Topological sensitivities via a Lagrangian approach for semi-linear problems
aa r X i v : . [ m a t h . O C ] M a r Topological sensitivities via a Lagrangian approachfor semilinear problems
Kevin Sturm ∗ Abstract
In this paper we present a methodology that allows the efficient computation of the topo-logical derivative for semilinear elliptic problems within the averaged adjoint Lagrangianframework. The generality of our approach should also allow the extension to evolutionaryand other nonlinear problems. Our strategy relies on a rescaled differential quotient of theaveraged adjoint state variable which we show converges weakly to a function satisfying anequation defined in the whole space. A unique feature and advantage of this framework isthat we only need to work with weakly converging subsequences of the differential quotient.This allows the computation of the topological sensitivity within a simple functional analyticframework under mild assumptions.
Primary 49Q10; Secondary 49Qxx,90C46.
Keywords: shape optimisation; topology optimisation; asymptotic analysis; shape sensitiv-ity; averaged adjoint approach.
Shape functions (also called shape functionals) are real valued functions defined on sets ofsubsets of the Euclidean space R d . The field of mathematics dealing with the minimisation ofshape functions that are constrained by a partial differential equation is called PDE constrainedshape optimisation . Numerous applications in the engineering and life sciences, such as theaircraft and car design or electrical impedance / magnetic induction tomography, underline itsimportance; [
24, 25 ] . Among other approaches [
9, 12, 16, 31, 35 ] the topological derivative ap-proach [ ] constitutes an important tool to solve shape optimisation problems for whichthe final topology of the shape is unknown. We refer to the monograph [ ] and referencestherein for applications of this approach.The idea of the topological derivative is to study the local behaviour of a shape function J with respect to a family of singular perturbations ( Ω ǫ ) . Two important singular perturbationsare obtained by translating and scaling of an inclusion ω which contains the origin by ω ǫ ( z ) : = z + ǫω ; then the singular perturbations are given by Ω ǫ : = Ω ∪ ω ǫ ( z ) for z ∈ Ω c and Ω ǫ : = Ω \ ω ǫ ( z ) for z ∈ Ω . Both singular perturbations are examples of the class of perturbations ∗ Technische Universität Wien, Wiedner Hauptstr. 8-10, 1040 Vienna, Austria, E-Mail: [email protected] [ ] . The topological derivative of a shape function J with respect to perturbations ( Ω ǫ ) is defined by ∂ J ( Ω ) : = lim ǫ ց J ( Ω ǫ ) − J ( Ω ) ℓ ( ǫ ) , (1.1)where ℓ : [ τ ] → R , τ >
0, is an appropriate function depending on the perturbation chosen.If Ω is perturbed by a family of transformations Φ ǫ : = Id + ǫ V : R d → R d generated by a Lipschitzvector field V : R d → R d , that is, Ω ǫ : = Φ ǫ ( Ω ) , then we can choose ℓ ( ǫ ) = ǫ and (1.1) reduces tothe definition of the shape derivative [ ] . So the topological derivative can be seen as an gen-eralisation of the shape derivative. In some cases, notably when shape functions are constrainedby elliptic partial differential equations, the topological derivative can be obtained as the singu-lar limit of the shape derivative as presented in the monograph [
31, pp. 12 ] . While the shapederivative can be interpreted as the Lie derivative on a manifold, the topological derivative ismerely a semi-differential defined on a cone, which makes its computation a challenging topic;see [ ] .The goal of this paper is to give a coincide way to compute topological sensitivities for thefollowing class of semilinear problems. Given a bounded domain D ⊂ R d , d ∈ {
2, 3 } , withLipschitz boundary ∂ D we want to find the topological derivative of the objective function J ( Ω ) : = ˆ D j ( x , u ( x )) d x ª (C)in an open set Ω ⊂ D subject to u = u Ω solves the semilinear transmission problem − div ( β ∇ u + ) + ̺ ( u + ) = f in Ω − div ( β ∇ u − ) + ̺ ( u − ) = f in D \ Ω u − = ∂ D ( β ∇ u + ) ν = ( β ∇ u − ) ν on ∂ Ω u + = u − on ∂ Ω (S)where u + , u − denote the restriction of u to Ω and D \ Ω , respectively. The function ν denotes theoutward pointing unit normal field along ∂ Ω . The technical assumptions for the matrix valuedfunctions β , β and the scalar functions j , ̺ , ̺ , f , f will be introduced in Section 4. A relatedwork is [
31, Ch. 10, pp. 277 ] , which is based on the research article [ ] , where a semilinearproblem without transmission conditions in a Hölder space setting is studied.There are two main approaches to compute topological derivatives for PDE constrained shapefunctions. A typical and general strategy to obtain the topological sensitivity is to derive theasymptotic expansion of the partial differential equation with respect to the singular perturba-tion of the shape [
29, 30 ] . For our problem above this would amount to prove that an expansionof the form (see [
31, p. 280 ] ) u ǫ ( x ) = u ( x ) + ǫ K ( ǫ − x ) + ǫ ( K ( ǫ − x ) + u ′ ( x )) + R ǫ ( x ) (1.2)exists. Here K , K are so-called called boundary layer correctors, which solve certain exteriorboundary value problems and u ′ is called regular corrector and solves a linearised system. Theopologicalderivativevia Lagrange 3function u ǫ denotes the solution to (S) for the singular perturbed domain Ω ǫ and R ǫ ( x ) is anappropriate remainder. However, the proof of an expansion like (1.2) can technically involvedand depends very much on the problem; [ ] .A second approach to compute the topological derivative is presented in [ ] and based on aperturbed adjoint equation, see also [
5, 6, 11, 22, 23 ] and [ ] . A key of this method is to prove u ǫ ( x ) = u ( x ) + ǫ K ( ǫ − x ) + R ǫ ( x ) , p ǫ ( x ) = p ( x ) + ǫ Q ( ǫ − x ) + R ǫ ( x ) , (1.3)where K is the same as in (1.2), Q is the solution to an exterior problem, and R ǫ , R ǫ are ap-propriate remainder that have to go to zero in some function space. Here p ǫ is the solution to acertain perturbed adjoint equation depending on the derivative of J ; see [ ] . As a by-product ofthis approach one obtains the topological sensitivity for non-transmission type problems whereNeumann boundary conditions on the inclusion are imposed. However, the proof of the ex-pansions (1.3), particularly for nonlinear problems, can be technically involved and necessitateknowledge of the asymptotic behaviour of Q and K at infinity.In this paper we will show that neither the expansion (1.2) nor (1.3) are necessary to obtainthe topological sensitivity for (S). For this purpose, we use a Lagrangian approach which usesthe averaged adjoint variable q ǫ [
15, 36, 37 ] . The key ingredient, which leads to the existenceof the topological derivative of (C), is the convergence property ∇ (cid:129) q ǫ ( z + ǫ x ) − q ( z + ǫ x ) ǫ ‹ * ∇ Q weakly in L ( R d ) d , (1.4)where Q is the same function as in (1.3). The averaged adjoint variable reduces to the usualadjoint in the unperturbed situation, that is, q = q = p = p . We emphasise that the weakconvergence property (1.4) is a relaxation of (1.2) and (1.3), since no remainder estimates arenecessary. In addition no further knowledge about the asymptotic behaviour of Q at infinity isneeded. We will demonstrate that the proof of (1.4) is constructive in that it reveals the equation Q must satisfy. This is particularly important when dealing with other more complicated non-linear equations, e.g., quasilinear equations. We will show that our strategy also allows, withminor changes, to treat the extremal case where β , ̺ , f =
0, i.e., the transmission problem(S) reduces to a semilinear equation with homogeneous Neumann boundary conditions on ∂ Ω .Compared to previous works we can prove the existence of the topological derivative undermilder assumptions on the regularity of the inclusion. Notation and definitions
Notation for derivatives
Let ( ǫ , u , q ) G ( ǫ , u , q ) : [ τ ] × X × Y → R be a function defined onreal normed vector spaces X , Y , and τ >
0. When the limits exist we use the following notation: v ∈ X , ∂ u G ( ǫ , u , q )( v ) : = lim t ց G ( ǫ , u + t v , q ) − G ( ǫ , u , q ) t (1.5) w ∈ Y , ∂ q G ( ǫ , u , q )( w ) : = lim t → G ( ǫ , u , q + t w ) − G ( ǫ , u , q ) t . (1.6)The notation t ց t goes to 0 by strictly positive values. K. Sturm Miscellaneous notation
Standard L p spaces and Sobolev spaces on an open set Ω ⊂ R d aredenoted L p ( Ω ) and W kp ( Ω ) , respectively, where p ≥ k ≥
1. In case p = k ≥ H k ( Ω ) : = W k ( Ω ) . Vector valued spaces are denoted L p ( Ω ) d : = L p ( Ω , R d ) and W kp ( Ω ) d : = W kp ( Ω , R d ) . We write ffl A f d x : = | A | ´ A f d x to indicate the average of f over ameasurable set A with measure | A | < ∞ . The Hölder conjugate of p ∈ [ ∞ ) is defined by p ′ : = p / ( p − ) . For 1 ≤ p < d we denote by p ∗ : = d p / ( d − p ) the Sobolev conjugate of p . Givena normed vector space V we denote by L ( V , R ) the space of linear and continuous functions on V . We denote by B δ ( x ) the ball centred at x with radius δ > B δ ( x ) : = B δ ( x ) . The following material can be found in [ ] . We begin with the definition of a Lagrangianfunction. Definition 2.1.
Let X and Y be vector spaces and τ >
0. A parametrised Lagrangian (or shortLagrangian) is a function ( ǫ , u , q ) G ( ǫ , u , q ) : [ τ ] × X × Y → R ,satisfying for all ( ǫ , u ) ∈ [ τ ] × X , q G ( ǫ , u , q ) is affine on Y . (2.1)The next definition formalises the notion of state and perturbed state variable associatedwith G . Definition 2.2.
For ǫ ∈ [ τ ] we define the state equation by: find u ǫ ∈ X , such thatfind u ǫ ∈ X such that ∂ q G ( ǫ , u ǫ , 0 )( ϕ ) = ϕ ∈ X . (2.2)The set of solution of (2.2) (for ǫ fixed) is denoted by E ( ǫ ) . For ǫ =
0, the elements of E ( ǫ ) arecalled unperturbed states (or short states) and for ǫ > perturbed states . Definition 2.3.
We introduce for ǫ ∈ [ ǫ ] the set of minimisers X ( ǫ ) = { u ǫ ∈ E ( ǫ ) : inf u ∈ E ( ǫ ) G ( ǫ , u , 0 ) = G ( ǫ , u ǫ , 0 ) } . (2.3)Notice that X ( ǫ ) ⊂ E ( ǫ ) and that X ( ǫ ) = E ( ǫ ) whenever E ( ǫ ) is a singleton. We associatewith the parameter ǫ the parametrised infimum ǫ g ( ǫ ) : = inf u ∈ E ( ǫ ) G ( ǫ , u , 0 ) : [ τ ] → R . (2.4)We now recall sufficient conditions introduced in [ ] under which the limit d ℓ g ( ) : = lim ǫ ց g ( ǫ ) − g ( ) ℓ ( ǫ ) (2.5)exists, where ℓ : [ τ ] → R is a given function satisfying ℓ ( ) = ℓ ( ǫ ) > ǫ ∈ ( τ ] .The key ingredient is the so-called averaged adjoint equation . The definition of the averagedadjoint equation requires that the set of states is nonempty:opologicalderivativevia Lagrange 5 Assumption (H0) . For all ǫ ∈ [ τ ] the set X ( ǫ ) is nonempty.Before we can introduce the averaged adjoint equation we need the following hypothesis. Assumption (H1) . For all ǫ ∈ [ τ ] and ( u , u ǫ ) ∈ X ( ) × X ( ǫ ) we assume:(i) For all q ∈ Y , the mapping s G ( ǫ , su ǫ +( − s ) u ) , q ) : [
0, 1 ] → R is absolutely continuous.(ii) For all ( ϕ , q ) ∈ X × Y and almost all s ∈ (
0, 1 ) the function s ∂ u G ( ǫ , su ǫ + ( − s ) u , q )( ϕ ) : [
0, 1 ] → R (2.6)is well-defined and belongs to L (
0, 1 ) . Remark 2.4.
Notice that item (i) implies that for all ǫ ∈ [ τ ] , ( u , u ǫ ) ∈ X ( ) × X ( ǫ ) and q ∈ Y , G ( ǫ , u ǫ , q ) = G ( ǫ , u , q ) + ˆ ∂ u G ( ǫ , su ǫ + ( − s ) u , q )( u ǫ − u ) ds . (2.7)This follows at once by applying the fundamental theorem of calculus to s G ( ǫ , su ǫ + ( − s ) u , q ) on [
0, 1 ] .The following gives the definition of the averaged adjoint equation; see [ ] . Definition 2.5.
Given ǫ ∈ [ τ ] and ( u , u ǫ ) ∈ X ( ) × X ( ǫ ) , the averaged adjoint state equation is defined as follows: find q ǫ ∈ X , such that ˆ ∂ u G ( ǫ , su ǫ + ( − s ) u , q ǫ )( ϕ ) ds = ϕ ∈ X . (2.8)For every triplet ( ǫ , u , u ǫ ) the set of solutions of (2.8) is denoted by Y ( ǫ , u , u ǫ ) and its elementsare referred to as adjoint states for ǫ = averaged adjoint states for ǫ > Y ( u ) : = Y ( u , u ) is the usual set of adjoint states associated with u , Y ( u ) = { q ∈ Y : ∀ ϕ ∈ X , ∂ u G ( u , q )( ϕ ) = } . (2.9)An important consequence of the introduction of the averaged adjoint state is the followingidentity: for all ǫ ∈ [ τ ] , ( u , u ǫ ) ∈ X ( ) × X ( ǫ ) and q ǫ ∈ Y ( ǫ , u , u ǫ ) , g ( ǫ ) = G ( ǫ , u ǫ , q ǫ ) = G ( ǫ , u , q ǫ ) . (2.10)This is readily seen by substituting q ǫ into equation (2.7). The following result is an extensionof [
15, Thm. 3.1 ] . We refer the reader to [
14, 38 ] for further results on the averaged adjointapproach and [ ] for more examples involving the shape derivative. Theorem 2.6 ( [ ] ) . Let Hypothoses (H0) and (H1) and the following conditions be satisfied.(H2) For all ǫ ∈ [ τ ] and ( u , u ǫ ) ∈ X ( ) × X ( ǫ ) the set Y ( ǫ , u , u ǫ ) is nonempty. K. Sturm(H3) For all u ∈ X ( ) and q ∈ Y ( u ) the limit ∂ ℓ G ( u , q ) : = lim ǫ ց G ( ǫ , u , q ) − G ( u , q ) ℓ ( ǫ ) exists. (2.11)(H4) There exist sequences ( u ǫ ) and ( q ǫ ) , where u ǫ ∈ X ( ǫ ) and q ǫ ∈ Y ( ǫ , u , u ǫ ) , such that thelimit R : = lim ǫ ց G ( ǫ , u , q ǫ ) − G ( ǫ , u , q ) ℓ ( ǫ ) exists. (2.12)Then we have d ℓ g ( ) = ∂ ℓ G ( u , q ) + R . (2.13)Moreover, R = R ( u , q ) does not depend on the choice of the sequences ( u ǫ ) and ( q ǫ ) , but onlyon u and q . Proof.
Thanks to Hypothoses (H0)-(H2) the sets X ( ǫ ) and Y ( ǫ , u , u ǫ ) are nonempty for all ǫ .Therefore in view of (2.10) we have for all ǫ ∈ [ τ ] , ( u , u ǫ ) ∈ X ( ) × X ( ǫ ) and q ǫ ∈ Y ( ǫ , u ǫ , u ) , g ( ǫ ) − g ( ) = G ( ǫ , u , q ǫ ) − G ( u , q )= G ( ǫ , u , q ǫ ) − G ( ǫ , u , q ) + G ( ǫ , u , q ) − G ( u , q ) . (2.14)Thus selecting ( u ǫ ) and ( q ǫ ) from Hypothosis (H4) and using Hypothosis (H3) we obtain d ℓ g ( ) = lim ǫ ց G ( ǫ , u , q ) − G ( u , q ) ℓ ( ǫ ) + lim ǫ ց G ( ǫ , u , q ǫ ) − G ( ǫ , u , q ) ℓ ( ǫ )= ∂ ℓ G ( u , q ) + R . (2.15)It follows from (2.15) that R only depends on u and q . Remark 2.7.
An important application of Theorem 2.6 is the computation of shape derivativesfor which one chooses ℓ ( ǫ ) = ǫ , see e.g., [
36, 38 ] . In this case one typically has R ( u , q ) = d ǫ g ( ) = ∂ ǫ G ( u , q ) . (2.16)However for the topological derivative, in which case ℓ ( ǫ ) = ǫ , the term R ( u , q ) is typicallynot equal to zero as shown by the one dimensional example of [ ] . d In preparation for the study of the semilinear problem (S), we first recall existence and unique-ness results for the following exterior problem. Let ω ⊂ R d be an open and bounded set, andlet ζ ∈ R d be a vector. Given a suitable vector space V of functions R d → R we consider: find Q ζ ∈ V such that ˆ R d A ∇ ψ · ∇ Q ζ d x = ˆ ω ζ · ∇ ψ d x for all ψ ∈ V . (3.1)opologicalderivativevia Lagrange 7Here A : R d → R d × d is a measurable, uniformly coercive (not necessarily symmetric) matrix-valued functions, that is, there are constants M , M >
0, such that M | v | ≤ A ( x ) v · v ≤ M | v | for a.e x ∈ R d and all v ∈ R d . (3.2)The well-posedness of (3.1) can be achieved by several choices of V . The most popular onesare weighted Sobolev spaces; see [ ] . In the next section we discuss a more straight forwardchoice for V . Definition 3.1.
For d ≥ B L ( R d ) : = { u ∈ H ( R d ) : ∇ u ∈ L ( R d ) d } . (3.3)Then the Beppo-Levi space is defined by˙
B L ( R d ) : = B L ( R d ) / R , (3.4)where / R means that we quotient out the constant functions. We denote by [ u ] the equivalenceclasses of ˙ B L ( R d ) . The Beppo-Levi space is equipped with the norm k [ u ] k ˙ H ( R d ) : = k∇ u k L ( R d ) d , u ∈ [ u ] . (3.5)The Beppo-Levi space is a Hilbert space (see [
17, 32 ] and also [ ] ) and C ∞ c ( R d ) / R is densein ˙ B L ( R d ) . Lemma 3.2.
Let d ≥ A satisfies (3.1). Then there exists a unique equivalenceclass [ Q ] ∈ ˙ B L ( R d ) solving ˆ R d A ∇ ψ · ∇ Q d x = ˆ ω ζ · ∇ ψ d x for all ψ ∈ B L ( R d ) . (3.6) Proof.
This is a direct consequence of the theorem of Lax-Milgram.As shown in [ ] , in dimension d ≥ [ u ] of ˙ B L ( R d ) contains anelement u ∈ [ u ] that is in turn contained in the Banach space E ( R d ) : = { u ∈ L ∗ ( R d ) : ∇ u ∈ L ( R d ) d } (3.7)equipped with the norm k u k E : = k u k L ∗ + k∇ u k ( L ) d . (3.8)This follows at once since C ∞ c ( R d ) / R is dense in ˙ B L ( R d ) and from the Gagliardo-Nirenberg-Sobolev inequality; see [ ] . As a result for d ≥ E ( R d ) and can even consider a more general problem. Lemma 3.3.
Let d ≥
3. Suppose that A satisfies (3.1) and A = A ⊤ a.e. on R d . Then for every F ∈ L ( E ( R d ) , R ) there exists a unique solution Q ∈ E ( R d ) to ˆ R d A ∇ ψ · ∇ Q d x = F ( ψ ) for all ψ ∈ E ( R d ) . (3.9) Proof.
A proof can be found in the appendix.So for d ≥ V : = E ( R d ) since obviously F ( ϕ ) : = ´ ω ζ · ∇ ϕ d x ∈ L ( E ( R d ) , R ) . K. Sturm Since the exterior equation (3.1) is, as we will see later, of paramount importance for the firsttopological derivative we review here an alternative choice for the space V , namely, a weightedSobolev Hilbert space. We follow the presentation of [ ] , where a more general situation thanthe following is considered.For this purpose we introduce the weight function w ∈ L ( R d ) defined by w ( x ) : = ( + | x | ) − γ d (3.10)where γ d : = d + δ and δ ∈ (
0, 1 / ) is arbitrary, but fixed. Since the weight satisfies | w | p ≤ | w | on R d for p ∈ [ ∞ ) it also follows that w ∈ L p ( R d ) for all p ∈ [ ∞ ) . Definition 3.4.
The weighted Hilbert Sobolev space H w ( R d ) is defined by H w ( R d ) : = { u : R d → R measurable : p wu ∈ L ( R d ) , ∇ u ∈ L ( R d ) d } . (3.11)The norm on H w ( R d ) is given by k u k H w : = kp wu k L + k∇ u k ( L ) .The weight w is chosen in such a way that the set of constant functions on R d are contained in H w ( R d ) . Therefore it is clear that (3.1) can only be uniquely solvable in H w ( R d ) up to a constant.A remedy is to consider the quotient space˙ H w ( R d ) : = H w ( R d ) / R (3.12)and equip this space, as in [ ] , with the quotient norm k [ u ] k ˙ H w : = inf c ∈ R k u + c k H w ( R d ) , (3.13)where [ u ] denote the equivalence classes of ˙ H w ( R d ) . In [
7, Cor. C.5, p. 23 ] it is shown that thereis a constant c >
0, such that, k [ u ] k ˙ H w ≤ c k∇ u k ( L ) d for all u ∈ H w ( R d ) . Therefore existence of asolution to (3.1) follows directly from the theorem of Lax-Milgram.In the following lemma let us a agree that the Sobolev conjugate of 2 in dimension two isgiven by ∞ , i.e. 2 ∗ : = ∞ if d = Lemma 3.5.
We have E ( R d ) , → H w ( R d ) for all d ≥
2, i.e., there is a constant C >
0, such that k u k H w ≤ C k u k E for all u ∈ E ( R d ) . (3.14) Proof.
Let u ∈ E ( R d ) be given so that u ∈ L ∗ ( R d ) . In case d =
2, we have 2 ∗ = dd − . Thereforethe Hölder conjugate of 2 ∗ / ∗ ∗ − = d / ˆ R d wu d x ≤ k w k L d / ( R d ) k u k L p ∗ ( R d ) . (3.15)Since d ≥ p wu ∈ L ( R d ) and since by definition also ∇ u ∈ L ( R d ) d we deduce E ( R d ) ⊂ H w ( R d ) and the continuity of the embedding follows from (3.15). In case d = ∗ = ∞ and thus Hölder’s inequality directly gives (3.15) and thus the continuousembedding.opologicalderivativevia Lagrange 9 In this section we show how Theorem 2.6 of Section 2 can be used to compute the topologicalderivative for a semilinear transmission problem. Our approach is related to the one of [ ] (seealso [ ] ), where also a perturbed adjoint equation is used, too. However the main differencehere is that we only need to work with weakly converging subsequences and do not need toknow any asymptotic behaviour of the limiting function. In the following exposition we restrict ourselves to the shape function J ( Ω ) = ˆ D u d x , (4.1)where u = u Ω ∈ H ( D ) ∩ L ∞ ( D ) is the weak solution of (S): ˆ D β Ω ∇ u · ∇ ϕ + ̺ Ω ( u ) ϕ d x = ˆ D f Ω ϕ d x for all ϕ ∈ H ( D ) , (4.2)where β Ω : R d → R d × d and f Ω : R d → R are defined by β Ω ( x ) : = (cid:26) β ( x ) for x ∈ Ω β ( x ) for x ∈ R d \ Ω , f Ω ( x ) : = (cid:26) f ( x ) for x ∈ Ω f ( x ) for x ∈ R d \ Ω , (4.3)and similarly ̺ Ω is defined by ̺ Ω ( u ) : = (cid:26) ̺ ( u ) for x ∈ Ω ̺ ( u ) for x ∈ R d \ Ω . (4.4)Notice that β Ω = β χ Ω + β χ R d \ Ω , f Ω = f χ Ω + f χ R d \ Ω and ̺ Ω ( u ) = ̺ ( u ) χ Ω + ̺ ( u ) χ R d \ Ω .It can be checked that the following proofs remain true when the shape function (4.1) isreplaced by (C) from the introduction under the assumption that j is sufficiently smooth. How-ever, in favour of a clearer presentation we use the simplified cost function (4.1). The functions β i , ̺ i and f i are specified in the following assumption. The extremal case where β , ̺ , f arezero will be discussed in the last section. Assumption 1. (a) For i =
1, 2, we assume that β i ∈ C ( R d ) d × d and that there are constants β m , β M >
0, such that β m | v | ≤ β i ( x ) v · v ≤ β M | v | for all x ∈ R d , v ∈ R d . (4.5)(b) For i =
1, 2, we assume ̺ i ∈ C ( R ) , ̺ i ( ) = ( ̺ i ( x ) − ̺ i ( y ))( x − y ) ≥ x , y ∈ R . (4.6)(c) For i =
1, 2, we assume f i ∈ H ( D ) if f = f and f i ∈ H ( D ) ∩ C ( D ) if f = f .0 K. SturmNotice that since for x ∈ D the matrix β Ω ( x ) is either equal to β ( x ) or β ( x ) and in view ofthe bound (4.5), we have β m | v | ≤ β Ω ( x ) v · v for all x ∈ R d , v ∈ R d . (4.7)Similarly, in view of the monotonicity property (4.6) and ̺ i ( ) =
0, we get0 ≤ ̺ Ω ( x ) x for all x ∈ R d . (4.8) Lemma 4.1. (i) Let f ∈ L r ( D ) , r > d /
2. Then for every measurable set Ω ⊂ D there is aunique solution u Ω of (4.2). Moreover, there is a constant C independent of Ω , such that k u Ω k L ∞ ( D ) + k u Ω k H ( D ) ≤ C k f k L r ( D ) . (4.9)(ii) For every z ∈ D \ Ω , we find δ >
0, such that u Ω ∈ H ( B δ ( z )) . Proof. (i) Our assumptions imply that we can apply [
39, Theorem 4.5 ] which gives the existenceof a solution to (4.2) and also the apriori bound (4.9). As pointed out in this reference theconstant C is independent of the nonlinearity ̺ Ω .(ii) Let U : = D \ Ω and z ∈ U . The restriction of u to U solves ˆ U β ∇ u · ∇ ϕ d x = ˆ U ˜ f ϕ d x for all ϕ ∈ H ( U ) , (4.10)with right-hand side ˜ f ( x ) : = f ( x ) − ̺ ( u ( x )) . Since ∇ ˜ f = ∇ f − ̺ ′ ( u ) ∇ u ∈ L ( U ) d we have˜ f ∈ H ( U ) . Hence u ∈ H ( U ) by standard regularity theory for elliptic PDEs; see, e.g., [
20, Thm.2, p. 314 ] . Since U is open we can choose δ > B δ ( z ) ⋐ U . This finishes the proof. Remark 4.2.
Although we restrict ourselves to Dirichlet boundary conditions in (S) other bound-ary conditions, e.g., Neumann boundary conditions, can be considered as well. This only requiresminimal changes in the following analysis and we will make remarks at the relevant places.
From now on we fix: • an open and bounded set ω ⊂ R d with 0 ∈ ω , • an open set Ω ⋐ D and a point z ∈ D \ Ω , • the perturbation Ω ǫ : = Ω ∪ ω ǫ ( z ) , where ω ǫ ( z ) : = z + ǫω and ǫ ∈ [ τ ] , τ > ω ǫ instead of ω ǫ ( z ) . Let X = Y = H ( D ) and introducethe Lagrangian G : [ τ ] × X × Y → R associated with the perturbation Ω ǫ by G ( ǫ , u , q ) : = ˆ D u d x + ˆ D β ǫ ∇ u · ∇ q + ̺ ǫ ( u ) q d x − ˆ D f ǫ q d x , (4.11)where we use the abbreviations β ǫ : = β χ Ω ǫ + β χ R d \ Ω ǫ , f ǫ : = f χ Ω ǫ + f χ R d \ Ω ǫ , ̺ ǫ ( u ) : = ̺ ( u ) χ Ω ǫ + ̺ ( u ) χ R d \ Ω ǫ . (4.12)We are now going to verify that Hypotheses (H0)-(H4) are satisfied with ℓ ( ǫ ) = | ω ǫ | . Moreover,we will determine the explicit form of R ( u , p ) .opologicalderivativevia Lagrange 11 Remark 4.3 (Removing an inclusion) . We only treat the case of "adding" a hole here, i.e., Ω ǫ : = Ω ∪ ω ǫ ( z ) for z ∈ D \ Ω . The second case of "removing" a hole, i.e., Ω ǫ : = Ω \ ω ǫ ( z ) for z ∈ Ω can be dealt with in the same way. The perturbed state equation reads: find u ǫ ∈ H ( D ) such that ∂ q G ( ǫ , u ǫ , 0 )( ϕ ) = ϕ ∈ H ( D ) , (4.13)or equivalently u ǫ ∈ H ( D ) satisfies, ˆ D β ǫ ∇ u ǫ · ∇ ϕ + ̺ ǫ ( u ǫ ) ϕ d x = ˆ D f ǫ ϕ d x for all ϕ ∈ H ( D ) . (4.14)Henceforth we write u : = u to simplify notation. Since (4.14) is precisely (4.2) with Ω = Ω ǫ ,we infer from Lemma 4.1 that (4.14) admits a unique solution. This means that E ( ǫ ) = { u ǫ } isa singleton and thus E ( ǫ ) = X ( ǫ ) and Hypothesis (H0) is satisfied. From this and Assumption 1we also infer that Hypothesis (H1) is satisfied. We proceed by shoing a Hölder-type estimate for ( u ǫ ) . Lemma 4.4.
There is a constant C >
0, such that for all small ǫ > k u ǫ − u k H ( D ) ≤ C ǫ d / . (4.15) Proof.
We obtain from (4.14) ˆ D β ǫ ∇ ( u ǫ − u ) · ∇ ϕ d x + ˆ D ( ̺ ǫ ( u ǫ ) − ̺ ǫ ( u )) ϕ d x = − ˆ ω ǫ ( β − β ) ∇ u · ∇ ϕ d x | {z } = : I ( ǫ , ϕ ) − ˆ ω ǫ ( ̺ ( u ) − ̺ ( u )) ϕ d x | {z } = : I I ( ǫ , ϕ ) + ˆ ω ǫ ( f − f ) ϕ d x | {z } = : I I I ( ǫ , ϕ ) (4.16)for all ϕ ∈ H ( D ) . Hence, since u ∈ C ( B δ ( z )) for δ > | I ( ǫ , ϕ ) | ≤ k β − β k C ( B δ ( z )) d × d k∇ u k C ( B δ ( z )) d Æ | ω ǫ |k∇ ϕ k L ( D ) d | I I ( ǫ , ϕ ) | ≤ k ̺ ( u ) − ̺ ( u ) k C ( B δ ( z )) Æ | ω ǫ |k ϕ k L ( D ) (4.17)and | I I I ( ǫ , ϕ ) | ≤ k f − f k L ∞ ( B δ ( z )) Æ | ω ǫ |k ϕ k L ( D ) . (4.18)Now testing (4.16) with ϕ = u ǫ − u and using (4.17) together with Assumption 1,(a)-(b) leadto the desired estimate.2 K. Sturm We introduce for ǫ ∈ [ τ ] the (not necessarily symmetric) bilinear form b ǫ : H ( D ) × H ( D ) → R by b ǫ ( ψ , ϕ ) : = ˆ D β ǫ ∇ ψ · ∇ ϕ + (cid:18) ˆ ̺ ′ ǫ ( su ǫ + ( − s ) u ) ds (cid:19) ϕψ d x , (4.19)where ̺ ′ ǫ ( u ) : = ̺ ′ ( u ) χ Ω ǫ + ̺ ′ ( u ) χ R d \ Ω ǫ Then the averaged adjoint equation (2.8) for the La-grangian G given by (4.11) reads: find q ǫ ∈ H ( D ) such that b ǫ ( ψ , q ǫ ) = − ˆ D ( u + u ǫ ) ψ d x (4.20)for all ψ ∈ H ( D ) . In view of Assumption 1 we have ̺ ′ ǫ ≥ β ǫ ≥ β m I and thus b ǫ is coercive, b ǫ ( ψ , ψ ) ≥ β m k∇ ψ k L ( D ) d for all ψ ∈ H ( D ) , ǫ ∈ [ τ ] . (4.21)As for the state equation, we use the notation q : = q . Lemma 4.5. (i) For each ǫ ∈ [ τ ] equation (4.20) admits a unique solution.(ii) We find for every z ∈ D \ Ω a number δ >
0, such that q ∈ H ( B δ ( z )) ⊂ C ( B δ ( z )) for d ∈ {
2, 3 } . Proof. (i) Since b ǫ is coercive and continuous on H ( D ) , the theorem of Lax-Milgram shows that(4.20) admits a unique solution.(ii) The proof is the same as the one for item (ii) of Lemma 4.1 and therefore omitted.The previous lemma shows that Y ( ǫ , u , u ǫ ) = { q ǫ } is a singleton and therefore Hypothe-sis (H2) is satisfied. We proceed with a Hölder-type estimate for ǫ q ǫ . Lemma 4.6.
There is a constant C >
0, such that for all small ǫ > k q ǫ − q k H ( D ) ≤ C ( k u ǫ − u k L ( D ) + ǫ d / ) . (4.22) Proof.
Using (4.20) we obtain b ǫ ( ψ , q ǫ − q ) = b ǫ ( ψ , q ǫ ) − b ǫ ( ψ , q ) (4.20) = − ˆ D ( u ǫ − u ) ψ d x − ( b ǫ − b )( ψ , q ) (4.23)for all ψ ∈ H ( D ) . Since furthermore ( b ǫ − b )( ψ , q ) = − ˆ ω ǫ ( β − β ) ∇ q · ∇ ψ d x − ˆ ω ǫ (cid:18) ˆ ( ̺ ′ − ̺ ′ )( su ǫ + ( − s ) u ) ds (cid:19) q ψ d x ,(4.24)we obtain using Hölder’s inequality and q ∈ C ( B δ ( z )) , | ( b ǫ − b )( ψ , q ) | ≤k β − β k C ( B δ ( z )) d × d k∇ q k C ( B δ ( z )) d Æ | ω ǫ |k∇ ψ k L ( D ) d + k ̺ ′ − ̺ ′ k L ∞ ( B C ( )) k q k L ( D ) k ψ k L ( D ) . (4.25)opologicalderivativevia Lagrange 13where C > k u ǫ k L ∞ ( D ) ≤ C for all ǫ ∈ [ τ ] . So inserting ψ = q ǫ − q astest function in (4.23) and using (4.25) yields β m k∇ ( q ǫ − q ) k L ( D ) d ≤ b ǫ ( q ǫ − q , q ǫ − q ) ≤ C ( p ω ǫ + k u ǫ − u k L ( D ) ) k q ǫ − q k H ( D ) . (4.26)Now the result follows from the Poincaré inequality and | ω ǫ | = ǫ d | ω | . Remark 4.7.
The proof of estimate (5.22) requires q ∈ C ( B δ ( z )) , but not q ǫ ∈ C ( B δ ( z )) , whichis false in general, since ∇ q ǫ has a jump across ∂ ω ǫ .Let us finish this section with the verification of Hypothesis (H3). Lemma 4.8.
We havelim ǫ ց G ( ǫ , u , q ) − G ( u , q ) | ω ǫ | =( β − β )( z ) ∇ u ( z ) · ∇ q ( z )+ ( ̺ ( u ( z )) − ̺ ( u ( z ))) q ( z ) − (( f − f ) q )( z ) . (4.27) Proof.
The change of variables T ǫ shows that for ǫ > G ( ǫ , u , q ) − G ( u , q ) | ω ǫ | = | ω | ˆ ω (( β − β ) ∇ u · ∇ q )( T ǫ ( x )) d x + | ω | ˆ ω (( ̺ ( u ) − ̺ ( u )) uq ) ( T ǫ ( x )) d x − | ω | ˆ ω (( f − f ) q )( T ǫ ( x )) d x . (4.28)Recalling that f , f ∈ C ( B δ ( z )) and u , q ∈ C ( B δ ( z )) for a small δ > T ǫ ( ω ) ⊂ B δ ( z ) for all small ǫ >
0, we can pass to the limit in (4.28) to obtain (4.27).
The goal of this section is to verify Hypothesis (H4), that is, to show that R ( u , q ) : = lim ǫ ց G ( ǫ , u , q ǫ ) − G ( ǫ , u , q ) | ω ǫ | (4.29)exists and, if possible, to determine its explicit form. In contrast to previous works we considerthe variation of the averaged adjoint state variable which we will show converges weakly to afunction Q defined on the whole space R d . For this purpose we need the following definition. Definition 4.9.
The inflation of D \ Ω around z ∈ D \ Ω is defined by D ǫ : = T − ǫ ( D \ Ω ) , wherethe transformation T ǫ is defined by T ǫ ( x ) : = ǫ x + z .Notice that ∪ ǫ> D ǫ = R d and that ǫ D ǫ is monotonically decreasing, that is, ǫ < ǫ ⇒ D ǫ ⊂ D ǫ . Lemma 4.10.
For ǫ > ϕ ∈ H ( D \ Ω ) if and only if ϕ ◦ T ǫ ∈ H ( D ǫ ) .4 K. Sturm D ε ω Ω ǫ D ε ω Ω ǫ D ε ω Ω ǫ Figure 1: Depicted are several inflated domains D ǫ = T − ǫ ( D \ Ω ) and Ω ǫ : = T − ǫ ( Ω ) with ǫ decreasing from left to right. The original inclusion ω ǫ appears as the fixed inclusion ω centeredat the origin in the inflated domain. It can be seen that the domain Ω ǫ is gradually pushed toinfinity the smaller ǫ gets. Proof.
Since T ǫ is bi-Lipschitz continuous for ǫ >
0, this follows from [
40, Thm. 2.2.2, p.52 ] .The next step is to consider the variation of the averaged adjoint state. For this purpose letus extend q ǫ to zero outside of D , that is,˜ q ǫ ( x ) : = § q ǫ ( x ) for a.e. x ∈ D ,0 for a.e. x ∈ R d \ D . (4.30)In the same way we extend u ǫ to a function ˜ u ǫ : R d → R . Notice that ˜ u ǫ , ˜ q ǫ ∈ H ( R d ) for all ǫ >
0. We will use the notation q ǫ : = ˜ q ǫ ◦ T ǫ . Remark 4.11 (Neumann boundary conditions) . If we had imposed Neumann conditions in (S),then it would be sufficient to replace (4.30) by ˜ q ǫ : = Eq ǫ , where E : H ( D ) → H ( R d ) is acontinuous extension operator; see [
20, Thm. 1, pp. 254 ] . The subsequent analysis were stillthe same. Definition 4.12.
The variation of the averaged adjoint state q ǫ is defined pointwise a.e. in R d by Q ǫ ( x ) : = ˜ q ǫ ( T ǫ ( x )) − ˜ q ( T ǫ ( x )) ǫ . (4.31)Notice that for every ǫ > Q ǫ ∈ H ( R d ) .Our next task is to show that ( Q ǫ ) converges in ˙ B L ( R d ) to a equivalence class of functions [ Q ] and determine an equation for it. The first step is to prove the following apriori estimates. Lemma 4.13.
There is a constant C >
0, such that for all small ǫ > ˆ R d ( ǫ Q ǫ ) + |∇ Q ǫ | d x ≤ C . (4.32)opologicalderivativevia Lagrange 15 Proof.
Obviously, the Lemmas 4.6 / C > k q ǫ − q k H ( D ) ≤ C ǫ d / for all small ǫ >
0. This and definition (4.30) imply ˆ R d ( ˜ q ǫ − ˜ q ) + |∇ ( ˜ q ǫ − ˜ q ) | d x ≤ C ǫ d . (4.33)Hence invoking the change of variables T ǫ in (4.33) yields the bound (4.32).Notice that for ǫ > Q ǫ belongs to H ( R d ) , but it is not bounded with respectto ǫ . However, the bound (4.32) is sufficient to show the following key theorem. Theorem 4.14.
For d ∈ {
2, 3 } , we have ∇ Q ǫ * ∇ Q in L ( R d ) d , ǫ Q ǫ * H ( R d ) , (4.34)where [ Q ] ∈ ˙ B L ( R d ) is the unique solution to ˆ R d A ∇ ψ · ∇ Q d x = ˆ ω ζ · ∇ ψ d x for all ψ ∈ B L ( R d ) , (4.35)where A : = β ( z ) χ ω + β ( z ) χ R d \ ω and ζ : = − ( β ( z ) − β ( z )) ∇ q ( z ) ; see (3.1). Proof.
Fix ¯ ǫ > < ǫ < ¯ ǫ . We first notice that using (4.20) we have b ǫ ( ψ , q ǫ − q ) = − ˆ D ( u ǫ − u ) ψ d x − ( b ǫ − b )( ψ , q ) (4.36)for all ψ ∈ H ( D ) . The idea is now to choose appropriate test functions in (4.36) and then passto the limit. For this purpose let ¯ ψ ∈ H ( D ¯ ǫ ) be arbitrary and define ψ : = ǫ ¯ ψ ◦ T − ǫ . Thanks toLemma 4.10 we have ψ ∈ H ( D \ Ω ) and the latter space embeds via (4.30) into H ( D ) . Hencewe readily check that for such a test function, using a change of variables, we have b ǫ ( ψ , q ǫ − q ) = ǫ d ˆ D ¯ ǫ A ǫ ∇ ¯ ψ · ∇ Q ǫ d x + ǫ d + ˆ D ¯ ǫ ( ̺ ′ ǫ ( u ) ◦ T ǫ ) ǫ Q ǫ ¯ ψ d x | {z } = : I ( ǫ , ¯ ψ ) (4.37a) ( b ǫ − b )( ψ , q ) = ǫ d ˆ ω ( β − β )( T ǫ ( x )) ∇ ¯ ψ · ∇ q ( T ǫ ( x )) d x | {z } = : I I ( ǫ , ¯ ψ ) (4.37b) + ǫ d + ˆ D ¯ ǫ (cid:18) ˆ ( ̺ ′ ǫ ( su ǫ + ( − s ) u ) ◦ T ǫ − ̺ ′ ( u ) ◦ T ǫ ds (cid:19) q ( T ǫ ( x )) ¯ ψ d x | {z } = : I I I ( ǫ , ¯ ψ ) ˆ D ( u ǫ − u ) ψ d x = ǫ d + ˆ D ¯ ǫ ( u ǫ ◦ T ǫ − u ◦ T ǫ ) ¯ ψ d x | {z } = : I V ( ǫ , ¯ ψ ) , (4.37c)6 K. Sturmwhere A ǫ ( x ) : = β ( T ǫ ( x )) χ ω ( x ) + β ( T ǫ ( x )) χ R d \ ω . Therefore inserting (4.37a)-(4.37c) into(4.36) we obtain ˆ D ¯ ǫ A ǫ ∇ ¯ ψ · ∇ Q ǫ d x + ˆ ω ( β − β )( T ǫ ( x )) ∇ ¯ ψ · ∇ q ( T ǫ ( x )) d x = − ǫ ( I − I I − I I I + I V )( ǫ , ¯ ψ ) (4.38)for all ǫ < ¯ ǫ and all ¯ ψ ∈ H ( D ¯ ǫ ) . The next step is to show that I-IV are bounded. Using theboundedness of u ǫ on D we see that ̺ ′ ǫ ( s ˜ u ǫ + ( − s ) ˜ u ) ◦ T ǫ and ̺ ′ ( ˜ u ) ◦ T ǫ are bounded (inde-pendently of ǫ ) on R d , too. Therefore Hölder’s inequality yields | I ( ǫ , ¯ ψ ) | ≤ c k ǫ Q ǫ k L ( R d ) k ¯ ψ k L ( R d ) (4.32) ≤ C k ¯ ψ k L ( R d ) , (4.39) | I I ( ǫ , ¯ ψ ) | ≤ c k∇ q k C ( B δ ( z )) k∇ ¯ ψ k L ( R d ) d (4.32) ≤ C k∇ ¯ ψ k L ( R d ) d , (4.40) | I I I ( ǫ , ¯ ψ ) | ≤ c k q k C ( B δ ( z )) k ¯ ψ k L ( R d ) (4.41) | I V ( ǫ , ¯ ψ ) | ≤ c k ˜ u ǫ ◦ T ǫ − ˜ u ◦ T ǫ k L ( R d ) k ¯ ψ k L ( R d ) (4.15) ≤ C k ¯ ψ k L ( R d ) (4.42)for all ¯ ψ ∈ H ( D ¯ ǫ ) and ǫ ∈ [
0, ¯ ǫ ] . Thanks to Lemma 4.13 the family ( Q ǫ ) is bounded in ˙ B L ( R d ) .The latter space is a Hilbert space and therefore for every null-sequence ( ǫ n ) we find a sub-sequence ( ǫ n k ) and [ Q ] ∈ ˙ B L ( R d ) , such that ∇ Q ǫ nk * ∇ Q in L ( R d ) d , where Q ∈ [ Q ] . Henceselecting ǫ = ǫ n k in (4.38) and taking into account (4.39)-(4.41) we can pass to the limit k → ∞ and obtain ˆ D ¯ ǫ A ∇ ¯ ψ · ∇ Q d x = − ( β ( z ) − β ( z )) ∇ q ( z ) · ˆ ω ∇ ¯ ψ d x for all ¯ ψ ∈ H ( D ¯ ǫ ) . (4.43)The mapping ¯ ǫ D ¯ ǫ is monotonically decreasing and we have H ( D ¯ ǫ ) ⊂ H ( R d ) . This shows,recalling that ¯ ǫ > D ¯ ǫ appearing in (4.43) may be replaced by R d . But thismeans that Q is the unique solution of (4.35).Let us now show that ǫ Q ǫ * H ( R d ) as ǫ ց
0. From the first part of the proof it is clearthat ∇ ( ǫ Q ǫ ) * L ( R d ) d . To see the weak convergence of ( ǫ Q ǫ ) in L ( R d ) we fix r >
0. ThenPoincaré’s inequality for a ball yields k ( ǫ Q ǫ ) r − ǫ Q ǫ k L ( B r ( )) ≤ ǫ C ( r ) k∇ Q ǫ k L ( B r ( )) d , (4.44)where ( ǫ Q ǫ ) r : = ffl B r ( ) ǫ Q ǫ d x denotes the average over the ball B r ( ) . Since the gradient k∇ Q ǫ k L ( R d ) d is uniformly bounded (see Lemma 4.13), the right hand side of (4.44) goes tozero as ǫ ց
0. But also ǫ Q ǫ is bounded in L ( R d ) and therefore we find for any null-sequence ( ǫ n ) a subsequence ( ǫ n k ) and ˆ Q ∈ L ( R d ) , such that ǫ n k Q ǫ nk * ˆ Q in L ( R d ) . It is clear that ( ǫ n k Q ǫ nk ) B r ( ) → ( ˆ Q ) B r ( ) in R . Therefore we obtain from (4.44) together with the weak lowersemi-continuity of the L -norm k ( ˆ Q ) r − ˆ Q k L ( B r ( )) ≤ lim inf k →∞ k ( Q ǫ nk ) B r ( ) − Q ǫ nk k L ( B r ( )) ≤
0. (4.45)This shows that ˆ Q = ( ˆ Q ) r a.e. on B r ( ) and thus ˆ Q is constant on B r ( ) . Since r > Q must be constant on R d . Further ˆ Q ∈ L ( R d ) implies ˆ Q = R ( u , q ) and thereby verify the second part of Hypothesis (H2). Lemma 4.15.
We have R ( u , q ) = ( β ( z ) − β ( z )) ∇ u ( z ) · ω ∇ Q d x , (4.46)where [ Q ] is the solution to (4.35). Proof.
Testing the state equation (4.14) (for ǫ =
0) with ϕ = q ǫ − q gives ˆ D β ∇ u · ∇ ( q ǫ − q ) + ̺ ( u )( q ǫ − q ) d x = ˆ D f ( q ǫ − q ) d x . (4.47)Therefore we can write for ǫ > G ( ǫ , u , q ǫ ) − G ( ǫ , u , q ) = ˆ D β ǫ ∇ u · ∇ ( q ǫ − q ) + ̺ ǫ ( u )( q ǫ − q ) d x − ˆ D f ǫ ( q ǫ − q ) d x (4.47) = ˆ ω ǫ ( β − β ) ∇ u · ∇ ( q ǫ − q ) + [( ̺ − ̺ )( u ) − ( f − f )]( q ǫ − q ) d x .(4.48)Invoking the change of variables T ǫ in (4.48) we obtain for ǫ > G ( ǫ , u , q ǫ ) − G ( ǫ , u , q ) | ω ǫ | = | ω | ˆ ω [( ̺ − ̺ )( u ( T ǫ ( x ))) − ( f − f )( T ǫ ( x ))] ǫ Q ǫ d x + | ω | ˆ ω (( β − β ) ∇ u )( T ǫ ( x )) · ∇ Q ǫ d x → ( β ( z ) − β ( z )) ∇ u ( z ) · ω ∇ Q d x , (4.49)where in the last step we used Theorem 4.14, f , f ∈ C ( B δ ( z )) , and u ∈ C ( B δ ( z )) for δ > Topological derivative
Now we are in a position to formulate our main result. In the previoussections we have checked that Hypotheses (H0)-(H4) of Theorem 2.6 are satisfied for the La-grangian G given by (4.11). Therefore Theorem 4.16 can be applied and we obtain the followingresult. Theorem 4.16.
The topological derivative of J at Ω in z ∈ D \ Ω is given bylim ǫ ց J ( Ω ∪ ω ǫ ( z )) − J ( Ω ) | ω ǫ ( z ) | = ∂ ℓ G ( u , q ) + R ( u , q ) , (4.50)where ∂ ℓ G ( u , q ) = (cid:0) ( β − β ) ∇ u · ∇ q + ( ̺ ( u ) − ̺ ( u )) q − ( f − f ) q (cid:1) ( z ) (4.51)and R ( u , q ) = ( β ( z ) − β ( z )) ∇ u ( z ) · ω ∇ Q d x , (4.52)where Q depends on z and is the solution to (4.35).8 K. SturmNext we rewrite the term R ( u , q ) with the help of the so-called polarisation matrix. For thispurpose we fix z ∈ D \ Ω in the following and denote by [ Q ζ ] , ζ ∈ R d , the solution to (3.6) with A : = A ω : = β ( z ) χ ω + β ( z ) χ R d \ ω . Also we denote by Q ζ an arbitrary representative of [ Q ζ ] . Definition 4.17.
The matrix representing the linear averaging operator ζ ω ∇ Q ζ d x , R d R d (4.53)is called weak polarisation matrix and will be denoted P z ∈ R d × d . Notice that this matrix dependson β ( z ) and β ( z ) .We use the term weak polarisation matrix here, because it is defined via the weak formulation(3.6) and therefore does not require any regularity assumptions on ∂ ω or Ω . We give anotherdefinition of a polarisation matrix later and relate it to the weak polarisation matrix. We alsorefer to [ ] and the monograph [
31, Sec. 9.4.4, pp. 273 ] . Corollary 4.18.
We havelim ǫ ց J ( Ω ∪ ω ǫ ( z )) − J ( Ω ) | ω ǫ | =((( β − β ) ∇ u ) · ( I − P z ( β − β )) ∇ q )( z )+ (( ̺ ( u ) − ̺ ( u )) q − ( f − f ) q )( z ) . (4.54) Proof.
This follows at once from (4.50) noting that P z ζ = ffl ω ∇ Q ζ d x , where ζ : = − ( β ( z ) − β ( z )) ∇ q ( z ) . Further properties of the polarisation matrix
Next we derive further properties of the polar-isation matrix. Furthermore we relate our polarisation matrix to previous definitions. We referthe reader to [ ] for further information on polarisation matrices. Lemma 4.19. If β ( z ) = β ⊤ ( z ) and β ( z ) = β ⊤ ( z ) , then the polarisation matrix is symmetric,that is, P z = P ⊤ z . Proof.
We compute for the ( i , j ) -entry of the polarisation matrix: e i · P z e j = e i · ω ∇ Q e j d x (4.35) = ˆ R d A ω ∇ Q e j · ∇ Q e i d x sym. of A ω = ˆ R d ∇ Q e j · A ω ∇ Q e i d x (4.35) = e j · ω ∇ Q e i d x = e j · P z e i . (4.55)This shows the symmetry.The polarisation matrix is also positive definite (even in the nonsymmetric case). Lemma 4.20.
The matrix P z is positive definite.opologicalderivativevia Lagrange 19 Proof.
Let w = ( w , . . . , w d ) ∈ R d be an arbitrary vector. Set W : = P di = w i Q e i . Then we readilycheck using (4.55), w · P z w = ˆ R d A ω ∇ W · ∇ W d x ≥ β m ˆ R d |∇ W | d x . (4.56)This shows that P z is positive semidefinite. Suppose now w is such that w · P z w =
0. Then, inview of (4.56), we must have [ W ] = [ ] . Hence (4.35) gives w · ω ∇ ϕ d x = ϕ ∈ BL ( R d ) . (4.57)Let V ⊂ R d be a bounded and open set, such that ω ⋐ V . Choose a smooth function ρ , such that ρ = ω , 0 ≤ ρ ≤ V \ ω and ρ = V . Then we define ϕ ( x ) : = e i · x ρ ( x ) for i ∈ {
1, . . . , d } , which belongs to BL ( R d ) . Hence we may test (4.57) with this function andconclude w i =
0. This shows w = β = γ I and β = γ I for γ , γ >
0. We select Q ζ ∈ [ Q ζ ] and supposethat it can be represented by a single layer potential: there is a function h ζ ∈ C ( ∂ ω ) , such that Q ζ ( x ) = ˆ ∂ ω h ζ ( y ) E ( x − y ) ds ( y ) , ˆ ∂ ω h ζ ds =
0, (4.58)where E denotes the fundamental solution of u
7→ − ∆ u ; [
21, Chap. 3 ] . It is readily checkedusing (4.58) that | Q ζ ( x ) | = O ( | x | − d ) . Definition 4.21.
The strong polarisation matrix is the matrix ˜ P z = ( ˜ P z ) i j ∈ R d × d with entries ( ˜ P z ) i j = ˆ ∂ ω x j h e i ds . (4.59)The strong and weak polarisation matrices are related as shown in the following lemma. Lemma 4.22.
Assume that ∂ ω is C . Then we have P z = − | ω | β β − β ˜ P z + β − β I . (4.60) Proof.
At first we obtain by partial integration, noting that e i = ∇ x i , e i · P z e j = ω ∇ x i · ∇ Q e j ds = | ω | ˆ ∂ ω x i ∂ ν Q e j ds − ω ∆ Q e j |{z} =
0, in view of (4.35) d x . (4.61)Next we express ∂ ν Q e j in terms of h e j . For this recall (see, e.g., [ ] ) that the jump condition ∂ ν Q + e i − ∂ ν Q − e i = h e i on ∂ ω (4.62)is satisfied. In addition we get from (4.35), β ∂ ν Q + e i − β ∂ ν Q − e i = e i · ν on ∂ ω . (4.63)0 K. SturmCombining (4.62) and (4.63) we obtain ∂ ν Q + e i = − β β − β h e i + β − β e i · ν . (4.64)Inserting this expression into (4.61) yields e i · P z e j = − β β − β | ω | ˆ ∂ ω x i h e j ds + β − β | ω | ˆ ∂ ω ( e i · ν ) x j ds . (4.65)This is equivalent to formula (4.60), since by Gauss’s divergence theorem1 | ω | ˆ ∂ ω ( e i · ν ) x j ds = ω div ( e i x j ) | {z } = δ ij d x = δ i j . (4.66) Remark 4.23.
In some cases, see, e.g., [
3, 5, 27 ] , the polarisation matrix can be computed ex-plicitly: for instance when β = γ I , β = γ I , β , β >
0, and ω is a circle or more generally anellipse. However for general inclusions ω the exterior equation (4.35) has to be solved numer-ically in order to evaluate formula (4.50). In this last section we discuss the extremal situation in which β = ̺ = f = [ ] . Since the extremal case is similar to the considerations from the previoussection, we will only work out the main differences in detail. We suppose as before that D ⊂ R d is a bounded Lipschitz domain. For a simply connected domain Ω ⋐ D with Lipschitz boundary ∂ Ω , we consider the shape function J ( Ω ) = ˆ D \ Ω u d x (5.1)subject to u = u Ω ∈ H ∂ D ( D \ Ω ) solves ˆ D \ Ω β ∇ u · ∇ ϕ + ̺ ( u ) ϕ d x = ˆ D \ Ω f ϕ d x for all ϕ ∈ H ∂ D ( D \ Ω ) , (5.2)where H ∂ D ( D \ Ω ) : = { v ∈ H ( D \ Ω ) : v = ∂ D } . This setting corresponds to the limitingcase of (4.2) in which β = ̺ = f = J at Ω = ; with respect to the inclusion ω (which will be specified below), i.e.,lim ǫ ց J ( ω ǫ ) − J ( ; ) | ω ǫ | . (5.3)opologicalderivativevia Lagrange 21We will see that almost all steps are the same as in the last section with two main differences.The first main difference being that X ( ǫ ) is not a singleton and that we have to introduce a newequation on the inclusion, which requires a more detailed explanation and a thorough analysis.The second difference concerns the required assumptions on the regularity of the inclusion ω .While in the previous section it was sufficient to assume that ω is merely an open set, here westrengthen the assumption and assume that ω is a simply connected Lipschitz domain. Assumption 2.
We assume that either(a) β ∈ R d × d is symmetric, positive definite and ̺ satisfies Assumption 1, (b) and it isbounded, or(b) β satisfies Assumption (1), (a) and ̺ satisfies Assumption 1, (b) and additionally ̺ ′ > λ for some λ > f ∈ H ( D ) ∩ C ( D ) .Under these assumptions we can prove, using similar arguments as in Lemma 4.1, that (5.2)admits a unique solution and that there is a constant C > Ω ), such that k u Ω k L ∞ ( D \ Ω ) + k u Ω k H ( D \ Ω ) ≤ C k f k L r ( D \ Ω ) (5.4)for r > d / d /
2. Moreover, for every z ∈ D \ Ω , we find δ >
0, such that u Ω ∈ H ( B δ ( z )) . From now on we fix: • a simplify connected Lipschitz domain ω ⊂ R d with 0 ∈ ω , • a point z ∈ D , • the perturbation Ω ǫ : = ω ǫ : = ω ǫ ( z ) , where ω ǫ ( z ) : = z + ǫω and ǫ ∈ [ τ ] , τ > X = Y = H ( D ) and introduce the Lagrangian G : [ τ ] × X × Y → R associated with theperturbation Ω ǫ by G ( ǫ , u , q ) : = ˆ D \ ω ǫ u d x + ˆ D \ ω ǫ β ∇ u · ∇ q + ̺ ( u ) q d x − ˆ D \ ω ǫ f q d x . (5.5)We will verify that Hypotheses (H0)-(H4) are satisfied with ℓ ( ǫ ) = | ω ǫ | .2 K. Sturm The perturbed state equation reads: find u ǫ ∈ H ( D ) such that ∂ p G ( ǫ , u ǫ , 0 )( ϕ ) = ϕ ∈ H ( D ) , or equivalently u ǫ ∈ H ( D ) satisfies, ˆ D \ ω ǫ β ∇ u ǫ · ∇ ϕ + ̺ ( u ǫ ) ϕ d x = ˆ D \ ω ǫ f ϕ d x for all ϕ ∈ H ( D ) . (5.6)Henceforth we write u : = u to simplify notation. Since (5.2) admits a unique solution ¯ u ǫ for Ω = ω ǫ , which can be extended to H ( D ) , (5.6) admits a solution, too, whose restriction to D \ Ω is unique. This means that E ( ǫ ) = { u ∈ H ( D ) : u = ¯ u ǫ a.e. on D \ ω ǫ } , (5.7)where ¯ u ǫ is the unique solution to (5.2). It also follows that X ( ǫ ) = E ( ǫ ) since the Lagrangianonly depends on the restriction of functions to D \ ω ǫ . Note that the set X ( ) is a singleton.Moreover for all ǫ ∈ [ τ ] , g ( ǫ ) = inf u ∈ E ( ǫ ) G ( ǫ , u , 0 ) = ˆ D \ ω ǫ ¯ u ǫ d x . (5.8)This shows that Hypothesis (H0) and, in view of Assumption 2, also Hypothesis (H1) is satisfied.The next step deviates from the transmission problem case (of Section 4). We constructfunctions u ǫ ∈ X ( ǫ ) and q ǫ ∈ Y ( ǫ , u , u ǫ ) that satisfy Hypothesis (H4). For this purpose weassociate with u ǫ ∈ H ∂ D ( D \ ω ǫ ) a function u + ǫ ∈ H ( ω ǫ ) defined as the unique weak solution tothe Dirichlet problem − div ( β ∇ u + ǫ ) + ̺ ( u ) = f in ω ǫ u + ǫ = u ǫ on ∂ ω ǫ . (5.9)With this function we can extend u ǫ to a function u ǫ ∈ H ( D ) by setting u ǫ : = § u + ǫ in ω ǫ u ǫ in D \ ω ǫ . (5.10)Now we prove the following analogue of Lemma 4.4. Lemma 5.1.
There is a constant C >
0, such that for all small ǫ > k u ǫ − u k H ( D ) ≤ C ǫ d / . (5.11) Proof.
We first establish an estimate for u ǫ − u on ω ǫ . For this purpose we fix a bounded domain S ⊂ D containing ω . We note that the difference e ǫ ( x ) : = u ǫ ( T ǫ ( x )) − u ( T ǫ ( x )) satisfies − div ( β ◦ T ǫ ∇ e ǫ ) = ω and e ǫ = u ǫ ( T ǫ ( x )) − u ( T ǫ ( x )) on ∂ ω . Hence by standard elliptic regularityand the trace theorem we find k e ǫ + λ k H ( ω ) ≤ c k e ǫ + λ k H / ( ∂ ω ) ≤ c k e ǫ + λ k H ( S \ ω ) (5.12)for all λ ∈ R . Since the quotient norms on the spaces H ( ω ) / R and H ( S \ ω ) are equivalent tothe seminorms | v | H ( ω ) : = k∇ v k L ( ω ) d and | v | H ( S \ ω ) : = k∇ v k L ( S \ ω ) d , respectively, we concludeopologicalderivativevia Lagrange 23 k∇ e ǫ k L ( ω ) d ≤ c k∇ e ǫ k L ( S \ ω ) d . Therefore estimating the right hand side and changing variablesshows k∇ ( u ǫ − u ) k L ( ω ǫ ) d ≤ c k∇ ( u ǫ − u ) k L ( D \ ω ǫ ) d . (5.13)A fortiori using (5.13) and a similar argument shows that (5.12) implies k u ǫ − u k L ( ω ǫ ) ≤ c ( ǫ k∇ ( u ǫ − u ) k L ( D \ ω ǫ ) d + k u ǫ − u k L ( D \ ω ǫ ) ) . (5.14)This finishes the first step of the proof. We now establish an estimate for the right hand side of(5.13). Following the steps of Lemma 4.4 we find ˆ D \ ω ǫ β ∇ ( u ǫ − u ) · ∇ ϕ d x + ˆ D \ ω ǫ ( ̺ ( u ǫ ) − ̺ ( u )) ϕ d x = ˆ ω ǫ β ∇ u · ∇ ϕ d x + ˆ ω ǫ ̺ ( u ) ϕ − f ϕ d x (5.15)for all ϕ ∈ H ( D ) . Let us first assume that Assumption 2, (a) holds. Fix ¯ ǫ > < ǫ < ¯ ǫ .Changing variables in (5.15) yields (recalling that we denote by ˜ u ǫ the extension of u ǫ to R d ) ˆ R d \ ω β ∇ K ǫ · ∇ ϕ d x = − ǫ ˆ R d \ ω ( ̺ ( ˜ u ǫ ( T ǫ )) − ̺ ( ˜ u ( T ǫ ))) ϕ d x | {z } →
0, since ̺ is bounded + ǫ ˆ ω β ∇ u ( T ǫ ) · ∇ ϕ d x + ǫ ˆ ω ̺ ( u ( T ǫ )) ϕ − f ( T ǫ ) ϕ d x | {z } →
0, since u ∈ C ( B δ ( z )) , f ∈ C ( D ) and ̺ ∈ C ( R ) , (5.16)for all ϕ ∈ H ∂ D ( D ¯ ǫ \ ω ) , where K ǫ : = ( u ǫ − u ) ◦ T ǫ . Since ¯ ǫ > K ǫ * ( R d \ ω ) . But this means that K ǫ must be bounded in ˙BL ( R d \ ω ) andhence we find C >
0, such that k∇ K ǫ k L ( R d \ ω ) d ≤ C or equivalently after changing variables k∇ ( u ǫ − u ) k L ( D \ ω ǫ ) ≤ C ǫ d / . Combining this estimate with (5.13) and using Poincaré’s inequalitygives (5.11).Let us now assume that Assumption 2, (b) is satisfied. Testing (5.15) with ϕ = u ǫ − u , using ̺ ′ > λ and applying Hölder’s inequality yield C k u ǫ − u k H ( D \ ω ǫ ) ≤ Æ | ω ǫ | ( k∇ u k C ( B δ ( z )) d k∇ ( u ǫ − u ) k L ( ω ǫ ) d + k ̺ ( u ) − f k C ( B δ ( z )) k u ǫ − u k L ( ω ǫ ) ) .Using (5.13) and (5.14) to estimate the right hand side and noting | ω ǫ | = | ω | ǫ d , we infer k u ǫ − u k H ( D \ ω ǫ ) ≤ C ǫ d / . Again combining this estimate with (5.13) yields (5.11). We introduce for ǫ ∈ [ τ ] the (not necessarily symmetric) bilinear form b ǫ : H ( D ) × H ( D ) → R by b ǫ ( ψ , ϕ ) : = ˆ D \ ω ǫ β ∇ ψ · ∇ ϕ + (cid:18) ˆ ̺ ′ ( su ǫ + ( − s ) u ) ds (cid:19) ϕψ d x . (5.17)4 K. SturmThen the averaged adjoint equation (2.8) for the Lagrangian G given by (5.5) reads: for ( u , u ǫ ) ∈ X ( ) × X ( ǫ ) find q ǫ ∈ H ( D ) , such that b ǫ ( ψ , q ǫ ) = − ˆ D \ ω ǫ ( u + u ǫ ) ψ d x (5.18)for all ψ ∈ H ( D ) . In view of Assumption 1 we have ̺ ′ ≥ β ≥ β m I and thus b ǫ satisfies, b ǫ ( ψ , ψ ) ≥ β m k∇ ψ k L ( D \ ω ) d (5.19)for all ψ ∈ H ( D ) and ǫ ∈ [ τ ] . As for the state equation, we use the notation q : = q . Lemma 5.2. (i) For each ǫ ∈ [ τ ] equation (5.18) admits a solution whose restriction to D \ ω ǫ is unique.(ii) For every z ∈ D \ Ω we find a number δ >
0, such that q ∈ H ( B δ ( z )) ⊂ C ( B δ ( z )) for d ∈ {
2, 3 } .The previous lemma shows that Y ( ǫ , u , u ǫ ) = { q ∈ H ( D ) : q = q ǫ a.e. on D \ ω ǫ } and thusHypothesis (H2) is satisfied. In the same way as done in (5.9) we extend the restriction q ǫ | D \ ω ǫ (which is unique) to a function q ǫ ∈ H ( D ) by solving the following Dirichlet problem: find q + ǫ ∈ H ( ω ǫ ) , such that − div ( β ⊤ ∇ q + ǫ ) + ̺ ′ ( u ) q = − u in ω ǫ q + ǫ = q ǫ on ∂ ω ǫ . (5.20)With this function we define again q ǫ : = § q + ǫ in ω ǫ q ǫ in D \ ω ǫ . (5.21)It is clear that q ǫ ∈ Y ( ǫ , u , u ǫ ) . We proceed with a Hölder-type estimate for the extension ǫ q ǫ . Lemma 5.3.
There is a constant C >
0, such that for all small ǫ > k q ǫ − q k H ( D ) ≤ C ( k u ǫ − u k L ( D ) + ǫ d / ) . (5.22) Proof.
The proof is the same as the one of Lemma 4.6 and therefore omitted.It is readily checked that Hypothesis (H3) is satisfied, too.
Lemma 5.4.
We havelim ǫ ց G ( ǫ , u , q ) − G ( u , q ) ℓ ( ǫ ) =( − β ∇ u · ∇ q − ̺ ( u ) q + f q )( z ) . (5.23) Proof.
Since f ∈ C ( B δ ( z )) and u , q ∈ C ( B δ ( z )) for a small δ >
0, we can repeat the steps ofthe proof of Lemma 4.8.opologicalderivativevia Lagrange 25
The next step is to consider the variation of the averaged adjoint state. The variation of theaveraged adjoint variable, denoted Q ǫ , is defined as in Definition 4.12.The following is the analogue of Lemma 4.13 with the main difference that we have anadditional equation which gives information of Q inside the inclusion ω . Lemma 5.5.
There is a constant C >
0, such that for all small ǫ > ˆ R d ( ǫ Q ǫ ) + |∇ Q ǫ | d x ≤ C . (5.24) Proof.
We follow the steps of Lemma 4.13, but use Lemmas 5.3,5.1 instead Lemmas 4.6,4.4.
Theorem 5.6.
We have ∇ Q ǫ * ∇ Q weakly in L ( R d ) d , (5.25a) ǫ Q ǫ * H ( R d ) , (5.25b)where [ Q ] ∈ ˙ B L ( R d ) is the unique solution to ˆ R d \ ω β ( z ) ∇ ψ · ∇ Q d x = ˆ ω ζ · ∇ ψ d x for all ψ ∈ B L ( R d ) , (5.26a) ˆ ω β ( z ) ∇ ψ · ∇ Q d x = ψ ∈ H ( ω ) , (5.26b)where ζ : = − ( β ( z ) − β ( z )) ∇ q ( z ) . Proof.
It follows from Lemma 5.5 that for every null-sequence ( ǫ n ) there is a subsequence (in-dexed the same) and Q ∈ BL ( R d ) such that (5.25a) and (5.25b) holds for this subsequence. Nowusing the same arguments as in the proof of Theorem 4.14 shows that Q satisfies (5.26a). Theuniqueness of Q | R d \ ω follows directly from (5.26a). To prove (5.26b) note that Q ǫ n satisfies ˆ ω β ( T ǫ n ( x )) ∇ ψ · ∇ Q ǫ n d x = ψ ∈ H ( ω ) . (5.27)Using (5.25a) and (5.25b) we may pass to the limit n → ∞ which shows that Q satisfies (5.26b).Since Q | ∂ ω is uniquely determined from (5.26a) also (5.26b) admits a unique solution, becauseit is a Dirichlet problem with boundary values Q | ∂ ω .We are now ready to compute R ( u , q ) . Lemma 5.7.
We have R ( u , q ) = − β ( z ) ∇ u ( z ) · ω ∇ Q d x , (5.28)where [ Q ] is the solution to (5.26b). Proof.
The proof follows the lines of Lemma 4.15 and Lemma 5.1.Collecting all previous results we see that Theorem 2.6 can be applied and we obtain thefollowing result.6 K. Sturm
Theorem 5.8.
The topological derivative of J given by (5.1) in z ∈ D is given bylim ǫ ց J ( ω ǫ ( z )) − J ( Ω ) | ω ǫ | = ∂ ℓ G ( u , q ) + R ( u , q ) , (5.29)where ∂ ℓ G ( u , q ) = ( − β ∇ u · ∇ q − ̺ ( u ) q + f q )( z ) (5.30)and R ( u , q ) = − β ( z ) ∇ u ( z ) · ω ∇ Q d x (5.31)and Q depends on z and is the unique solution to (5.26a).Let Q ζ denote the solution to (5.26a)-(5.26b) for fixed z ∈ D and for ζ ∈ R d . Since Q ζ depends linearly on ζ we can proceed as in Subsection 4.6 and introduce a polarisation matrix P ∈ R d × d (depending on β ( z ) ) such that P ζ = ffl ω ∇ Q ζ d x to simplify (5.29). Finally in thesame way done as in Lemmas 4.19,4.20 we can show that P is symmetric if β is symmetric andthat it is always positive definite. Since the considerations are almost identical with the ones ofSubsection 4.6 the details are left to the reader. Concluding remarks
In this paper we showed that the Lagrangian averaged adjoint framework of [ ] provides anefficient and fairly simple tool to compute topological derivatives for semilinear problems. Weillustrated that using standard apriori estimates for the perturbed states and averaged adjointvariables are sufficient to obtain the topological sensitivity under comparatively mild assump-tions on the inclusion. Our work also provides a second examples (the first was given by [ ] )for which the R term in [
15, Thm. 3.1 ] is not equal to zero and thus underlines the flexibility ofthis theorem.There are several problems that remain open for further research. Firstly, it would be inter-esting to consider quasilinear equations, but also other types of equations, such as Maxwell’sequation. Secondly, an important point we have not addressed here is the topological derivativewhen Dirichlet boundary conditions are imposed on the inclusion. This case is know to be muchdifferent from the Neumann case and needs further investigations. p ( R d ) Define for 1 < p < d the space E p ( R d ) : = { u ∈ L p ∗ ( R d ) : ∇ u ∈ L ( R d ) d } (6.1)with the norm k u k E p : = k u k L p ∗ ( R d ) + k∇ u k L ( R d ) d . (6.2)opologicalderivativevia Lagrange 27 Lemma 6.1.
Let d ≥
3. Let A satisfy (3.1) and A = A ⊤ a.e. on R d . Then for every F ∈ L ( E ( R d ) , R ) ,there is a unique solution Q ∈ E ( R d ) to ˆ R d A ∇ ϕ · ∇ Q d x = F ( ϕ ) for all ϕ ∈ E ( R d ) . (6.3) Proof.
Let us introduce the energy E : E ( R d ) → R by E ( ϕ ) : = ˆ R d A ∇ ϕ · ∇ ϕ d x − F ( ϕ ) . (6.4)We are now going to prove that the minimisation probleminf ϕ ∈ E ( R d ) E ( ϕ ) , (6.5)admits a unique solution. We have to show that E is coercive on E ( R d ) , that is,lim E ( ϕ ) = + ∞ for ϕ ∈ E ( R d ) , with k ϕ k E → ∞ (6.6)and that the energy is lower semi-continuous; see [
18, Prop. 1.2, p.35 ] . For the coercively it issufficient to show that there are constants C , C > E ( ϕ ) ≥ C k ϕ k E ( R d ) − C k ϕ k E ( R d ) for all ϕ ∈ E ( R d ) . (6.7)Using the NSG inequality we can estimate as follows12 ˆ R d A ∇ ϕ · ∇ ϕ d x ≥ M k∇ ϕ k ( L ) d ≥ C N M k ϕ k L ∗ + M k∇ ϕ k ( L ) d ≥ C ( k ϕ k L ∗ + k∇ ϕ k ( L ) d ) , (6.8)where C : = min { C N M , M } . On the other hand using again the NSG inequality yields ( k ϕ k L ∗ + k∇ ϕ k ( L ) d ) = k ϕ k L ∗ + k∇ ϕ k ( L ) d + k ϕ k L ∗ k∇ ϕ k ( L ) d ≤ k ϕ k L ∗ + k∇ ϕ k ( L ) d + C N k ϕ k L ∗ ≤ ˜ C ( k ϕ k L ∗ + k∇ ϕ k ( L ) d ) (6.9)where ˜ C : = min { + C N , 1 } . Combining (6.8) and (6.9) yields12 ˆ R d A ∇ ϕ · ∇ ϕ d x ≥ C ˜ C k ϕ k E ( R d ) . (6.10)Finally the continuity of F gives F ( ϕ ) ≥ −k F k L ( E , R ) k ϕ k E ( R d ) . (6.11)Combining (6.10) and (6.11) yields (6.7) with C = C / ˜ C and C = k F k L ( E p , R ) .8 K. SturmRecall the Gagliardo-Nirenberg-Sobolev inequality (short NSG inequality) k u k L p ∗ ( R d ) ≤ C N k∇ u k L p ( R d ) (6.12)valid for all u ∈ C ∞ c ( R d ) . The constant C N does not depend on the support of the function u .Notice also that for p = d the inequality fails. Thanks to Lemma 6.2 we know that C ∞ c ( R d ) isdense in E p ( R d ) . Hence it follows that (6.12) holds for all test functions u ∈ E p ( R d ) . For instancefor d = E ( R ) we have k u k L ( R ) ≤ C k∇ u k L ( R ) . (6.13) Lemma 6.2.
For all 1 < p < d the space ( E p ( R d ) , k · k E p ) is a Banach space. For every sequence ( u n ) in E p ( R d ) we find a subsequence ( u n k ) and an element u ∈ E p ( R d ) , such that u n k * u weakly in L p ∗ ( R d ) as n → ∞ , ∇ u n k * ∇ u weakly in L p ( R d ) d as n → ∞ . (6.14)Moreover, C ∞ c ( R d ) is dense in E p ( R d ) . Proof.
Let ( u n ) be a bounded sequence in E p ( R d ) . Since the L p ( R d ) -spaces are reflexive for all p ∈ ( ∞ ) , we find elements η ∈ L p ∗ ( R d ) and ζ ∈ L p ( R d ) d and a subsequence ( u n k ) , such that u n k * η weakly in L p ∗ ( R d ) as n → ∞ , ∇ u n k * ζ weakly in L p ( R d ) d as n → ∞ . (6.15)Now we claim that ζ = ∇ η , which then implies η ∈ E p ( R d ) . To see this notice that by definitionof the weak derivative ˆ R d ∂ x i ϕ u n k d x = − ˆ R d ϕ∂ x i u n k d x (6.16)for all ϕ ∈ C ∞ c ( R d ) . Now we pass to the limit in (6.16) and obtain ˆ R d ∂ x i ϕη d x = − ˆ R d ϕζ d x (6.17)for all ϕ ∈ C ∞ c ( R d ) , which proves the claim. Since a linear and continuous functional on aBanach space is continuous if and only if it is weakly continuous the claim follows.To prove the completeness of E p ( R d ) let ( u n ) be a Cauchy sequence in E p ( R d ) . Then ( u n ) is aCauchy sequence in L p ∗ ( R d ) and ( ∇ u n ) is a Cauchy sequence in L p ( R d ) d . Since ( u n ) is a Cauchysequence in L p ∗ ( R d ) and ( ∇ u n ) is a Cauchy sequence in L p ( R d ) we find elements η ∈ L p ∗ ( R d ) and ζ ∈ L p ( R d ) d so that u n → ζ strongly in L ∗ ( R d ) and ∇ u n → ζ strongly in L p ( R d ) d . Now wecan follow the previous argumentation to show that ∇ η = ζ which shows that η ∈ E p ( R d ) andthus shows that E p ( R d ) is complete.Let us now show the density of C ∞ c ( R d ) in E p ( R d ) . As shown in [
1, Thm. 3.22, p. 68 ] it sufficesto show every u ∈ E p ( R d ) with bounded support can be approximated by function in C ∞ c ( R d ) .Suppose that the support of u is compact. Denote by u ǫ : = ̺ ǫ ∗ u the standard mollification of u with a mollifier ̺ ǫ ∈ C ∞ ( R d ) , ǫ >
0; see [
1, pp. 36 ] . Then u ǫ ∈ L p ∗ ( R d ) and ∂ i u ǫ = ̺ ǫ ∗ ∂ i u ∈ L ( R d ) . Moreover according to [
1, Thm. 2.29, p. 36 ] we have lim ǫ ց k u ǫ − u k L ∗ ( R d ) = ǫ ց k ∂ x i ( u ǫ − u ) k L ( R d ) =
0. This finishes the proof.opologicalderivativevia Lagrange 29
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