Charles DAPOGNY Michael S.VOGELIUS

(Dedicated to Haim Brezis on the occasion of his 70th birthday)

1 Introduction

Asymptotic expansions of the voltage potential in terms of the “radius” ε of a diametrically small(or several diametrically small)material inhomogeneity(ies)are by now quite well-known(see[4,11]).Let ωεdenote the inhomogeneity,and let 0

As was shown in[10],the existence of the first two terms of the asymptotic expansion carries over to a situation much more general than that of a finite collection of diametrically small inhomogeneities,namely that of an arbitrary set ω?whose Lebesgue measure converges to zero.The convergence statement there is modulo the extraction of a subsequence,and so it is really a compactness result.Furthermore,the convergence is not generally uniform with respect to the inhomogeneity conductivity aε.

Thin inhomogeneities,whose limit set is a smooth,codimension 1 manifold,are indeed examples of inhomogeneities for which the convergence to the background potential u0or the standard expansion cannot be valid uniformly in aε.Indeed,by taking aεclose to 0 or to∞,one obtains either a nearly homogeneous Neumann condition or nearly constant Dirichlet condition at the boundary∂ωεof the inhomogeneity.This boundary,however,does not shrink to a single point as ε→ 0,as is the case when the inhomogeneity is of small radius,but rather it“converges” to a codimension 1 manifold,σ,which has positive capacity.Neither the problem with homogeneous Neumann boundary condition nor the one with constant Dirichlet condition on σhas u0as its solution.Consequently,the convergence of uεtowards u0cannot take place uniformly in aε.

The purpose of this paper is to find a “simple” replacement for u0,say,with the following properties:

(1)may be(simply)calculated from the limiting domain Ω σ,the boundary data on∂Ω,and the right hand side.

(2)depends on ε and aεthrough its boundary conditions on σ.

(3)uε−converges to 0 uniformly in aε,as ε tends to 0.

Such a convergence result is useful for theoretical as well as for practical purposes as follows:

(i)For theoretical purposes,it easily allows one to identify the(ε independent)limit of the potential uε,when the behavior of aεis more precisely known.

(ii)For numerical purposes,it allows to trade a problem posed on a very thin domain,which may be difficult to simulate due to the requirements of a very small mesh size,for a problem posed on a fixed domain with a single additional interphase boundary condition(see the numerical experiments in[22]).

We also briefly discuss the derivation of the next term in a“uniform”asymptotic expansion of uε.From a practical point of view,knowledge of the first two terms would give a very effective tool for the determination of ω?from the knowledge of far field data of uε,in a fashion that would work independently of the conductivity aε(see[3]for the description of such a reconstruction algorithm in the context where the conductivity inside the inhomogeneity is constant and does not depend on ε:aε=a,where 0

There are other studies of asymptotic expansions,specifically related to thin inhomogeneities.In[7],the authors established a first-order asymptotic expansion of uεwhen the conductivity coefficient aεis independent of ε.They considered both the case of a closed,and an open curve σ as far as the limiting set of the inhomogeneity is concerned.They relied on very sharp regularity estimates for uεnear the boundary of the inhomogeneity.This analysis was carried over to the Helmholtz equation in[6].In[5],a(closed)thin conductivity inhomogeneity was considered and analyzed in the case,where the coefficient aεdegenerates to 0 as ε→ 0,by using Γ-convergence techniques.This situation was also investigated in[1]in the context of the minimization of non-linear energy functionals,and in[9]in a situation where the boundary of the inhomogeneity was oscillating.In[22],the resistive limit→0 was considered,a case of particular relevance as an approximation to the behavior of the membrane of a biological cell.In this very particular situation,the authors established the existence of a limiting potential.The analysis is very different from the one presented here and relies on matched asymptotic expansions in all three subdomains:The interior region,the membrane,and the exterior region.It seems difficult to extend such an analysis to the general case studied here.

The technique which we use here to verify the uniform approximation property ofestimates the norm distance between uεand uin terms of the gap between the corresponding energies,by using both the primal and dual formulation.This technique goes back to at least the reference[20].It has the additional nice feature that it only relies on uniform regularity estimates for the approximate solutionnot for uε.

2 Preliminaries and Main Notations

2.1 Setting of the problem

Let Ω ⊂ R2be a bounded domain with smooth boundary,and σbe a closed C2,αcurve,included in Ωand lying at positive distance from ∂Ω.The closed curve σdivides Ωinto two subdomains Ω−and Ω+.Ω−(resp.Ω+)denotes the subdomain interior(resp.exterior)to the curve σ,and unless otherwise specified,n stands for the normal vector to σ,pointing outward from Ω−.For any subset V ⊂ Ω,we denote V±:=V ∩ Ω±(remark that,with this notation,∂V±?= ∂(V±)).If u is any function defined on Ω,we denote by u±its restriction to Ω±.If u+and u−have traces u+|σand u−|σon σ,we denote by[u]:=u+|σ−u−|σthe jump of u across σ.Moreover,when u is sufficiently regular,we denote by

the exterior and interior normal components of∇u at x∈ σ.The associated normal jump across σis denoted by

Except for the thin inhomogeneity the domain Ω is occupied by a conductive material,with conductivity 1.The thin inhomogeneity(with mid-surface σ,and width 2ε(see Figure 1))is

and it has conductivity aε.The conductivity γεin the entire domain is therefore given by

We assume that aε∈ (0,∞)is a scalar constant,but this constant may change with ε.In particular,aεmay go to 0 or∞ as ε→ 0.

A potential ϕ ∈H12(∂Ω)is applied to ∂Ω,and Ω has a charge distribution f ∈ L2(Ω).The electric potential uεin Ω is the solution to

It is well-known that under the above hypotheses,the system(2.2)has a unique solution uε∈ H1(Ω).The following notations will be useful:

(i)For any open subset U⊂R2,(U)denotes the subspace of L2(U)composed of functions u such thatUu dx=0.There is a natural mappingBy a small abuse of notation,for any function u∈L2(U),we shall write

(ii)For sufficiently small δ>0,Fδdenotes the following closed subspace of L2(Ω):

This Hilbert space may also be identified as

Figure 1 Setting of the thin inhomogeneity problem.

The goal of this paper is to understand the uniform asymptotic behavior of the potential uε,as the width 2ε of the thin inhomogeneity goes to 0,uniform,that is,with respect to the conductivity aεinside the inclusion.More precisely,we will derive an approximate problem posed on the fixed domain Ωσ (with boundary conditions on σ,depending on ε and aε),whose solutionis uniformly close to uεas ε → 0,independently of the behavior of the sequence aε.

Remark 2.1Let us briefly comment on the hypotheses of the above model and the possible generalizations of our results.

(i)We assume that the background conductivity γ0,that is,the conductivity outside the inhomogeneity,is equal to 1.This is only a matter of convenience,and it would be straightforward to replace it by a smooth,variable conductivity distribution γ0(x)with 0

(ii)We consider the case of only one internal inhomogeneity,but our analysis immediately carries over to the case of finitely many well separated,internal inhomogeneities.

(iii)We have chosen for simplicity to restrict our analysis to the case of two space dimensions,but with some additional work,it carries over to thin inhomogeneities in higher dimension as well.The curve σ then gets replaced by a closed,smooth(codimension 1)hypersurface.

(iv)We also assume that aεis constant inside ωε.As we will show,the limit behavior of uεis completely different depending on whether aεdegenerates to 0 or to ∞ as ε→ 0(and at what rate).We do not currently know how to(rigorously)generalize the analysis presented here to the situation where aεis variable inside ωεand degenerates to 0 on some parts of ωεand to ∞on other parts.A somewhat related problem would be to consider the case of a simple open curve σ.

(v)Our present results pertain to the conductivity problem(zero frequency).It should be interesting to study the same geometric setting in the context of the Helmholtz problem.We expect the generalization to a single fixed frequency to be rather straightforward,a more challenging problem would be to obtain results that are also uniform over a broad range of frequencies.

2.2 Some facts about distances and projections

In this subsection,we present some material about distances and projections,as well as a version of the coarea formula that will prove very useful when calculating integrals on a set of the form ωε.The context is the same as in Section 2.1: σ is a closed curve of class C2,αdefining two subdomains Ω−,Ω+of a larger(smooth)bounded domain Ω ⊂ R2.For any x ∈ Ω,letbe the Euclidean distance from x to σ.The signed distance function dΩ−to the interior subdomain Ω−is defined as

It is well-known that the projection mapping

is well-defined on a sufficiently small tubular neighborhood ωδof σ (see,e.g.,[18,Proposition 5.4.14]).The maximum thickness of such a neighborhood depends on the curvature of σ.In the remainder of this note,we shall assume that

This hypothesis is only a matter of scaling,and all the analysis adapts mutatis mutandis to the general case.Property(2.3)allows us to define an extension of the normal vector field n: σ → S1to the whole ω1as:n(x):=n(pσ(x));other quantities which are intrinsically defined on σ can be extended likewise.Thus,for any point x ∈ ω1,we shall denote by κ(x)the curvature of σat the point pσ(x).

The derivatives of dΩ−and pσare as follows(see,e.g.,[2]):where the above matrix identities are expressed in the orthonormal basis(τ(x),n(x))of R2.Here τ denotes the 90 degree clockwise rotate of n(x),in other words the extension of a smooth tangent field on σ,and ∇2u stands for the Hessian matrix of a function u.

These observations,together with the coarea formula(see[12])yield the following proposition.

Proposition 2.1Let g∈ L1(Ω).Then,

where dμ1is the one-dimensional Hausdor ffmeasure on the pre-images(y)∩ωε,and ds(y)is the Hausdor ffmeasure on the codimension 1 subset σ.

Remark 2.2This formula may seem ill-defined at first glance,since g is only integrable over Ω.It is a priori that is not defined on all the one-dimensional sets(y),y∈ σ.However,it turns out to be defined on almost every such set(see[15,Subsection 3.4.3,Theorem 2]),and that is sufficient.

As explained above,the normal vector field n and the tangent vector field τ on σcan be extended as orthonormal vector fields to a tubular neighborhood of σ.The coordinates(ξ·τ,ξ·n)of a vector ξ in this basis will be denoted by(ξτ,ξn).

It is convenient to express the two-dimensional divergence operator in the local basis(τ,n).

Lemma 2.1Let ξ be a vector field of class C1defined on a tubular neighborhood of σ.Then,

ProofWe calculate

and similarly,(ξn)=(∇ξ n)·n.For the latter identity,we relied on the fact that ∇n n=∇nTn=0(which follows,e.g.,from(2.4)).Since div(ξ)=tr(∇ξ)can be evaluated in any orthonormal basis,

By differentiation of τ·τ=1,one obtains(∇ττ)·τ=(∇τTτ)·τ=0.Similarly,by differentiation of n·τ=0,using(2.4),one obtains

The desired result follows from a combination of these two observations with(2.5).

Remark 2.3Arguments similar to those of the last proof reveal that

for any function g of class C2on a neighborhood of σ.Thus,for any such function,Lemma 2.1 allows us to conclude that the vector fieldn is divergence-free.

3 A General Argument to Estimate the Difference Between Energy Minimizers

In this section,we introduce our main tool for assessing the convergence of minimizers of variational problems,defined on possibly varying domains.We also present the special considerations required to apply this tool to inhomogeneous Dirichlet problems,which are of most relevance to the present studies.

3.1 An energy lemma

The following lemma may be viewed as a generalization of a rather standard fact about the difference between minimizers of quadratic functionals.

Lemma 3.1Let Vε,Wεbe two families of Hilbert spaces,and let H be another Hilbert space,which continuosly contains all the Vεand Wε.Consider also aε:Vε× Vε→ R and bε:Wε×Wε→R,two families of symmetric bilinear forms that are continuous and coercive.For any?∈H?,define the energy functionals Eεand Fε(whose dependence on?is omitted)by

Eεand Fεadmit unique minimizers∈Vε,∈Wε,due to the usual Lax-Milgram theorem.The gap betweenandcan be controlled in terms of the gap between the corresponding energies as follows:

ProofLet?be an arbitrary linear form in H?.By the standard Lax-Milgram theorem,we know thatandare characterized by the fact that

This in particular implies that

Consequently,for any?∈H?,one has

Now,define the bilinear form q:H?×H?→R by

Using(3.2),we obtain that

from which it is clear that q is symmetric.We are thus in position to use the polarization identity for q,

to conclude that

In combination with(3.4),this last inequality yields

which immediately gives

This completes the proof of the lemma.

Remark 3.1Suppose that the spaces Vεand Wεare only“weakly”contained in H,in the sense that there exist linear continuous mappings Pε:Vε→ H and Qε:Wε→ H through which they may be identified with subspaces of H(we might even allow for the possibility that these mappings are not injective).Change the quadratic functionals slightly to accommodate for the following mappings:

withandbeing the adjoints of Pεand Qε,respectively.The equivalent of Lemma 3.1 now asserts that

Remark 3.2Some comments are in order about the meaning of Lemma 3.1,and the way we intend to use it.Our purpose is to prove an estimate for the difference(vε− wε)between the minimizers vε∈ Vεand wε∈ Wεof two energy functionals Eεand Fε.In the applications ahead,vεand wεare solutions to some elliptic PDEs whose coefficients,or domains of definition,depend on ε.Of course,such an estimate can only be realized in terms of the norm? ·?Hof a“larger” space H,which “contains” all the Vε,Wε.Lemma 3.1 states that such an estimate can be obtained in terms of the difference between the corresponding minimized energies(a quantity which should in principle be simpler to compute).To be more precise,such an estimate may be obtained,provided that we are able to calculate the energy differences in a slightly more general context,namely in the case when a(common)additional and rather arbitrary linear term? ∈ H?has been added to the energies Eε,Fε.Somehow,this additional linear term plays the role of a “sentinel”,and is meant to “observe” functions in Vεand Wε,or at least the features of these that are expressed in the space H through which they are “seen”.

3.2 Extension of Lemma 3.1 to the case of inhomogeneous Dirichlet boundary conditions

The purpose of this subsection is to describe the adjustments needed to the framework of the previous lemma when we deal with inhomogeneous Dirichlet boundary conditions.

3.2.1 A short remark about minimization of functionals over sets of functions satisfying an inhomogeneous Dirichlet boundary condition

Let Ω ⊂ R2be a bounded Lipschitz domain,and V be a Hilbert space of functions over Ω,such that the trace mapping

is well-defined,continuous,and has a continuous right inverse(e.g.V=H1(Ω)).Let V0={u ∈V,v=0 on∂Ω}be the associated homogeneous space.Let a:V×V → R be a continuous and coercive bilinear form over V,and?:V→R be a continuous linear form over V.We are interested in the following minimization problem:

the solution,u,of which solves the variational problem

As is well-known,(3.7)(and thus the minimization problem(3.6))has a unique solution u=+uϕ∈ V,where uϕ∈ V is a right inverse of ϕ for the trace operator(i.e.,uϕ= ϕ on∂Ω),and?u∈V0is defined by

The existence and uniqueness ofare straightforward consequences of the Lax-Milgram theorem.From a slightly different point of view,can also be regarded as the unique solution to the following minimization problem:

By using(3.8),we actually have

We return to(3.6).As a straightforward consequence of the definition of uϕ,

Note that the quantity E0(v)differs from F(v)by a term which is independent of v.Owing to the previous considerations,E0has a unique minimum point v=?u,and

or,by using(3.9),

This last formula is particularly convenient since it is an affine expression of E(u)in terms of u,depending on the data?and ϕ of the problem(3.7).It is the equivalent of(3.3)in the context of variational problems of the form(3.7),posed on affine function spaces.

3.2.2 The energy lemma,the Dirichlet version

The following result adapts Lemma 3.1 to the case when inhomogeneous Dirichlet boundary conditions are considered.

Lemma 3.2Let Ωbe a bounded domain in R2,and let Vε,Wεbe two families of Hilbert spaces of functions defined on Ω,such that,for any ε>0,the trace operator

is well-defined,continuous,and has a linear continuous right inverse ϕ ?→ vϕ(similarly for Wεwith a mapping ϕ ?→ wϕ).Let H be another Hilbert space,which continuously contains all the Vεand Wε.Denote also by aε:Vε×Vε→ R and bε:Wε×Wε→ R two families of symmetric bilinear forms that are continuous and coercive.For any ϕ ∈ H12(∂Ω),? ∈ H?,consider the minimization problems

which admit unique minimizers∈Vε,∈Wε(again,the dependence of Eε,Fεon?is omitted).Then,for any s≥the following estimate holds:

ProofFor any elements ϕ ∈ Hs(∂Ω)and?∈ H?,(3.10)implies that

Consider the space H:=H?× Hs(∂Ω)equipped with the norm

and introduce the bilinear form q:H × H → R,defined for(?1,ϕ1),(?2,ϕ2) ∈ H by the expression

The form q is symmetric.Indeed,introducingand,one obtains

The polarization identity now yields

and therefore by the same technique as in the proof of Lemma 3.1

This is the desired estimate.

Remark 3.3(1)For the estimates(3.1)and(3.11)of Lemma 3.1 and Lemma 3.2,respectively,it is sufficient(on the right-hand side)to envoke the supremum for?belonging to a dense subset of H?,due to the continuity of the mappings??→,??→

(2)Lemmas 3.1 and 3.2 do not generally hold when the energies Eεand Fεcontain additional linear terms cε∈and dε∈(i.e.,contain linear terms from a larger class than H?)In this case,it may still be possible to control the differencein terms of the differencebetween the corresponding energies.However,this control will in general be“weaker”,and may require assumptions that are not so naturally formulated in an abstract framework.

4 Derivation of the 0thOrder Approximation of uε

In this section,we formally construct a uniform 0th-order approximation to the solution uεto(2.2).This approximationis,as explained earlier,the solution to a “simpler” problem with the same data f,ϕ,but posed on a fixed domain.Some of the coefficients of this“simpler”problem depend on ε and aε,and as we have explained in the introduction this is inevitable.Later,in Section 6,we shall rigorously prove a uniform approximation estimate for.To be more precise at that point,we shall prove that there exists a constant C which only depends on the data Ω,σ,f and ϕ,and not on ε and aε,such that

The norm? ·?,and the dependence of C on f and ϕ will be specified later.

To construct the approximation,we rely on the fact that uεis the minimizer of an energy functional Eε,and that the flux(γε∇uε)is the maximizer of a dual energy.We begin with the construction of an approximate energyto Eε,and then we shall search the desired approximationas the minimizer of.We also analyze the dual energyto obtain additional information about the behavior of the flux(γε∇uε),which we shall need for the proof of the estimate of(uε−).

4.1 Asymptotic expansions of the energy functionals associated with uε

4.1.1 Asymptotic expansion of the primal Dirichlet energy

As is well-known,the solution uεto(2.2)is the unique solution of the minimization problem

First,we transform part of this energy expression by means of the mapping Hε:ω1→ ωε,defined by

A straightforward calculation based on(2.4)yields

where the above matrix is expressed in the local basis(τ,n)of the plane.For any function u ∈ H1(ωε),we denote:=u◦Hε.A change of variables now leads to

Using this change of variables,we may now equivalently restate problem(4.1)as

where the setis defined as

and the rescaled energyis given by

Obviously,the equalities featured in the above definition of the spaceare understood in the sense of traces.We now proceed to formally simplify this problem.Retaining only the leading order contribution in the definition of the energy functional(and of the space),we are led to the approximate problem

where we have introduced the function space

and the approximate energy

This problem can be further simplified,by performing the “inner”minimization in v and expressing the result in terms of u.The problem(4.5)can thus be rewritten as

where

This problem can be solved in terms of u which would give rise to an explicit expression forInstead of doing so,we note that the two terms of the energy are of different orders when ε→ 0.One might therefore naturally expect that the behavior of the minimizer v of the previous expression to leading order should be dictated by the termdx.From the Euler-Lagrange equation associated with this minimization,it follows that v should satisfy

If we introduce the coarea formula of Proposition 2.1,this simplifies to

Choosing a test function w of the form w(x+tn(x))= φ(x)ψ(t),with arbitrary φ ∈ C∞(σ)and ψ ∈(−1,1),we now arrive at

from which we conclude that for any x ∈ σ,and any function ψ ∈(−1,1),

As a consequence,for any x ∈ σ,the function t?→ v(x+tn(x))is affine.Introducing the boundary conditions for v(see 4.8),we now arrive at

Substituting this expression for the minimizer in(4.8),we obtain

where Proposition 2.1 was used for the first identity.Finally,after integration in t

Let us draw some conclusions of these formal calculations.(4.7)and(4.9)suggest to search for an approximationto uεby solving

where Vσdenotes the space

and the approximate energyreads

We also note that according to these calculations,the(rescaled)potential(uε◦Hε),inside the inhomogeneity ω1,should be approximated by the function∈H1(ω1),given by

4.1.2 Asymptotic expansion of the dual energy and its maximizer

Before turning to a rigorous study of the functionand its distance to uε,we perform in this section a formal study of the dual energycorresponding to Eεin the spirit of[19].

The dual energy principle associated withasserts that

with

The last extremal problem admits(γε∇uε)as the unique maximal argument.We shall now apply the same strategy as in the previous subsection,namely,to split the integraldx into two,one over Ω he other over ωε,and rescale the second one by using a change of variables.The following lemma provides a hint of what is the relevant rescaling when the objects in question are vector fields.

Lemma 4.1Let U,V be two smooth subdomains of R2,ψ:U→V be a diffeomorphism of class C1.Let ξ∈ L2(V)2be a vector field,and f ∈ L2(V).Then the(weak)divergence of ξ equals f if and only if the vector field|det(∇ψ)|(∇ψ)−1(ξ◦ ψ) ∈ L2(U)2has divergence|det(∇ψ)|f◦ψ.In particular,ξ is(weakly)divergence-free,if and only if|det(∇ψ)|(∇ψ)−1(ξ◦ψ)is divergence-free.

ProofWe have,successively,

which proves the desired result.

Remark 4.1In the same way,we established Lemma 4.1.We may establish that if ξ∈with ξ·n=g on ∂V in a weak sense,thenon∂U.

For any ξ∈ L2(Ω)2,

where we denote?ξ=det(∇Hε)(∇Hε)−1(ξ◦Hε).We also calculate that

Performing a change of variables on ωε,and using these two identities in combination with(4.3),Lemma 4.1 and Remark 4.1,we are led to rewrite the maximization problem forin the form

where

and the functionalis given by

Here we use that the support of f is away from ωε(since f ∈ Fδfor some fixed δ>0).

As before,only the leading order terms in the definitions ofandare now retained in the construction of the approximate extremal problem

The approximate set Vc0is

and the approximate energyis

Note that we have included the integral constraint=0 as part of the description of the set Vc0.This additional constraint is a consequence of the interface conditions imposed on ξ and η,and the constraint div(η)=0,and so it leaves the maximization unchanged.To simplify(4.17)further,we remark as in Subsection 4.1.1 that the extremal problem in η can be solved explicitely(at least approximately)in terms of ξ.Indeed,we rewrite(4.17)as

where

Here the set Wc0is given by

We then proceed to calculate explicitely the expression(4.20).Intuitively,the minimizer η should be characterized to leading order by the minimization of the termThe associated Euler-Lagrange equation reads

for any ζ∈Hdiv(ω1)s.t.−div(ζ)=0,and(1±κ(x))ζn(x±n(x))=0.Since for any ψ ∈the fieldis divergence-free(see Remark 2.3),and has a vanishing normal componenton ∂ω1,we obtain

and now by using Proposition 2.1,we have

Due to the same argument as in Subsection 4.1.1,we conclude that the quantity ητ(x+tn(x))is independent of t∈(−1,1),that is,there exists a function a:σ → R,such that

We now rely on the divergence-free property of η to complete the calculation.Using Lemma 2.1,one has,for any fixed x∈σ and t∈(−1,1),

that is,letting z(t)=ηn(x+tn(x)),

which is nothing but an ODE for z.A simple calculation now gives that there exists a function b:σ→R,such that

Owing to the boundary conditions for ηnin the definition of the set Wc0,the functions a and b must satisfy

which after straightforward manipulations leads to

These expressions are unfortunately not as explicit as those obtained in Subsection 4.1.1,and in particular they do not lead to a similarly simple variational problem for ξ.However,they do(approximately)connect the exterior and interior components,ξ and η,of the maximizer of,which hopefully is close to that of

5 Study of the Approximate Function :Uniform Energy and Regularity Estimates

In this section,we study properties of the solutionto(4.10),which is our candidate for the 0thorder term of the asymptotic expansion of uε.

We assume the data to be such that f ∈ L2(Ω)with support away from σ,and withf dx=0(this is expressed by requiring f ∈ Fδfor some fixed δ>0,see the definitions in Subsection 2.1).We also assume that ϕ ∈(∂Ω).After first proving existence and uniqueness of the solution u0ε,our main purpose is to establish energy and regularity estimates for(and its derivatives)which are uniform with respect to ε and the sequence aε(see Subsections 5.3–5.4).

5.1 Existence,uniqueness,and a classical formulation of(4.10)

Let Vσ,0be the subspace of Vσ(the latter being defined by(4.11))composed of functions with vanishing trace on ∂Ω.We define the following semi-norm and norm on Vσ:

We note that due to a standard Poincaré inequality,the seminorm|·|Vσis actually a norm on Vσ,0,equivalent to.The variational formulation associated to(4.10)is as follows.

Find∈ Vσwith= ϕ,such that

Proposition 5.1The minimization problem(4.10),or equivalently the variational problem(5.1),has a unique solution

ProofThe existence and uniqueness offollow from the standard Lax-Milgram theorem,the only point which deserves comment is the(nonuniform in ε and aε)coercivity of the bilinear form involved in(5.1)on the space Vσ,0.This coercivity follows from the inequality

and the fact(noted above)that the seminorm|·|Vσis a norm on Vσ,0,equivalent to

Problem(4.10)can be stated in a “classical”form.Indeed,using smooth test functions v∈(Ω σ)in(5.1),we first see thatsatisfies

in the sense of distributions.If f and ϕ are smooth,then it is fairly easy to prove thatis actually C2,αup to the boundary ∂Ωand up to the curve σ,and it solves the equation−=f in a classical sense.The proof of regularity is a very standard elliptic regularity argument,that we leave to the reader,however,in Subsections 5.3–5.4(and the appendix),we shall show exactly what a priori estimates hold uniformly in εand aε.Now using again(5.1),and an integration by parts,we obtain that

for all functions v∈ Vσ,0.Using this last equality with test functions v∈ H1(Ωσ),such that v=0 on ∂Ω,v+is smooth on σ,and v−=0 on σ,we obtain that

Symmetrically,by exchanging the roles of v−and v+,one obtains

In summary,is a solution to the following problem on Ω σ:

Let us also notice that insertion of v∈(Ω),v ≡ 1 in a neighborhood of σ,into(5.3)yields

This identity,in combination with the fact thatds=0,gives

5.2 The dual energy maximization problem for

In this paper,it will prove convenient on several occasions to use the dual energy maximization principle forWe remind the reader that the hypotheses for f and ϕ are

We write

where the maximum in the last expression is achieved uniquely at ξ= ∇u,w+=w−=and z=(u+−u−).We can now exchange the min and max in the above formula(see[14])to rewrite

In this last expression,the maximum is taken over all functions ξ∈ L2(Ω σ)2,w+,w−,z ∈L2(σ),such that

We note that,in this particular context,the above exchange of the minimum and maximum can be justified very simply,since the functionals at stake are quadratic,and we know explicitly the associated minimizer and maximizer.

This last maximum is achieved uniquely atWe thus end up with the following convenient alternative expression for the minimum energy

5.3 Uniform energy estimates for

The following lemma provides preliminary energy estimates for the function

Lemma 5.1Let Ω ⊂ R2be a bounded Lipschitz domain,and σbe a closed C2,αcurve in Ω,lying at positive distance from ∂Ω.Let ϕ ∈ H12(∂Ω)and f ∈ Fδfor some δ>0.Then,

(1)There exists a constant C>0,independent of ε and aε(but dependent on Ω andσ),such that

(2)There exists a constant C>0 independent ofεand aε(but dependent on Ω andσ),such that

Proof(1)By definition of ϕ ∈(∂Ω),there exists uϕ∈ H1(Ω)which we may assume to have compact support in Ω+for some δ>0,such that uϕ= ϕ on ∂Ω and≤The variational formulation of problem(4.10)may be expressed in terms of

Inserting v=wεas a test function,and relying on the inequality(5.2),we immediately obtain

for any value m∈R(since=0).Due to the Poincaré inequality for functions on Ω+which vanish on ∂Ω,we have

and from the PoincarWirtinger inequality on Ω−

It follows from a combination of these estimates and(5.9)that

The desired result follows from this estimate and the facts that=wε+uϕ,≤,and uϕvanishes on σ.

(2)The first inequality is a consequence of(5.10)and the decomposition=wε+uϕ,combined with the Poincarinequality for functions on Ω+which vanish on ∂Ω.The second inequality similarly follows from(5.10)and the Poincaré-Wirtinger inequality on the domain Ω−.Note that this latter estimate concerns thesemi-norm,not the L2(Ω−)norm.

5.4 Uniform regularity estimates for

We now proceed to state the uniform regularity estimates for the functionwhich we shall require for our later analysis.The results needed are stated in the following theorem,whose proof is postponed to Section 9.

Theorem 5.1Assume that Ωand σare of class C2,α,that the source term f belongs to Fδfor some δ>0,and that ϕ ∈ H32(∂Ω).Then the unique solution u0εto the problem(4.10)belongs to H2(Ωσ)∩H2(σ),and the following estimates hold:

wherestands for the H2semi-norm of a function u∈H2(V),and the constant C depends only on Ωand σ (and not on ε and aε).

Remark 5.1(1)The proof of Theorem 5.1 can be iterated,if one assumes higher regularity of Ω, σ,f and ϕ.More precisely,if Ωand σare of class Cm,α,f ∈ Fδ∩ Hm−2(Ω)and ϕ ∈(∂Ω)for some m ≥ 2,then

for any multi-index β of length ≤ m.Note also that these results are local.Thus,even if f only belongs to Fδfor some δ>0,but σis a Cm,αcurve,thenis of class Cm(Vσ)for any open set V,such thatand

for any multi-index β of length ≤ m.

(2)The two estimates(5.11)–(5.12)are of a quite different nature.They are complementary in the sense that,depending on the behavior of the sequence aε,one may prove more precise than the other.Estimate(5.11)expresses the fact that all the derivatives of u0εare uniformly bounded with respect to ε and aε,provided that the data of the problem have enough regularity.On the other hand,the estimate(5.12)is analogous to the preliminary estimates of Lemma 5.1.It does not carry much information in the low conductivity regime(i.e.,aε? ε),but it is in some sense much stronger than(5.11)in the high conductivity regime(i.e.,aε? ε).

(3)Recall that,due to Lemma 5.1,u0ε|Ω+(and not just its derivatives)also turns out to be uniformly bounded with respect to ε and aε.However,in general,this is not the case of u0ε|Ω−,which is only uniformly bounded up to a constant.

6 Proof of the Asymptotic Exactness of

We are now in position to verify the asymptotic exactness ofin other words to show that the gaptends to zero as ε tends to zero.The precise estimate we establish is the following.

Theorem 6.1Assume that the“center”curve σ is of class C∞,and that ϕ ∈(∂Ω).Let δ>0 be a fixed positive real number,and suppose f ∈ Fδ.Let uε∈ H1(Ω)(resp.∈ Vσ)be the unique solution to the minimization problem(4.1)(resp.(4.10)).Then the following estimates hold,for ε>0 sufficiently small:

where the constant C is independent of ε,and of aε.

ProofThe technique used here is very close to that used in[21](a main idea of which is already found in[20]).It relies on two key ingredients:

(i)The uniform energy and regularity estimates forandpresented in Subsections 5.3–5.4.Interestingly enough,neither energy nor regularity estimates for the exact solution uεare required.

(ii)The general argument of Lemma 3.2,which controls the discrepancy between uεandin terms of the discrepancy between the minimum values of the corresponding energies Eεand

Using the notation of Lemma 3.2,we choose Vε=H1(Ω),Wε=Vσand H=Fδ(we identifywith Fδ).The natural mapping Pε:Vε→ H is

The operator Pε(which,like Vεand Wε,in this case actually does not depend on ε)also naturally maps Wεinto H.According to Lemma 3.2(and Remark 3.3)the following estimates hold:

The idea is then to estimate the discrepancy(Eε(uε)−)between the minimum values of the energies by using particular“test functions”in place of uε(or its gradient),which make Eε(or its dual)mimic the behavior of the functionalnear the limiting curve σ.The existence of such test functions is made possible by the regularity estimates forstated in the Subsection 5.3.Subsections 6.1–6.2 below are devoted to establishing the desired control over this energy discrepancy.

In the following,for the sake of brevity,we denote by C a constant,possibly changing from one instance to the other,which only depends on Ωand σ,but is independent of ε,aε,f and ϕ.We also use the shorthand

6.1 Proof of the upper bound Eε(uε)−()≤C(f,ϕ)2ε

As a straightforwardconsequence of the definition(4.1),one has,for any function u∈ H1(Ω),such that u=ϕ on∂Ω,

We proceed to construct a“test function”u,which makes the right-hand side of the above inequality small.To this end,a natural idea is to exploit the equivalent form(4.4)of the problem,and use the pair()as a test function,whereis the unique solution to(4.10),andis given by(4.13).This is unfortunately not possible,since the pair()does not belong to the spaceIndeed,it does not satisfy the boundary conditions

but satisfies instead

To remedy this,let us define zε∈ H1(Ω)as the unique solution to

By construction,the pair(+zε,)belongs to.Let us now work toward estimating the function zε.As an easy consequence of definitions,

Here V is a neighboorhood contained in ωδfor a fixed δ withAccording to Theorem 5.1(and Remark 5.1),it follows that

By a very simple construction,we may extend the trace zε|∂ωεto a function Zεdefined on the whole domain Ωwith Zε=0 on ∂Ω and

A simple calculation gives that

in other words,

and so

Now,using the pair(+zε,)as a“test function”in(4.4),we calculate

Here

where we used(6.3)and the uniform energy estimate of Lemma 5.1.Similarly,one has

because of our assumptions about f,and the estimate(6.3),in combination with the fact that zεvanishes on ∂Ω.Concerning the terms on ω1,

where the first line is a consequence of the uniform energy estimates of Lemma 5.1,and the second line follows by the exact same calculation that we performed in Subsection 4.1.1.Similarly,we obtain

To conclude,let∈ H1(Ω)denote the function

Combining all these estimates,we finally get

6.2 Proof of the lower bound:−Eε(uε)≤C(f,ϕ)2ε,and end of proof of Theorem 6.1

In order to prove the lower bound,we rely on the use of the dual energies associated to EεandMore precisely,based on the equivalent,rescaled form(4.15)of the dual problem to Eε,

for every vector couple(ξ,η)in the spacedefined by(4.16),and satisfying −div(ξ)=f,−div(η)=0.Using the definition ofand the alternative expression(5.7)for,we may rewrite this as

In light of the discussions in Subsections 4.1.2 and 5.2,and particularly due to the formulas(4.21),it is tempting to define a test flux ξ∈ Hdiv(Ωσ)by ξ=,and to define η ∈ Hdiv(ω1)in such a way,that for x∈ σ,t∈ (−1,1),

and insert(ξ,η)into(6.4).Using the pointwise expression(5.4)for the boundary conditions forwe are led to

Unfortunately,such a choice of“test couple”is not admissible,since it does not belong to the spaceNevertheless,it “almost” belongs to this space,and we may use a “small” additive correction to remedy that situation.We define zε∈ H1(Ω )as the unique solution(up to a constant)to the problem

Recall that in the last two boundary conditions,n stands for the normal vector to,oriented in the direction from Ω−to Ω+.The functionis defined by

andis defined by the similar formula

so that the couple(ξ+∇zε,η)belongs toThe requirement that0 is guaranteed by the identity(5.5)and the fact that f vanishes in ωε,so thatas well.Using the uniform regularity estimates of Theorem 5.1(and Remark 5.1),we obtain that

and a standard regularity argument(as for the Dirichlet problem in the previous section)now gives

It is now possible to use(ξ+ ∇zε,η)as a test couple in(6.4).Doing so,we obtain first

and

Besides,

where for the last estimate we used the algebraic inequality

Using the uniform energy estimates of Lemma 5.1,we conclude

On the other hand,

Due to the uniform energy estimates of Theorem 5.1,the first two integrals in the last expression are easily controlled by C(f,ϕ)2ε2.When it comes to the third integral,one has

since the integral terms in the product are each bounded by C(f,ϕ).We thus obtain the estimate

where we again make use of the uniform energy estimate in Lemma 5.1.Application of the auxiliary estimates(6.5)–(6.8)to(6.4)with the test couple(ξ+ ∇zε,η)finally yields

which is the desired lower bound on Eε(uε)−().

The End of Proof of Theorem 6.1By a combination of the upper bound of the previous subsection and the lower bound of this subsection,we obtain

or

Insertion of this estimate into(6.1)and(6.2)respectively finally gives

This completes the proof of Theorem 6.1.

Remark 6.1The 0thorder uniform approximation to uεis only unique modulo a function that is of the order O(ε),uniformly in ε and aε.As a reflection of this,the energetic expression E0ε(of(4.12)is not unique either.A proof very similar to the one presented above(together with corresponding uniform regularity and energy estimates)would reveal that the unique minimizer to

is also a uniform 0thorder approximation of uε.

7 Limit Behavior of

So far,we have only discussed the approximation of uεin terms of the solutionto another simpler minimization problem,which,however,still depends on ε and aε.When the behavior of the sequence aεis known more precisely as ε→ 0,then explicit,ε and aεindependent limit behaviors of(and thus of uε)can be derived.

7.1 The general case

Let us assume that both εaεandhave a limit as ε→ 0,including possible limits of 0 and ∞.Remark that,in the general case,there always exists a subsequence εn→ 0,such that this is achieved.Since εaε?,the limiting pairhas one of the five possible forms(∞,∞),(a0,∞),(0,∞),(0,b0)and(0,0),where 0

Proposition 7.1Let aεbe any sequence of positive real numbers,and∈ Vσbe the unique solution to the minimization problem(4.10).Suppose f ∈ Fδfor some δ>0 and ϕ ∈ H12(∂Ω),and suppose that both εaεandhave a limit as ε→ 0,including possible limits of 0 and∞.The following five cases describe the associated limiting behaviour of.

Case 1εaε→ ∞ (thus→ ∞).The limit ofis∈(Ω):={u ∈ H1(Ω),u=cst on σ},the unique solution to the minimization problem

and there exists a constant C independent of ε and aε,such that

Case 2εaε→ a0for a certain real value 0

and there exists a constant C independent of ε and aε,such that

Case 3εaε→ 0 and∞.The limit ofis∈ H1(Ω),the unique solution to the minimization problem

and there exists a constant C independent of ε and aε,such that

Case 4→ b0for a certain real value 0

and there exists a constant C independent of ε and aε,such that

Case 5→ 0(thus εaε→ 0).The limit ofis∈ H1(Ω σ),a solution to the minimization problem

This solution is unique up to an additive constant on Ω−.There exists a constant C independent of ε and aε,such that

The proof of this proposition again relies on Lemma 3.2.It is in many ways very similar to the proof of Theorem 6.1,but simpler,so we only provide a sketch.A complete proof would notably involve uniform estimates for the limit problems in the spirit of Theorem 5.1.Before we proceed to the sketch of the proof,some remarks are in order.

(i)The functional spaces involved in the minimization problems(7.1)–(7.3)feature functions that belong(at least)to H1(Ω),and thus do not jump across σ.As a consequence,the derivation of uniform energy estimates in the spirit of Lemma 5.1 does not require any assumption about f other than f ∈ L2(Ω).The natural choice for the space H in the application of Lemma 3.2 is then L2(Ω),and so we obtain L2(Ω)estimates of the discrepancy betweenand its limits.The assumption=0 is not necessary in order to establish the results of Proposition 7.1 in Cases 1 through 3.

(ii)In Case 4,the assumption=0 is not required to ensure that the minimization problem(7.4)has a unique solution.It is needed in order to ensure that one may obtain energy estimates forthat are uniform with respect to b0(see the proof of Lemma 5.1).Lemma 3.2 then provides a uniform estimate for(−)on Ω+,and a uniform estimate for the same difference on Ω−,modulo aconstant.

(iii)In Case 5,the assumption=0 is required to ensure that the minimization problem(7.5)has a unique solution,which is defined up to a constant in Ω−.Note that the convergence result expressed in this case is independent of this constant.

Proof(1)We use Lemma 3.2 with Vε=Vσ,Wε=(Ω)and H=L2(Ω),and proceed to estimate the difference−Since(Ω)⊂Vσ,we have

To obtain an upper bound for(()−()),we first rewrite()as

Sinceamounts to a constant on σ,and since(which is easily derived from the fact thatis the minimizer to(7.1)),we conclude that

Now,introducing the dual energy principle forestablished in Subsection 5.2,we obtain

for any ξ∈ L2(Ω σ)2and w+,w−,z ∈ L2(σ)satisfying the relations(5.6).Insertion of ξ=,z=0,and the two indefinite σintegrals

in the above relation yields

The result follows by using energy estimates for

(2)We rely again on Lemma 3.2 with Vε=Vσ,Wε=H1(Ω)∩ Vσand H=L2(Ω).As H1(Ω)∩Vσ⊂ Vσ,we have on one hand,

The factords is bounded byuniformly with respect to a0(as follows easily from standard energy estimates for the problem(7.2)).

On the other hand,the dual energy maximization principle forreads

where the maximum is taken over the set of functions ξ∈ L2(Ω)2,w ∈ L2(σ),such that

The maximum is uniquely attained at ξ=and w=We thus obtain

for any ξ∈ L2(Ω σ)2and w+,w−,z ∈ L2(σ)satisfying(5.6).We now inserttogether with

and z given by

The last identity holds true because of(7.6),and it insures that this choice of ξ,w±,z satisfies(5.6).As a result

These upper and lower bounds for−(),in combination with the appropriate a priori estimate forlead to the desired conclusion.

(3)It is in every aspect simpler to handle than the other cases,and is left to the reader.

(4)Here we take Vε=Vσ,Wε=H1(Ωσ)and

We obtain an upper bound for(−))by using v=as a “test function”in the minimization of

for a constant C,which does not depend on b0and ε,aε.Here we used the fact that

and an appropriate a priori estimate forIn order to establish a satisfactory lower bound on()−()),we first observe that,as an immediate consequence of the variational problem satisfied byone has

Now,using the dual energy maximization principle for(see Subsection 5.2),and the fact that

we obtain

for any ξ∈ L2(Ω σ)2and w+,w−,z ∈ L2(σ)satisfying(5.6).Due to(7.7),we may choose ξ=,w+=w−=0 and z=(−)for insertion into the last line of the previous inequality.This yields

for some constant C which is independent of b0and ε,aε.Here we used the same algebraic inequality as before,and an appropriate a priori estimate for.In summary,we have proved

and by Lemma 3.2 this yields the desired estimate for

(5)In this last case,we take(a set over which the minimization problem(7.5)has a unique solution)and

The proof proceeds along the same lines as in the previous case(s),and is left to the reader.

7.2 A closer look at the case aε =a,independent of ε

In this section,we make some observations pertaining to the case when the coefficient aεis independent of ε,in other words when

Following the discussions in Section 6 and Subsection 7.1,two 0th-order approximations of the solution uεto(2.2)are available in this case,namely,

which we shall refer to as the 0thorder uniform expansion of uε,and

which we shall refer to as the 0thorder“natural asymptotic” expansion of uε.The latter is just the one term Taylor expansion of uεwith respect to ε(at zero).is the unique solution to

The particular form of the remainder term in(7.10)follows from(7.9)and Case 3 of Proposition 7.1.We recall that∈ Vσis the unique solutions to(4.10)(or(5.4)).

From Proposition 7.1,we know that and so a Taylor expansion ofwith respect to ε also starts with the termWe would like to understand a little better the answer to the following question“in the process of correctingto make it into auniform approximation to uεin terms of the conductivity coefficient a,will it suffice to add just a finite number of terms in the Taylor series(of)?”.For that purpose we now derive the specific form of the first-order Taylor expansion

To this end,we follow the strategy employed before.As a first step,we define the(εdependent)function∈ Vσby the relation=+and write a minimization problem satisfied byWe then approximate this problem by using heuristic arguments,and define u1as the solution to this simplified problem.In spite of the heuristic nature of our derivation,it is possible to prove that=u1+O(ε)(we shall,however,omit the proof here).

Step 1Derivation of a minimization problem forDue to the definition of,arises as the unique minimizer in Vσ,0of the following energy:

A simple calculation gives

Step 2Simplification of the minimization problem ofIt seems reasonable to assume that the minimization process ofwill principally seek to minimize the terms of order 0 as ε→ 0,that is,the two terms

The minimum of this last expression is achieved when(u+−u−)=on σ.Subject to this relation,the minimization process should then concentrate on the first order terms

Using the corresponding Euler-Lagrange equations,we are led to a candidate u1∈ Vσ,0(for the 0th-order approximation tothat is characterized as the solution to the following problem:

It is indeed possible to prove the following proposition.

Proposition 7.2Let u1∈ H1(Ωσ)be the unique solution to(7.12).There exists a constant C,which only depends on Ω,σand a,such that

The proof of this is fairly straightforward,and follows by carefully considering the boundary value problem satisfied by−−εu1= ε(−u1).We leave the details to the reader.The fact that u1degenerates like a andwhen a tends to∞and 0,respectively,strongly indicates that the estimate−=O(aε+)is the best possible.Higher order terms in the Taylor series ofcould be calculated,and they would degenerate too when a tends to ∞ and 0.This would strongly indicate that no finite Taylor expansion of(at zero)would achieve a uniform approximation to(that is uniform with respect to a).

It is interesting to compare the above calculation of the first two terms in the Taylor Series ofto the calculation carried out in[7].In that paper,the authors considered the Neumann version of the problem(2.2)in the case that aε=a,and they calculated the first two terms in the ε→ 0 asymptotic expansion of the solution to the problem

(i.e.,the case f=0)which we shall also call uε,since the difference in the type of boundary conditions on ∂Ω and source in Ω plays no role for the discussion here. γεis defined by(2.1)as before.The result in[7]is

In this formula,the functionis defined in terms of the Neumann function N(x,y)of Ω,a polarization tensor M(x),and the harmonic function

The polarization tensor M(x)is for x∈σ given byin the local basis(τ(x),n(x)),and the Neumann function is the solution to

where δyis the Dirac distribution centered at x=y.Equivalently,due to the jump relations for single and double layer potentials(see,e.g.,[16,Chapter 3]),∈ H1(Ωσ)is the unique solution(modulo a constant)to the following problem:

We immediately notice that the boundary value problems satisfied by u1andimply that the difference u1−is uniformly bounded with respect to a.If the same thing were to happen for higher terms in the Taylor series,then it would be very consistent with the fact that the difference uε−is uniformly bounded with respect to a.It would also strongly suggest that no finite Taylor expansion of uεwould lead to a uniform approximation(that is uniform in a).

8 Derivation of the 1st-Order Approximation of uε

In the previous sections,we have derived a uniform 0th-order approximation()∈Vσ× H1(ω1)to the couple(,uε◦ Hε) ∈ H1(Ω )× H1(ω1).Properly speaking,we only proved thatis a uniform approximation of“far away from the curve σ”,that is,on subsets of Ωof the form Ω for some fixed δ>0.However,the proof of this fact made use of the heuristic approximate guessfor the potential(uε◦Hε)inside the rescaled inhomogeneity.

Relying on the same strategy,we now briefly outline the derivation of a uniform first-order approximation result for the solution uεto(2.2).We note that the 0th-and first-order analyses turn out to share a lot of common features.Thus for the sake of brevity,we shall omit some of the very tedious calculations related to the latter.

We start from the rescaled form of the problem(4.1)as established in Subsection 4.1.1.The couple(,uε◦ Hε)is the unique minimizer of the energy

among the elements of the space

that additionally satisfies u=ϕ on∂Ω.We have seen that a uniform 0th-order approximation of this couple(in the sense described above)is(,)∈V0,where V0is defined in(4.6),is defined as the solution to the minimization problem(4.10),andis given by(4.13).For technical convenience,we define the couple∈ H1(Ω )× H1(ω1)by the identity

where yε∈ H1(Ω)denotes the unique solution to the problem

and wε∈ H1(ω1)is given by the formula

We note that(x± εn(x))describesas x runs through σ.Due to the introduction of these two auxiliary functions yεand wε,the “unknown” couple()has no “jump” from ∂ωεto∂ω1,i.e.,()lies in.Note that,using the uniform regularity estimates of Theorem 5.1 and arguing as we did for the study of the function zεin Subsection 6.1,we may easily prove that

From its definition,()is the unique minimizer of the functional

among the couples(u,v)∈Vε0,such that u=0 on ∂Ω.To find a uniform 0th-order approximation to(),we expand the functional(u,v)as follows:

We observe that the quadratic part of this energy is the same as that of the 0th-order energy(modulo a factor of ε).The linear part has two components,correponding to the first three linear terms and the last four linear terms of(8.4),respectively.Following this splitting of the linear part,we decompose()as

where()and() ∈are the unique minimizers of the respective energies(u,v)and(u,v),defined by

and

Note that the definition ofslightly differs from the sum of the quadratic terms and thefirst three linear terms of(8.4)by an additive term that only depends on wε(and a factor of ε),which has no effect on the solution to the corresponding minimization problem.

8.1 0th-order approximation of the couple(,)

To obtain a 0th-order approximation(u1,ε,v1,ε)of(,),we follow the same strategy as in Section 4.We use a heuristic argument to build an approximate two-scale minimization problem

This problem can now(heuristically)be solved for v in terms of u,leading to a minimization problem featuring only u.This process yields a candidate(u1,ε,v1,ε)for a uniform 0th-order approximation of(,).Then we can rigorously prove a uniform approximation estimate,using arguments similar to those of Section 6.This estimate would assert that

with C independent of ε and aε.For brevity,we shall not present the proof of this estimate here,instead we limit ourselves to describing the heuristic derivation of the approximate energy

Arguing as in Section 4,and relying on the estimate(8.3),we approximate the quantityby

The problem(8.8)can now be rewritten as

where we define

We(heuristically)solve this minimization problem to get an explicit approximate expression for(u)in terms of u.To this end,we notice that(u)features two terms with different behavior as ε→ 0.Intuitively,the minimizer vuof this composite energy will to lowest order be determined by the termThe corresponding Euler-Lagrange equation asserts that vumust satisfy

Arguing as in Subsection 4.1.1(that is,taking w(x+tn(x))= φ(x)ψ(t)with arbitrary φ ∈C∞(σ)and ψ ∈(−1,1),and using Proposition 2.1),we conclude that the function t?→vu(x+tn(x))is affine for any fixed x∈σ.The boundary conditions of the problem(8.10)now give

Inserting this expression into(8.10),and using(8.2)as well as Proposition 2.1,we arrive at the minimization problem

where(u):=(u,vu)has the following expression:

The solution u1,εto this minimization problem is our candidate for a uniform approximation toThe function v1,ε∈ H1(ω1)defined in the rescaled inhomogeneity by

is our candidate for an approximation to

8.2 0th-order approximation of the couple(,)and the uniform first order approximation result

Let us now turn our attention to the uniform approximation of the solution(,)to the problem

where the energy(u,v)is given by(8.7).Performing calculations somewhat more complicated than those in the previous section it is possible heuristically to arrive at a candidate(u2,ε,v2,ε)for a uniform approximation.We shall not present these calculations here,but only state the result as follows.

The function u2,εis the solution to the problem

where the functionalis given by

The function v2,ε∈ H1(ω1)is defined as

the function w2,ε∈ H1(ω1)being given by

with

It is then possible to prove that

with C independent of ε and aε.Combining the decompositions(8.1)and(8.5)with(8.3)and the above estimates for−u1,εand−u2,ε,we would now arrive at the following theorem.

Theorem 8.1In the situation described in Section 2.1,let δ>0 be a fixed positive real number,f ∈ Fδand ϕ ∈ H12(∂Ω).Let uε∈ H1(Ω)be the unique solution of the minimization problem(4.1),let u0εbe the unique solution to(4.10)and u1,ε,u2,εbe the unique solutions to(8.11)and(8.13).Then the following estimates hold for ε>0 sufficiently small:

where the constant C depends only on Ω and σ,and is independent of f,ϕ,ε and the sequence aε.

Remark 8.1In view of these results it is interesting to expand a little on the discussion of Subsection 7.2,concerning the comparison between the uniform asymptotic expansion of uε(uniform,with respect to the conductivity aε)and the “natural” asymptotic expansion(7.13)in the particular case,where aεis a fixed real number a>0 independent of ε.

For fixed aε=a,arguing as in Subsection 7.2,one may show that the following expansion holds for the first-order term(u1,ε+u2,ε)of the uniform asymptotic expansion of uεas ε→ 0:

where U1∈ Vσ,0is characterized by the following equations

Hence,it is verified exactly how the first-order termof the a-dependent“natural” asymptotic expansion of uε(defined as in(7.14),but with a homogeneous Dirichlet boundary condition on ∂Ω)decomposes as the sum of the first-order term u1of the principal uniform expansion(defined by(7.12))and of the leading term U1of the first-order term(u1,ε+u2,ε)in the uniform expansion of uε.

9 Appendix Proof of the Uniform Regularity Estimates for

This appendix is devoted to the proof of Theorem 5.1.For the reader’s convenience,let usfirst recall a useful characterization of W1,pspaces.Let Ω⊂R2be an open set,and suppose 1

If Ω and V are both convex,then it is fairly simple to prove that

for any vector h ∈ R2with|h|

Proposition 9.1Let u ∈ Lp(Ω).Then the following assertions are equivalent:

(i)u belongs to W1,p(Ω).

(ii)There exists a constant C>0,such that

(iii)There exists a constant C>0,such that for any open subset V?Ω,

Furthermore,the smallest constant C satisfying(ii)or(iii)is C=

We are now in position to prove the desired result.

Proof of Theorem 5.1The proof of this result is an adaptation of that of Theorem 9.25 in[8],and relies on the method of translations.First we observe that,by a standard argument of partition of unity,it is enough to prove thatbelongs to H2(V σ)and that the estimate(5.11)holds with Vσ instead of Ωσ,where V is a sufficiently small(convex)neighborhood in Ω of an arbitrary point x0∈Three cases must be distinguished as follows:

(i)x0belongs to Ω σ.

(ii)x0lies on∂Ω.

(iii)x0lies on σ.

The uniform estimate(5.12)arises as a consequence of the treatment of Case(iii).

Case(i)Let V and W be open convex subsets of Ω+(or Ω−)with VWΩ+(or Ω−).Let χ ∈(Ωσ)be a smooth cuto fffunction with

Then,for any test function v∈ H1(Ωσ),

where we used the variational formulation(5.1)with a test function whose support is compact in Ω σ.Let us now define wε:=Our goal is to use the method of translations to show that∇wεbelongs to H1(Ωσ).Let h∈ R2be any vector of sufficiently small length,and let us insert D−hDhwε∈ H1(Ωσ)as a test function in(9.1).The result is

Here we use the following formula for the difference quotient of a product:

as well as “discrete integration by parts” for the difference quotients(which is nothing but change of variables in the corresponding integrals).We recall that for h sufficiently small?less thandist(W,∂(Ωσ))Dhwεhas compact support in some convex,with WΩ+(or Ω−).From(9.2),we now obtain

Using the Poincaré inequality forfunctions vanishing on,we have that there exists a constant C which only depends on,such that

From(9.3),we conclude that

IfΩ+,then,due to Lemma 5.1,

On the other hand,ifis a subset of Ω−,then we have a priori no bound onTo circumvent this,we note that from the very beginning,we could rewrite the entire argument by replacingin the various integral inequalities by−m,where m is an arbitrary constant.This includes the definition of wε,which now becomes wε= χ(−m).We selectand from the“revised”version of(9.4),we now obtain

owing to the Poincaré-Wirtinger inequality and Lemma 5.1.Whetheris a subset of Ω+or Ω−,Proposition 9.1 now allows us to conclude that all the entries of the Hessian matrixbelong to L2(W),and that the following inequality holds:

Case(ii)The proof in this case is similar to that of(i),modulo the usual changes of the method of translation due to the presence of the boundary(see[8,Theorem 9.25]again).We omit the details and concentrate instead on those of Case(iii).

Case(iii)Let VΩ be a sufficiently small convex neighborhood of the point x0∈σ.Let W be another convex open subset of Ω,such that VWΩ,and let χ ∈(Ω)be a smooth cuto fffunction,such that

To simplify notations,we assume that σ∩W is flat(the general case being no more difficult,but more involved as far as notations are concerned).The tangent vector τ to σis the coordinate vector ex,and the normal vector n,pointing outward from Ω−,is ey.Following the steps of the proof of(i),let wε= χ(−m)for some constant m to be specified later.A simple calculation reveals that wεsatisfies

for all v ∈ Vσ,0.Here gε=fχ −·∇χ and hε=(− m)∇χ.

Let us introduceand letbe defined asWe now use the method of translations to show that the tangential derivativesrespectively.To this end,let h=tτ=tex,for t>0 sufficiently small,and choose v=D−hDhin W−and v=0 in W+,and then v=0 in W−and v=D−hDhin W+as test functions in(9.5).This yields

where=(− m0)∇χ,and

whereNote that,by performing an integration by parts on the first integral in the right-hand side of(9.6),we can rewrite

A similar identity holds for the first integral in the right-hand side of(9.7).Combining(9.6),(9.7)and(9.8),we obtain

and

Some of the terms in the right-hand sides of the above inequalities can be estimated further.Owing to Poincaré’s inequality,there exists a constant C(which only depends on W and σ),such that for any function u∈H1(Wσ)with u=0 on∂W,

Similarly,there exists a constant C(still depending only on W and σ),such that for any function u∈H1(σ)with u=0 on ∂W ∩ σ,

From Proposition 9.1(and the equivalent for σ)we conclude that

In particular,we deduce from(9.13)that

Using(9.11),we obtain that there exists a constant C,independent of ε and aε,such that

From the a priori estimates of Lemma 5.1,it also follows that

From the Poincar-Wirtinger inequality on σ,we have

We now sum(9.9)and(9.10),noticing thaton σ.Taking into account(9.14)–(9.17),we arrive at

Some terms in this last expression still need to be rewritten.We observe that

where we used the uniform a priori estimates of Lemma 5.1.This inequality,in combination with the fact that

allows us to rewrite the last integral in the left-hand side of(9.18)as follows:

It follows,using the algebraic identity(5.2)and(9.19),that there exist two positive constants C1and C2,which do not depend on ε or aε,such that

We now estimate the next to last integral in the left-hand side of(9.18).It may be rewritten

with

and so

Turning to the right-hand side of(9.18),we have

due to(9.19)and the uniform a priori estimates of Lemma 5.1.Here we have also used that εaε≤Combining(9.18),(9.20)–(9.22),and using Lemma 5.1,we finally get

In particular,

from which Proposition 9.1 allows us to conclude thatH1(W+),with the estimate

the constant C being independent of ε and aε.

We have to obtain the corresponding estimate forFirst

To get control ofwe go back to the original equation(5.4)satisfied by

These two observations lead to a uniform H1(Vσ)estimate forand thus to the desired uniform H2(V σ)seminorm estimate forFrom(9.23),it also follows that

and this completes the proof of Theorem 5.1.

Remark 9.1In this proof,we relied in a crucial way on the ordering εaε≤between the coefficients appearing in the approximate energy(4.12).We do not know whether the similar uniform regularity estimate holds in other regimes of coefficients.

PostscriptAny opinion,findings,and conclusions or recommendations expressed in this paper are those of the authors,and do not necessarily reflect the views of the National Science Foundation.

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