u 2 are two nonnegative solutions of (1.1 ) vanishing on ∂Ω \ {a}, there holds

u 1 (x) Cu 1 (y) ≤ u 2 (x) u 2 (y) ≤ Cu 1 (x) u 1 (y) ∀(x, y) ∈ Ω × Ω s.t. |x| = |y| . (1.9)
for some structural constant C > 0. Another consequence of the construction leading to (1.8 ) is the existence of a power-like a priori estimate: Assume Ω is a bounded C 2 domain with a ∈ ∂Ω, A ∈ Ω is an arbitrary point and d a measurable function such that |d(x)| ≤ C 0 |x -a| -p a. e. in Ω.

(1.10)

Then there exist two positive constants α > 0 depending on N , p, Ω and C 0 , and C depending on the same parameters and also on A such that, any nonnegative solution

u ∈ C(Ω \ {a}) ∩ W 1,p loc (Ω \ {a})) which vanishes on ∂Ω \ {a} verifies u(x) ≤ C ρ(x) |x -a| α+1 u(A) ∀x ∈ Ω \ {a}. (1.11)
The precise value of α is unknown and surely difficult to know explicitely, even in the simplest case when u is a p-harmonic function. In several cases the value of α is associated to the construction of separable p-harmonic functions called the spherical p-harmonics. Another striking applications of the boundary Harnack principle deals with the structure of the set of positive singular solutions. We prove the following: Let Ω be C 2 and bounded, a ∈ ∂Ω and d satisfies (1.10 ). Assume also that the operator

v → -div |Dv| p-2 Dv -d(x)v p-1
satisfies the comparison principle in Ω \ B ǫ (a) for any ǫ > 0, among nonnegative solutions which vanishes on ∂Ω \ B ǫ (a). If u and v are two positive solutions of (1.1 ) in Ω which vanish on ∂Ω \ {a}, there exists k > 0 such that

k -1 u(x) ≤ v(x) ≤ ku(x) ∀x ∈ Ω. (1.12)
Furthermore, if we assume also either p = 2, either p > 2 and u has no critical point in Ω, or 1 < p < 2 and d ≥ 0, there exists k > 0 such that

v(x) = ku(x) ∀x ∈ Ω. (1.13)
In the last section we give some partial results concerning the existence of singular solutions of equations of type (1.1 ) and their link with separable solutions which are solution under the form u(x) = |x| -γ φ(x/ |x|). If d ≡ 0 such specific solutions, studied by Kroll and Maz'ya [START_REF] Kroll | The lack of continuity and Hölder continuity of the solution of a certain quasilinear equation[END_REF], Tolksdorff [START_REF] Tolksdorff | On the Dirichlet problem for quasilinear equations in domains with conical boundary points[END_REF], Kichenassamy and Véron [START_REF] Kichenassamy | Singular solutions of the p-Laplace equation[END_REF], are called spherical p-harmonics.

Our paper is organized as follows: 1-Introduction.

The boundary Harnack principle

In this section we consider nonnegative solutions of

-div |Du| p-2 Du -d(x)u p-1 = 0 (2.1)
in a domain Ω which may be Lipschitz continuous or C 2 . The function d, is supposed to be measurable and singular in the sense that it satisfies

|d(x)| ≤ C 0 |x -a| -p a.e. ∈ Ω (2.2)
for some point a ∈ ∂Ω and some C 0 ≥ 0. By a solution of (2.1 ) vanishing on ∂Ω \ {a}, we mean a u ∈ C( Ω \ {a}) such that Du ∈ L p (K) for every K compact, K ⊂ Ω \ {a} which verifies

Ω |Du| p-2 Du.Dζ + d(x)u p-1 ζ dx = 0 (2.3)
for every ζ ∈ C 1 ( Ω), with compact support in Ω \ {a}.

Estimates near the boundary in Lipschitz domains

Let Ω be a bounded domain in R N with a Lipschitz continuous boundary. Then there exist m > 1 and r 0 > 0 such that for any Q ∈ ∂Ω there exists an isometry I Q in R N and a Lipschitz continuous real valued function φ defined in R N -1 such that

|φ(x) -φ(y)| ≤ m |x -y| ∀(x, y) ∈ R N -1 × R N -1 , (2.4)
and

B 2r0 (Q) ∩ {x = (x ′ , x N ) = (x 1 , ..., x N ) : x N ≥ φ(x ′ )} = I Q (Ω ∩ B 2r0 (Q)) .
For any A ∈ B r0(Q) ∩ ∂Ω, A = (a ′ , φ(a ′ ), r > 0 and γ > 0, we denote by C A,r,γ the truncated cone

C A,r,γ = {x = (x ′ , x N ) : x N > φ(a ′ ) , γ |x ′ -a ′ | ≤ x N -φ(a ′ )} ∩ B r (A)
The opening angle of this cone is θ γ = tan -1 (1/γ) Clearly, for every γ ≥ m and 0 < r ≤ r 0 ,

I -1 Q (C A,r,γ
) is included into Ω. Up to an orthogonal change of variable, we shall assume that I Q = Id. We denote also by ρ(x) the distance from x to ∂Ω. The next result is a standard geometric construction which can be found in [START_REF] Bauman | Positive solutions of elliptic equations in nondivergence form and their adjoints[END_REF] Lemma 2.1 Let Q ∈ ∂Ω and 0 < r < r 0 /5 and h > 1 an integer. There exists N 0 ∈ N depending only on m such that for any points x and y in Ω∩B 3r/2 (Q) verifying min{ρ(x), ρ(y)} ≥ r/2 h , there exists a connected chain of balls B 1 , ...B j with j ≤ N 0 h such that remains bounded by a constant depending on p and C 0 .

x ∈ B 1 , y ∈ B j , B i ∩ B i+1 = ∅ and 2B i ⊂ B 2r (Q) ∩ Ω for 1 ≤ i ≤ j -1. ( 2
Up to a translation, we shall assume that the singular boundary point a is the origin of coordinates.

Lemma 2.3

Assume Ω is as in Lemma 2.1 with Q = 0 ∈ ∂Ω, 0 < r ≤ min{r 0 , |Q| /4} and u is a nonnegative solution of (2.1 ) in B 2r (Q). Then there exists a positive constant c 2 > 1 depending on p, N , m and C 0 such that

u(x) ≤ c h 2 u(y), (2.7)
for every x and y in B 3r/2 (Q) ∩ Ω such that min{ρ(x), ρ(y)} ≥ r/2 h for some h ∈ N.

Proof. By Lemma 2.1 there exists N 0 ∈ N * and a connected chain of j ≤ N 0 h balls B i with respective radii r i and centers x i , satisfying (2.5 ). Thus

max Bi u ≤ c 1 min Bi u ∀i = 1, ..., j, (2.8)
by the previous lemma. Therefore (2.7 ) holds with

c 2 = c N0 1 . Lemma 2.4 Let 0 < r ≤ |Q| /4 and u be a nonnegative solution of (2.1 ) in B 2r (Q) ∩ Ω which vanishes on ∂Ω ∩ B 2r (Q). If P ∈ ∂Ω ∩ B r (Q) and 0 < s < r/(1 + m) so that B s (P ) ⊂ B 2r (Q)
, there exist two positive constants δ and c 3 depending on N , p, m and C 0 such that

u(x) ≤ c 3 |x -P | δ s δ M s,P (u) (2.9)
for all x ∈ B s (P ) ∩ Ω, where M s,P (u) = max{u(z) : z ∈ B s (P ) ∩ Ω}.

Proof. Since ∂Ω is Lipschitz, it is regular in the sense that there exists θ > 0,

s 1 > 0 such that meas (Ω c ∩ B s (y)) ≥ θ meas (B s (y)), ∀y ∈ ∂Ω, ∀0 < s < s 1 .
By [START_REF] Trudinger | On Harnack type inequalities and their applications to quasilinear elliptic equations[END_REF]Th. 4.2] there exists δ ∈ (0, 1) depending on p, N , C 0 , θ and s 1 , such that for any y ∈ ∂Ω, there holds

|u(z) -u(z ′ )| ≤ C s s 1 δ(1-γ) ∀(z, z ′ ) ∈ B s (y) ∩ Ω × B s (y) ∩ Ω, (2.10)
where C depends on p, N , C 0 and sup Bs 1 (y)∩Ω u = M s1,y (u). Because the equation is homogeneous with respect to u, this local estimate is invariant if we replace u by ũ = u/M s1,y (u). Thus the dependence is homogeneous of degree 1 with respect to M s1,y (u), which implies

|u(z) -u(z ′ )| ≤ C ′ s s 1 δ(1-γ) M s1,y (u) ∀(z, z ′ ) ∈ B s (y) ∩ Ω × B s (y) ∩ Ω. (2.11)
Taking z ′ = P = y, s = |x -P |, we derive (2.9 ).

If X ∈ B 2r0 (0)∩∂Ω and r > 0, we denote by A r (X) the point with coordinates (x ′ , φ(x ′ )+ r). The next result is the key point in the construction. Although it follows [START_REF] Bauman | Positive solutions of elliptic equations in nondivergence form and their adjoints[END_REF], we give the proof for the sake of completeness.

Lemma 2.5 Let 0 < r ≤ min{2r 0 , |Q| /8, s 1 , 2 5 } and u be a nonnegative solution of (2.1 ) in B 2r (Q) ∩ Ω which vanishes on ∂Ω ∩ B 2r (Q).
Then there exists a positive constant c 4 depending only on N , p, m and C 0 such that

u(x) ≤ c 4 u(A r/2 (Q)) ∀x ∈ B r (Q) ∩ Ω.
(2.12)

Proof. The proof is by contradiction. We first notice from (2.9 ) that if

P ∈ B 2r (Q) ∩ ∂Ω verifies B s (P ) ∩ Ω ⊂ B 2r (Q) ∩ Ω and if c 5 = (2c 3 ) 1/δ
, there holds

M s/c5,P (u) ≤ 1 2 M s,P (u). (2.13) By Lemma 2.3, if y ∈ B 3r/2 (Q) satisfies u(y) > c h 2 u A r/2 (Q) , it means that ρ(y) < r/2 h . Let M > 0 such that 2 M > c 2 (defined in Lemma 2.3), N = max{1+E(6+M ln c 5 / ln 2), M +5}, so that 2 N > 2 6 c M 5 , and c 4 = c N 2 .
Let u be a positive solution of (2.1 ) vanishing on

B 2r (Q) ∩ ∂Ω which satisfies u(Y 0 ) > c N 2 u(A r/2 (Q)), (2.14)
for some

Y 0 ∈ B r (Q) ∩ Ω. Then ρ(Y 0 ) < r/2 N . Let Q 0 ∈ ∂Ω such that ρ(Y 0 ) = |Y 0 -Q 0 |. Then |Q -Q 0 | ≤ |Y 0 -Q 0 | + |Y 0 -Q| ≤ r/2 N + r ≤ r(1 + 2 -5 ) Therefore Q 0 ∈ B 3r/2 (Q) ∩ ∂Ω. Set s 2 = c M 5 r/2 N , then B s2 (Q 0 ) ⊂ B r(1+2 -5 +2 -6 ) (Q) ⊂ B 3r/2 (Q) because s 2 ≤ 2 -6 by the choice of N . Applying (2.13 ) with s = s 2 yields to M s2,Q0 (u) ≥ 2 M M s2/c M 5 ,Q0 (u) ≥ 2 M u(Y 0 ) ≥ 2 M c N 2 u(A r/2 (Q)) ≥ c N +1 Now M s2/2,Q1 (u) ≥ 2 M M r/2 N +1 ,Q1 (u) ≥ 2 M u(Y 1 ) ≥ 2 2M c N 2 u(A r/2 (Q)) ≥ 2 N +2 A r/2 (Q).
Iterating this procedure, we construct two sequences

{Y k } of points such that ρ(Y k ) < r/2 k+N and {Q k } such that Q k ∈ ∂Ω and |Q -Q k | ≤ r(1+2(2 -5 +2 -6 +...+2 -5-k )) < 3r/2 and u(Y k ) ≥ 2 N +k A r/2 (Q) ∀k ∈ N * . Since Y k ∈ B 3r/2 and ρ(Y k ) → 0 as k → ∞ we get a contradiction with the fact that A r/2 (Q) > 0.
Remark. The proof of the previous lemma shows that estimate (2.12 ) is valid for a much more general class of equations under the form

-divA(x, u, Du) + B(x, u, Du) = 0 (2.15)
where A and B are respectively vector and real valued Caratheodory functions defined on Ω × R × R N and verifying, for some constants γ > 0 and a 0 , a 1 , C 0 ≥ 0,

A(x, r, q).q ≥ γ |q| p , |A(x, r, q)| ≤ a 0 |q| p-1 + a 1 |r| p-1 , and |B(x, r, q)| ≤ C 0 |r| p-1 |x| -p , for (x, r, q) ∈ Ω × R × R N .

Estimates near the boundary in C 2 domains

From now we assume that Ω is a bounded domain with a C 2 boundary. For any x ∈ ∂Ω, we denote by ν x the normal unit outward vector to ∂Ω at x. Let R 0 > 0 be such that for any x ∈ ∂Ω, the two balls B R0 (x -R 0 ν x ) and B R0 (x + R 0 ν x ) are subsets of Ω and Ωc respectively. If P ∈ ∂Ω, we denote by N r (P ) and N r (P ) the points Prν P and = P + rν P . Notice that r ≤ R 0 implies ρ(N r (P )) = ρ(N r (P )) = r.

Lemma 2.6 Let Q ∈ ∂Ω \ {0}, 0 < r ≤ min{R 0 /2, |Q| /2} and u be a nonnegative solution of (2.1 ) in B 2r (Q) ∩ Ω which vanishes on B 2r (Q) ∩ ∂Ω.
Then there exist b ∈ (0, 2/3) and c 6 > 0 depending respectively on N , p and C 0 and N , p, R 0 and C 0 such that

t c 6 r ≤ u(N t (P )) u(N r/2 (Q)) ≤ c 6 t r (2.16
)

for any P ∈ B r (Q) ∩ ∂Ω and 0 ≤ t ≤ rb/2.
Proof. Up to a dilation, we can assume that |Q| = 1, since if we replace x by x/ |Q|, equation (2.1 ) and the estimates (2.16 ) are structuraly invariant (which means that C 0 and the C i are unchanged), while the curvature constant R 0 is replaced by R 0 / |Q| which is no harm since Ω is bounded.

Step 1 The lower bound. For a > 0 and α > 0 to be made precise later on, let us define

v(x) = V ( x -N r/2 (P ) ) = e -a(|x-N r/2 (P )|/r) α -e -a/2 α e -a/4 α -e -a/2 α for x ∈ B r/2 (N r/2 (P )) ∩ B r/4 (P ). We set s = x -N r/2 (P ) . Since |Q| = 1, the function d satisfies -C0 ≤ d(x) ≤ C0 . Next -div(|Dv| p-2 Dv) + C0 v p-1 = -|V ′ | p-2 ((p -1)V ′′ + (N -1)V ′ /s) + C0 V p-1 .
Therefore this last expression will be nonpositive if and only if (p -1)

aαs α r α + 1 -α + 1 -N ≥ C0 aα r α 1-p
e (p-1)a(s/r) α s θ e -a(s/r) αe -a/2 α p-1

(2.17) where θ = p + (1p)α. But θ = 0 by choosing α = p/(p -1), thus (2.17 ) is equivalent to (p -1)

aαs α r α + 1 -α + 1 -N ≥ C0 aα r α 1-p 1 -e a(1/4 α -1/2 α ) p-1 . If r/4 ≤ s ≤ r/2 ≤ 1/4, (p -1) aαs α r α + 1 -α + 1 -N ≥ (p -1) aα 4 α + 1 -α + 1 -N = pa 4 α -N, while aα r α 1-p 1 -e a(1/4 α -1/2 α ) p-1 ≤ aα r α 1-p ≤ a 1-p .
Therefore, if we choose a such that

a p-1 ( ap 4 α -N ) ≥ C0 , (2.18)
we derive -div(|Dv|

v(x) ≥ e -a/2 α (2 -α -(s/r) α ) e -a/4 α -e -a/2 α ≥ C(a, α)(1 -(1 -2t/r) α ) ≥ C ′ (a, α)t r if x = N t (P ) with 0 ≤ t ≤ r/2.
This gives the left-hand side of (2.16 ).

Step 2 The upper bound. Let b ∈ (0, 2/3] be a parameter to be made precise later on. By the exterior sphere condition, B 3br (N 3rb (P )) ⊂ Ωc . Let φ 1 be the first eigenfunction of the p-Laplace operator in B 3 \ B1 with Dirichlet boundary conditions and λ 1 the corresponding eigenvalue. It is well known that φ 1 is radial. We normalize φ 1 by φ 1 (y) = 1 on {y : |y| = 2} (notice that φ 1 is radial) and set

φ rb (x) = φ 1 |x -N rb (P )| rb , thus -div |Dφ rb | p-2 Dφ rb = λ 1 (rb) p φ p-1
rb in B 3rb (N rb (P )) \ Brb (N rb (P )) and vanishes on the boundary of this domain. For b small enough λ 1 /(rb) p ≥ 1 + C0 for any r ∈ (0, 1/2], thus

-div |Dφ rb | p-2 Dφ rb -C0 φ p-1 rb ≥ φ p-1 rb (2.21) in Ω ∩ B 3rb (N rb (P )) \ Brb (N rb (P )) ⊇ Ω ∩ B 2rb (N rb (P )) while u verifies -div |Du| p-2 Du -C0 u p-1 ≤ 0 (2.22)
in the same domain. We can also take b > 0 such that B 2br (N br (P )) ⊂ B r (Q), thus

u(x) ≤ c 4 u(N r/2 (Q)) for x ∈ ∂B 2rb (N rb (P )) ∩ Ω
Ω∩B 2rb (N rb (P ))     div |Du| p-2 Du u p-1 - div D φrb p-2 D φrb φp-1 rb     (u p -φp rb ) + dx ≤ 0, valid because (u p -φ p rb ) + ∈ W 1,p 0 (Ω ∩ B 2rb (N rb (P ))). Therefore Ω∩B 2rb (N rb (P )) (u p -φp rb ) + dx ≤ 0,
from which follows the inequality u ≤ φrb in Ω ∩ B 2rb (N rb (P )). In particular

u(N t (P )) ≤ c 4 φ 1 |N t (P ) -N rb (P )| rb u(N r/2 (Q)).
Since φ 1 (s) ≤ C(s -1) for s ∈ [1, 2], we obtain the right-hand side of (2.16 ).

The main result of this section is the following Theorem 2.7 There exists two constants α > 0 and c 7 > 0 depending on N , p, C 0 and N , p, C 0 and R 0 respectively such that if u is any nonnegative solution of (2.1 ) vanishing on ∂Ω \ {0} there holds

1 c 7 ρ(x) |x| α-1 u(A) ≤ u(x) ≤ c 7 ρ(x) |x| -α-1 u(A) (2.23) for any x ∈ Ω, where A is a fixed point in Ω such that ρ(A) ≥ R 0 . Proof. Step 1: Tangential estimate. Let x ∈ Ω such that |x| = 2r ≤ R 0 and ρ(x) = t < br/2. Let Q ∈ ∂Ω \ {0} such that |Q| = |x| and x ∈ B r (Q), the previous lemma implies 2 c 6 |x| ρ(x)u(N r/2 (Q)) ≤ u(x) ≤ 2c 6 |x| ρ(x)u(N r/2 (Q)).
(

2.24)

There exists a fixed integer k > 2 such that we can connect two points lying on ∂B 2r (0) ∩ ∂Ω by k connected balls B j (j = 1, ..., k) with radius r/4 and center on ∂B 2r (0). In particular we can connect N r/2 (Q) with N 2r (0) = -2rν 0 and all the balls can be taken such that the distance of their center to ∂Ω be larger that r/2. Since by Lemma 2.2 there holds sup

Bj u ≤ c 1 inf Bj u ∀j = 1, ...k, we derive 2 c k 1 c 6 |x| ρ(x)u(N 2r (0)) ≤ u(x) ≤ 2c k 1 c 6 |x| ρ(x)u(N 2r (0)). (2.25) Let A 0 = -R 0 ν 0 , b 1 = -2rν 0 = N 2r (0), for ℓ ≥ 2, b ℓ = -2(1 + 3(2 ℓ-1 -1)/2)rν 0 and r ℓ = 2 ℓ-1 r. Applying again Lemma 2.2 in B 2r ℓ (b ℓ ) ⊂ Ω, we have sup Br ℓ (b ℓ ) u ≤ c 1 inf Br ℓ (b ℓ ) u ∀ℓ = 1, 2, ... ( 2

.26)

Let τ be the solution of

2(1 + 3(2 τ -1 -1)/2)r = R 0 ⇐⇒ τ = ln(R 0 + r) -ln 3r ln 2 + 1
and ℓ 0 = E(τ ) + 1, then A 0 ∈ B r ℓ 0 (b ℓ0 ), and the combination of (2.25 ) and (2.26 ) ℓ 0 times) yields to

1 c k+ℓ0 1 c 6 |x| ρ(x)u(A 0 ) ≤ u(x) ≤ c k+ℓ0 1 c 6 |x| ρ(x)u(A 0 ). (2.27)
Since r ≤ R 0 /2, the computation of τ yields to

2 τ = 2(R 0 + r) 3r ≤ R 0 r =⇒ c τ 1 ≤ R 0 r ln c1/ ln 2 .
This implies (2.23 ) with α = ln c 1 / ln 2.

Step 2: Internal estimate. If x ∈ Ω satisfies |x| ≤ R 0 and ρ(x) ≥ b/4 absx, we can directly procede without using Lemma 2.6. Using internal Harnack inequality (2.6 ) and connecting x to N r (0) and then to A 0 we obtain ) by radii such that r ℓ+1 = βr ℓ , where β > 0 depends on the opening of the standard cone C associated to the inside cone property of Ω. This observation shows that in the general case p = 2 and d singular, the validity of Lemma 2.6 implies Theorem 2.7 when Ω is a bounded domain satisfying the inside cone property.

1 c 7 |x| α u(A) ≤ u(x) ≤ c 7 |x| -α u(A), (2.28
Remark. In the case p = 2 and lim x→0 |x| 2 d(x) = 0 the value of α is known and equal to N -1. When p = 2 the value of α is unknown, even in the case where d = 0.

The next result is a consequence of the method used in the proof of Theorem 2.7.

Theorem 2.8 Let u ∈ C 1 ( Ω \ {0}) be a positive solutions of (2.1 ) vanishing on B 2R0 ∩ (∂Ω \ {0}). Then there exists a constant c 9 > 0 depending on p, N , C 0 and R 0 such that Proof. By (2.25 ) we have

1 c 9 u(y) ρ(y) ≤ u(x) ρ(x) ≤ c 9 u(y) ρ(y) , ( 2
1 c ′ |x| u(N |x| (0)) ≤ u(x) ρ(x) ≤ c ′ |x| u(N |x| (0)).
1 c 1 c 8 |x| ρ(x)u(N s (0)) ≤ u(x) ≤ c 1 c 8 |x| ρ(x)u(N s (0)). ( 2
c 9 = c 2 1 c 2 8 .
Another consequence of this method and of Lemma 2.2 and Lemma 2.5 is the Theorem 2.9 There exists a constant c ′ 9 depending on N , p, C 0 and R 0 such that any u ∈ C 1 ( Ω \ {0}) be a positive solutions of (2.1 ) vanishing on

B 2R0 ∩ (∂Ω \ {0}) verifies u(x) ≤ c ′ 9 u(N r (0)) (2.32)
for every 0 < r ≤ R 0 /2 and any x ∈ Ω ∩ B 2r (0) \ B r/2 (0).

Remark

i ∈ C 1 ( Ωǫ ) (i = 1, 2) nonnegative solutions of (2.1 ) in Ω ǫ = Ω \ Be (0) which vanish on ∂ * Ω ǫ = ∂Ω \ B ǫ (0), u 1 (x) ≥ u 2 (x) on Ω ∩ ∂B ǫ (0) implies u 1 ≥ u 2 in Ωǫ .
Clearly, if d is nonpositive it satisfies the local comparison principle. However there are many other cases, depending either on the value of C 0 or the rate of blow-up of d near 0 which insure this principle. Theorem 2.13 Assume d satisfies the local comparison principle and there exists a nonnegative nonzero solution u to (2.1 ) in Ω which vanishes on ∂Ω \ {0}. If v is any other nonnegative solution of (2.1 ) in Ω vanishing on ∂Ω\{0} there exists k ≥ 0 such that v ≤ ku.

Proof. Since any nontrivial nonnegative solution is positive by Harnack inequalities we can assume that both u and v are positive in Ω. We denote by H the set of h > 0 such that v < hu in Ω and we assume that H is empty otherwhile the results is proved. Then for any n ∈ N * there exists x n ∈ Ω such that v(x n ) ≥ nu(x n ). We can assume that x n → ξ for some ξ ∈ Ω. Clearly ξ ∈ Ω is impossible. Let us assume first that ξ ∈ ∂Ω \ {0} and denote by ξ n the projection of x n onto ∂Ω. Thus

v(x n ) -v(ξ n ) ρ(x n ) ≥ n u(x n ) -u(ξ n ) ρ(x n ) . Because u and v are C 1 in Ω \ {0}, lim n→∞ v(x n ) -v(ξ n ) ρ(x n ) = ∂v ∂ν ξ (ξ) and lim n→∞ u(x n ) -u(ξ n ) ρ(x n ) = ∂u ∂ν ξ (ξ).
Since Hopf boundary lemma is valid (see [START_REF] Tolksdorff | On the Dirichlet problem for quasilinear equations in domains with conical boundary points[END_REF]), the two above normal derivative at ξ are negative, which leads to a contradiction. Thus we are left with the case x n → 0. Set

r n = |x n |. By Theorem 2.11 inf v(x) u(x) : |x| = r n ≥ c -1 11 v(x n ) u(x n ) ≥ c -1 11 n
By the local comparison principle assumption, v ≥ nc -1 11 u in Ω rn . This again leads to a contradiction.

The next statement is useful to characterize unbounded solutions Proposition 2.14 Assume u is a nonnegative solution of (2.1 ) vanishing on ∂Ω \ {0}, unbounded and without extremal points near 0. Then

lim x→0 |x| u(x) ρ(x) = ∞. (2.40)
Proof. Assume that (2.40 ) is not true. Then there exist a sequence {s n } converging to 0 and a constant M > 0 such that

sup |x| u(x) ρ(x) : |x| = s n ≤ M.
Therefore sup {u(x) : |x| = s n } ≤ M . Because u has no extremal points near 0, say in B s0 (0) for some s 0 > 0, the maximum of u in Ω ∩ (B s0 (0) \ B sn (0)) is achieved either on |x| = s 0 or on |x| = s n . Therefore max{u(x) :

x ∈ Ω ∩ (B s0 (0) \ B sn (0))} ≤ max {M, max{u(x) : x ∈ Ω ∩ ∂B s0 (0)}} = M.
Since this is valid for any n, it implies that u is bounded in Ω, contradiction.

Such a solution is called a singular solution. The next result, which extends a previous result in [START_REF] Borghol | Boundary singularites of solutions of quaslinear equations[END_REF], made more precise the statement of Theorem 2.13. Proof. Let us assume that v is not zero. By Theorem 2.13 there exists a minimal k > 0 such that v ≤ ku. As in the proof of Theorem 2.13 the following holds: (i) either the graphs of v and ku are tangent at some ξ ∈ Ω. If we set w = kuv, then w(ξ) = 0, and

-Lw -Dw = 0 (2.42)
where L is a linear elliptic operator and D = d(x)(k p-1 u p-1v p-1 )/w. Since ku(ξ) = v(ξ) > 0, D is locally bounded near ξ. If p > 2 and u admits no critical point in Ω, L is uniformly elliptic ([5], [START_REF] Borghol | Boundary singularites of solutions of quaslinear equations[END_REF] for details in a similar situation). Thus the strong maximum principles holds and w is locally zero. Since Ω is connected w ≡ 0 in Ω. If 1 < p ≤ 2, the strong maximum principle holds to and we have the same conclusion.

(ii) either the graphs of v and ku are not tangent inside Ω, but tangent on ∂Ω\{0}. Since the normal derivatives of ku and v at ξ coincide, L is uniformly elliptic. If p ≥ 2 the coefficient D is locally bounded. If 1 < p ≤ 2 this is not the case but D remains nonnegative. In both case Hopf maximum principle applies and yields to ∂w/∂ν ξ (ξ) < 0. This is again a contradiction.

(iii) or v < ku in Ω, ∂v/∂ν > k∂u/∂ν on ∂Ω \ {0} and there exists a sequence {x n } ⊂ Ω converging to 0 such that

lim n→∞ v(x n ) u(x n ) = k.
Furthermore we can assume that the degenerate elliptic equations theory [START_REF] Libermann | Boundary regularity for solutions of degenerate elliptic equations[END_REF] that, up to subsequences, u n and v n converge in the C 1 loc ( Ωn ∩ (B 2 (0) \ B1/2 (0)))-topology to functions U and V which satisfy

v(x n ) u(x n ) = sup v(x) u (
-div |Df | p-2 Df -d ∞ f p-1 = 0 in H ∩ (B 2 (0) \ B1/2 (0)))
, where H is the half space {η ∈ R N : η.ν 0 < 0} and d ∞ is some weak limit of d n in the weak-star topology of L ∞ . Moreover, if p > 2, (2.41 ), jointly with (2.43 ) and (2.44 ), implies that

|Du n (x)| = r n Du(r n x) a n ≥ γ (2.45)
for 2/5 ≤ |x| ≤ 8/5, where γ > 0. We put

ℓ n = inf v(x) u(x) : |x| = |x n | =: r n ≤ k
and ξ n = x n /r n . Up to another choice of subsequence, we can also assume that ℓ n → ℓ and

ξ n → ξ with ξ = 1. Furthermore V ≤ kU , V (ξ) = kU (ξ) and, if ξ ∈ ∂H ∩ (B 2 (0) \ B1/2 (0))), ∂V ∂ν 0 (ξ) = k ∂U ∂ν 0 (ξ) < 0.
In this case, and more generally if the coincidence set Ξ of V and kU has a nonempty intersection with ∂H ∩ (B 2 (0) \ B1/2 (0))), Hopf boundary lemma applies and implies that V = kV in the whole domain. If this is not the case we use (2.45 ) to conclude again by the strong maximum principle that V = kU in H ∩ (B 2 (0) \ B1/2 (0))). Therefore ℓ = k and for any ǫ > 0 there exists

n ǫ ∈ N such that n ≥ n ǫ implies (k -ǫ)u(x) ≤ v(x) ≤ ku(x) ∀x ∈ Ω ∩ ∂B rn (0).
By the local comparison principle the same estimate holds in Ω rn . Since this is valid for any n and any ǫ, we conclude that v = ku.

3 Existence of singular solutions

Separable solutions

The existence of N-dimensional regular separable p-harmonic functions associated to cones is due to Tolksdorff [START_REF] Tolksdorff | On the Dirichlet problem for quasilinear equations in domains with conical boundary points[END_REF]. Extension to singular function is proved in [START_REF] Véron | Some existence and uniqueness results for solution of some quasilinear elliptic equations on compact Riemannian manifolds[END_REF]. These solutions are obtained as follows: Let (r, σ) ∈ R + × S N -1 be the spherical coordinates in R N and S ⊂ S N -1 a smooth spherical domain. Then there exists two couples (γ S , ψ S ) and (β S , φ S ), where γ S and β S are positive real numbers and ψ S and φ S belong to C 2 ( S) and vanish on ∂S, such that U S = r γS ψ S and V S = r -βS φ S ,

are p-harmonic functions in the cone C S = {(r, σ) : r > 0, σ ∈ S}. These couples are unique up to homothety over ψ S and φ S . Furthermore the following equation holds

   -div (a 2 η 2 + |∇η| 2 ) (p-2)/2 ∇η = λ(a)(a 2 η 2 + |∇η| 2 ) (p-2)/2 η in S η = 0 on ∂S. (3.2)
where λ(a) = a(a(p -1) + p -N ) if (a, η) = (β S , φ S ), and λ(a) = a(a(p -1) + Np) if (a, η) = (γ S , ψ S ).

If p = 2 and N = 2, γ S and β S are unknown except if S = S N -1

+ = S N -1 ∩ {x = (x ′ , x N ) : x N > 0 ∈ R N }, in which case γ S = 1 and ψ S = x N . If N = 2, equation (3.2 )
is completely integrable and the values of the γ S and β S are known ([8], [START_REF] Kichenassamy | Singular solutions of the p-Laplace equation[END_REF]). When p = 2, the existence of solutions to (3.2 ) is not easy since this is not a variational problem on S. Tolksdorff's method ( [START_REF] Tolksdorff | On the Dirichlet problem for quasilinear equations in domains with conical boundary points[END_REF]) is based upon a N -dimensional shooting argument: he constructs the solution v of

-div |Dv| p-2 Dv = 0 in C 1 S = C S ∩ {x : |x| ≥ 1} v = (2 -|x|) + on ∂C 1 S . (3.3)
Then he proves, thanks to an equivalence principle, that the function v stabilizes at infinity under the asymptotic form v(x) ≈ |x| -β φ(x/ |x|), with β > 0, which gives (3.2 ) and the function V S . The domain S characterizes the exponent β. The same argument applies if (3.3 ) is replaced by

       -div |Dv| p-2 Dv + cu p-1 |x| p = 0 in C 1 S = C S ∩ {x : |x| ≥ 1} v = (2 -|x|) + on ∂C 1 S . (3.4)
with c > 0. This gives rise to a solution of (3. Remark. If N = p the set of p-harmonic functions is invariant under the Möebius group, and in particular under the transformation x → I(x) = x/ |x| 2 which preserves C S . In such a case β S N -1 + = 1. By using the transformation I it is possible to prove (see [START_REF] Borghol | Boundary singularites of solutions of quaslinear equations[END_REF]) that there exist positive N -harmonic functions in any bounded domain Ω having a singularity at a point a of the boundary and vanishing on ∂Ω \ {a}. We conjecture that such a result holds for (2.1 ) and p = 2 although the precise upper limit as x → a of |x -a| p d(x). We believe that at least if N ≥ p and lim sup x→a |x -a| 2 d(x) ≤ ((Np)/p) p , (the Hardy constant for W 1,p in R N ), such a singular solution do exist.

by Lemma 2 . 5 .

 25 Now the function φrb = c 4 u(N r/2 (Q))φ rb satisfies (2.21 ) in Ω ∩ B 2rb (N rb (P )) and dominates u on ∂(Ω ∩ B 2rb (N rb (P ))) = (∂B 2rb (N rb (P )) ∩ Ω) ∪ (B 2rb (N rb (P )) ∩ ∂Ω). By the Diaz-Saa inequality[START_REF] Diaz | Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires[END_REF]

  .29) for every x and y in B R0 (0) ∩ Ω satisfying |y| /2 ≤ |x| ≤ 2 |y|.

c n 1 u

 1 for any x ∈ Ω such that |x| ≤ R 0 /2 and ρ(x) ≤ b |x| /4. If we assume that x ∈ Ω ∩ B R0/2 (0) verifies |x| ≤ R 0 /2 and ρ(x) > b |x| /4 we can connect x to N |x| (0) by a fixed number n of balls of radius b |x| /8 with their center at a distance to ∂Ω larger than b |x|. The classical Harnack inequality yields to 1 (N |x| (0)) ≤ u(x) ≤ c n 1 u(N |x| (0)). Since ρ(x) ≤ |x| ≤ ρ(x)/b, we obtain, for any x ∈ B R0/2 (0) ∩ Ω, 1 c 8 |x| ρ(x)u(N |x| (0)) ≤ u(x) ≤ c 8 |x| ρ(x)u(N |x| (0)). (2.30) where c 8 depends on p, N , C 0 and R 0 . By Harnack inequality, we can replace u(N |x| (0)) by u(N s (0)) for any |x| /2 ≤ s ≤ 2 |x| and get

Theorem 2 . 15

 215 Assume d satisfies the local comparison principle and there exists a positive singular solution u to (2.1 ) in Ω vanishing on ∂Ω \ {0}. Assume also either 1 < p ≤ 2 and d ≥ 0, or p > 2, u admits no critical point in Ω and If v is any other positive solution of (2.1 ) in Ω vanishing on ∂Ω \ {0} there exist k ≥ 0 such that v = ku.

  x) : |x| = |x n | =: r n Put a n = max{u(x) : |x| = r n }. By Theorem 2.9 there exists c ′ 9 > 0 depending on N , p, C 0 and R 0 such that u(N rn (0)) ≤ a n ≤ c ′ 9 u(N rn (0)), (2.43) which implies max{u(x) : r n /2 ≤ |x| ≤ 2r n } ≤ c ′ 9 a n . (2.44) We set u n (x) = u(r n x)/a n , v n (x) = v(r n x)/a n and d n (x) = r p n d(r n x). Then both u n and v n are solutions of -div |Df | p-2 Dfd n f p-1 = 0 in Ω n = Ω/r n and vanish on ∂Ω n \ {0}. By (2.44 ), u n and v n are uniformly bounded in Ωn = Ω n ∩ (B 2 (0) \ B1/2 (0)). Since ∂Ω n ∩ (B 2 (0) \ B1/2 (0)) is uniformly C 2 we deduce by

1 + 1 + 1 +: 1 + 1 +- 2 1 + 1 +- 1 +

 111112111 [START_REF] Diaz | Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires[END_REF] in C S under the form V S,c = r -βc,S η where β c,S > 0 and   -div (β 2 c,S η 2 + |∇η| 2 ) (p-2)/2 ∇η + cη p-1 = λ(β c,S )(β 2 c,S η 2 + |∇η| 2 ) (p-2)/2 η in S η = 0 on ∂S.(3.5) With these considerations we can construct singular solutions of (2.1 ) under a very restrictive geometry for Ω, where S = S N -, the upper half unit sphere. Theorem 3.1 Assume d(x) = -c |x| -p with c ≥ 0 and Ω is a bounded domain with a C 2 boundary containing 0. Assume also ∂Ω is flat in a neighborhood of 0 and x.ν 0 ≤ 0 for any x ∈ Ω. Then there exists a positive solution of (2.1 ) which vanishes on ∂Ω \ {0} and satisfieslim x → 0 x/ |x| → σ |x| β c,S N -1 + u(x) = η(σ) (3.6) uniformly for σ ∈ S N -, where η is a positive solution of (3.5 ).Proof. Since Ω is located on one side of the hyperplane Λ = {x : x.ν 0 = 0}, the restriction to Ω of the functionV c,S N -Λ ∩ ∂Ω \ {0} and is positive on Ω \ H. Let K = max{V c,S N -(x) : x ∈ ∂Ω \ Λ}. Then V c,S N -K is a solution (or a subsolution if c > 0). For any ǫ > 0 let u ǫ be the solution of Du ǫ + c |x| p |u ǫ | p-2 u ǫ = 0 in Ω \ B ǫ (0) u ǫ = 0 on ∂Ω ∩ B c ǫ (0) u ǫ = V c,S N -on Ω ∩ ∂B ǫ (0). (3.8) Then V c,S N -K ≤ u ǫ ≤ V c,S N -and the mapping ǫ → u ǫ is increasing. Therefore u ǫ converges to u in the C 1 loc ( Ω \ {0}) topology and u is a solution of (2.1 ) which vanishes on ∂Ω \ {0} and satisfies(3.6 ).

  Remark. When p = 2 it is possible to prove the existence of a singular solution to-∆u + d(x)u = 0 in Ω u = 0 on ∂Ω \ {a}(3.9)where a ∈ ∂Ω, for any C 2 domain Ω and any d locally bounded in Ω \ {a} such that-∞ < lim inf x→a |x -a| 2 d(x) ≤ lim sup x→a |x -a| 2 d(x) < N 2 /4.

  2 The boundary Harnack principle. 3 Existence of singular solutions. 4 References.

  Remark. If p = 2 and d is regular, it is proved that Lemma 2.6 holds even if ∂Ω is Lipschitz continuous. The previous proof is adapted by replacing the doubling property of the radius on the connecting balls B r ℓ (b ℓ

) from which (2.24 ) is derived since ρ(x) ≥ b |x| /4. Finally, if |x| ≥ R 0 and ρ(x) ≤ R 0 , we can replace the singular point 0 by a regular point B ∈ ∂Ω such that |x -B| ≤ R 0 . The previous procedure leads to the same estimate. At end, if ρ(x) > R 0 we apply again the internal Harnack inequality (2.6 ). Since Ω is bounded, x and A 0 can be joined by at most d = 2 diam (Ω)/R 0 balls B i with radius R 0 /2 and center b i satisfying ρ(b i ) ≥ R 0 . Then using d times (2.6 ) yields to (2.24 ).

  . Since Lemma 2.2 and Lemma 2.5 are valid in Lipschitz continuous domains and the construction of connected chain of balls too by Lemma 2.1, the above inequality remains valid if Ω is Lipschitz continuous. Definition 2.12 A measurable function d satisfying (2.2 ) with a = 0 ∈ ∂Ω is said to satisfy the local comparison principle in Ω if, for any ǫ > 0 and any u

The next result is known as the boundary Harnack inequality.

u(A r/2 (Q)), since |Y 0 -Q 0 | ≤ r/2 N = s 2 /c M 5 and 2 M > c 2 . Hence we can choose Y 1 ∈ B s2 (Q 0 ) ∩ Ω which realizes M s2,Q0 (u) and this implies that ρ(Y 1 ) < r/2 N +1 . A point Q 1 ∈ ∂Ω such that ρ(Y 1 ) = |Y 1 -Q 1 | satisfies also |Q -Q 1 | ≤ |Q -Q 0 | + |Q 0 -Q 1 | ≤ r(1 + 2 -5 + 2 -6 ).

Theorem 2.10 Let Q ∈ ∂Ω, 0 < r ≤ min{R 0 /2, |Q| /2}, and u 1 and u 2 be two nonnegative solutions of (2.1 ) in B 2r (Q) ∩ Ω which vanish on B 2r (Q) ∩ ∂Ω. Then there exists c 10 > 0 depending respectively on N , p and C 0 such that

for any x, y ∈ B r (Q) ∩ Ω.

Proof. If x ∈ B r (Q)∩Ω satisfies ρ(x) ≤ br/2, we denote by P x = P the unique projection of x on ∂Ω and put t = ρ(x). By (2.16 ),

for i = 1, 2. Therefore (2.35 ) holds with c 2 6 replaced by c β 2 . Finally (2.33 ) is verified with c 10 = max{c 4 6 , c 2β 2 }.

The next result is another form of the boundary Harnack inequality

Theorem 2.11 Let u i ∈ C 1 ( Ω \ {0}) (i = 1, 2) be two nonnegative solutions of (2.1 ) vanishing on B 2R0 ∩ (∂Ω \ {0}). Then there exists c 11 > 0 depending respectively on N , p and C 0 such that for any r ≤ R 0

Proof. Applying twice Theorem 2.8, we get

for any x and y such that |x| /2 ≤ |y| ≤ 2 |x|. Equivalently

which the claim with c 11 = c 2 9 .

The set of singular solutions

We still assume that Ω is a bounded domain with a C 2 boundary containing the singular point 0. We introduce the following assumption on the function d.


est destinée au dépôt

Perception and impact of the nutritional recommendations: a sociological approach

Gap between recommendations and daily food practices:

The same message can mean: > a simple modification of previous practices or norms (in the upper strata: it is then well perceived) Or > a complete destabilization of lifestyle (in the poorer strata: it is then rejected with violence)

A link between food and health?

> The upper strata pay much attention to the nutritional advice, in relation to health, with a strong interest for prevention > In the poor strata, the relation between food and health is not clear

Helps and obstacles to the implementation of recommendations

>

  Perception of a risk for oneself or for a member of the family > Birth of a child > Intensity of social links > Negative perception of obesity and desire for social integrationObstacles:

To cite this version:

  Faustine Régnier, Ana Masullo, . Fédération Des Sociétés Européennes de Nutrition, . Union Française Pour La Nutrition Et L'Alimentation

	A sociological
	approach
	Faustine Régnier and Ana Masullo, INRA, Consumption Research Laboratory
	regnier@ivry.inra.fr; masullo@ivry.inra.fr
	Method 86 semi-directive interviews (800 pages of discourse) hal-02816098 3 operations of prevention (n=52): -Nutrition service in the hospital Hôtel-Dieu (Paris) -Roubaix, EPODE city ("Together, let's prevent childhood obesity")	1 p., 2007.	Representations of body: what does "obese" mean? > Severe control of weight and a fast reaction in case of gain of weight, in the upper categories > Overweight or obese women in modest categories: they are normal
	-Program "Eat better and move more", Val-de-Marne
	+ General population (n=34)
	A contrasted sample: rich/middle/working class social strata;
	People living in the centre of Paris/suburbs/country
			Upper strata	Working class strata
			Prevention	Cure
			Health, long term	Disease, short term
			Hygiene of life	Targeted treatment
			Recommendations	Recommendations for the
			apply to all	one who is obese
			Global changes of	Intermittent changes
			family food habits	(elimination of some products)
		This research was sponsored by Ferrero France

Faustine Régnier, Ana Masullo, . Fédération Des Sociétés Européennes de Nutrition, . Union Française Pour La Nutrition Et L'Alimentation. Perception and impact of the nutritional recommendations: a sociological approach. 10. Conférence européenne de nutrition, Jul 2007, Paris, France. > Tolerance with the children more significant and at a more advanced age in the modest categories Feeding one's child well: 2 different points of view > Upper strata: early education toward foods perceived as good for health > Working class strata: valorisation of abundance and choice > Controlling the infants' food: a nutritional point of view (upper strata) vs being sure they eat at home (poorer strata)

The diversity of implementation