TOPICS IN HOMOGENIZATION OF DIFFERENTIAL EQUATIONS 7

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Claribel Simpson
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1 TOPICS IN HOMOGENIZATION OF DIFFERENTIAL EQUATIONS 7. Two-scale asymptotic epansions method We will illustrate this method by applying it to an elliptic boundary value problem for the unknown u : æ R given by ] div A Òu = f in, () u = on ˆ. The above model can be an e cient model for various phenomena in continuum mechanics. In the below table, we list some of the scientific domains in which the above elliptic problem can be used and the meaning of the coe cient A() and the unknown u () in those respective fields. Domain Unknown u () Material coe cient A() Electrostatics Electric potential Dielectric coe cient Magnetostatics Magnetic potential Magnetic permeability Steady heat transfer Temperature Thermal conductivity As an helpful eample, let us consider a one-dimensional problem which I borrowed from a lecture by Daniel Peterseim (Bonn). Eample.. For the unknown u (), consider the boundary value problem _] d 3 4 du a = in (, ), () d d _ u () = u () =, with the conductivity coe which is plotted below cient given as a = + cos! " fi Figure 3. Plot of! + cos! "" fi with = (left), (right). He gave this eample during a scientific workshop at the Hausdor Research Institute for Mathematics, Bonn.

2 HARSHA HUTRIDURGA Observe that the coe cient a! " given above has more and more oscillations in the π regime. It is indeed of interest to learn the e ect of these high frequency oscillations on the solution u (). A lucky accident in this eample is that we can eplicitly integrate the di erential equation () yielding fi u () = + 4fi sin 3 fi fi sin which is plotted below 4 4fi cos 3 fi fi Figure 4. Plot of u () on, ] with = (left), (right). Remark that the solution u () has a parabolic profile with small amplitude oscillations of O() along the profile. Indeed we have the uniform convergence u () æ as æ. So, the above parabolic profile is the homogenized approimation of the solution profile u (). In the above eample, instead of characterizing the homogenized operator, we have eplicitly computed the homogenized profile. This is possible, thanks to the eplicit integration of the di erential equation (). There is no such luck in obtaining eplicit analytical epressions for solutions to di erential equations in general, even in one dimension. Let us briefly describe the setting of periodic homogenization. Denote the unit cube := (, ) d µ R d. Consider a -periodic function A : æ R d d,i.e., A( + ke i )=A() for each œ R d, k œ Z i œ{,...,d} where {e i } denotes the canonical basis in R d. Taking >as a parameter, consider the dilated sequence defined as A () :=A. Remark that the dilated function A () is -periodic because A kei ( + ke i )=A = A + ke i = A = A () for all k œ Z. In all the theory that is being presented in this course, a fundamental assumption is that of separated scales. To be precise, we consider to be a slow variable

3 TOPICS IN HOMOGENIZATION OF DIFFERENTIAL EQUATIONS 9 and introduce a new auiliary variable y := which we call a fast variable. The assumption of separated scales essentially implies that the variables and y are to be treated as independent variables. With the above understanding of the slow and fast variables, in the periodic setting, we let the variable y œ,theunitcubeinr d introduced earlier. Then, the periodic conductivity coe cient A can be treated as a function of the fast spatial variable y. We further assume that the conductivity coe cient A(y) is uniformly bounded and uniformly positive definite, i.e., there eist constants, > (uniform in the y variable) such that (3) Æ A(y) Æ y œ and œ R d \{}. Employing the La-Milgram theorem, we can prove that the boundary value problem () is well-posed. Theorem.. Suppose the source term f œ L ( ). Suppose that the periodic conductivity matri A(y) is positive definite and that it is uniformly bounded, i.e., it satisfies (3). Then for each fied >, there eists a unique solution u œ H ( ) to the elliptic boundary value problem (). Furthermore, we have the uniform estimate (4) Îu Î H ( ) Æ C ÎfÎ L ( ) with constant C> being independent of the parameter... A postulate and matched asymptotic. The model conductivity problem () has two spatial scales and it has a small parameter >. Hence the essential idea of the method of asymptotic epansions is to consider a power series epansion for the unknown u in the parameter and with coe cients that depend on both the spatial scales, i.e., on both the slow and fast variables. More precisely, we postulate that the solution to () can be written as (5) u () = ÿ iø i u i, under the assumption that the coe cient! functions " u i (, y) are -periodic in the y variable. The coe cient functions u i, are referred to as locally periodic functions. At least formally, for π, we have from (5) the approimation u () u (, y) - y= Then, in the π regime, one can argue that the y dependance of u gets averaged out over the periodic cell. One could then find the homogenized approimation for u as u hom () := u (, y)dy. The above heuristics essentially motivates the postulated asymptotic epansion (5) for the solution u (). These hand-wavy heuristics are not at all rigorous. More analysis oriented approach to derive homogenized limits will be the objective of the chapters to come. As far as this chapter is concerned, we admit the two-scale asymptotic epansion for the solution and carry on with the computations. Observe

4 HARSHA HUTRIDURGA that the eplicit solution obtained in Eample. has the structure postulated above in (5). That is a mere coincidence. The principle behind the asymptotic epansions method is to plug the epansion (5) in the model problem () and match the coe cients of various powers of. Before we proceed, we need to keep the following in mind while applying the asymptotic epansion method. (6) For any locally periodic function Â! ",, its total derivative becomes Ë Ò Â, È - = Ò Â(, y) + Ò - yâ(, y) - y= - y= Note that the above epression is a consequence of the chain rule for di erentiation. Suppose (, y) is a function of both the slow and fast variables. Then the following are equivalent (i), = œ, >, (ii) (, y) = œ, y œ. Let us get on with the task of plugging the asymptotic epansion (5) inthemodel problem () yielding div A Òu = - div --y= y A (y) Ò y u (, y) div y A (y)! Ò u (, y)+ò y u (, y) "- - -y= div A (y) Ò y u (, y) - --y= div y A (y)! Ò u (, y)+ò y u (, y) "- - -y= div A (y)! Ò u (, y)+ò y u (, y) "- - -y= div y A (y)! Ò u (, y)+ò y u 3 (, y) "- - -y= div A (y)! Ò u (, y)+ò y u (, y) "- - -y= where we will not eplicitly write down the O( ) terms and further on (but the readers can naturally etrapolate). In the computations to follow, only the O() terms and the terms of higher order play significant role. In the boundary value problem (), f() is a given source term of O(). Hence equating the coe cients of various powers of, we obtain a cascade of equations. As it will become apparent in the computations to follow, the orders in that are essential for arriving at the

5 TOPICS IN HOMOGENIZATION OF DIFFERENTIAL EQUATIONS homogenized equation are that of O( ), O( ) and O(). At O( ), we obtain ] div y A (y) Ò y u (, y) = for y œ, (7) y æ u (, y) is -periodic. Note that the periodic boundary conditions are because of our periodicity assumption on the coe cient functions in the postulated asymptotic epansion. The variable is treated as a parameter in (7) which is treated as a periodic boundary value problem in the y variable. Before proceeding further, let us remark that u is a function independent of the y variable. To see that, let us multiply the equation (7) byu and integrate over yielding A(y)Ò y u (, y) Ò y u (, y)dy A(y)Ò y u (, y) n(y)d (y) =. ˆ By assumption A( ) and u (, ) are periodic. As the normal n(y) takes opposite signs on the opposite ends of the boundary ˆ,wehave A(y)Ò y u (, y) n(y)d (y) =. ˆ Furthermore, the coercivity assumption on A(y) implies ÎÒ y u (, )Î L ( ) Æ A(y)Ò y u (, y) Ò y u (, y)dy with the right hand side of the above inequality vanishing by the above energy calculations. Hence we indeed have that u (, y) u (). Continuing our method of matched asymptotic, at O( ), we have ] div y A (y) Ò y u (, y) =div y A (y) Ò u () for y œ, (8) y æ u (, y) is -periodic. Note that we have dropped the term involving the Ò y u (, y) from (6) even though it was apparently of order O( ). This is because of our earlier observation that u is independent of the y variable. Equation (8) should be treated as an equation for the unknown u (, y). Because of linearity, observe that we can separate the slow and fast variables in u as (9) u (, y) = dÿ i= Ê i (y)ˆu ˆ i ()

6 HARSHA HUTRIDURGA where the unknown functions w i (y) for each i œ{,...,d} solve the so-called cell problems ] div y A (y) Ò y Ê i (y) =div y A (y) e i for y œ, () y æ Ê i (y) is -periodic. Here {e i } d i= denotes the canonical basis of Rd.Thedequations in () are called cell problems as they are posed on the periodicity cell. The solutions to () are henceforth called cell solutions. Finally, at O(), we obtain () div y A (y) Ò y u (, y) =div y A (y) Ò u (, y) _] +div A (y) Ò u ()+Ò y u (, y) + f(), _ y æ u (, y) is -periodic. So far, we have not spoken about the solvability of the periodic boundary value problems (7)-(8)-()-() that we derived by plugging the asymptotic epansion in the model problem. The following lemma will address this issue. Lemma.3. Suppose the periodic conductivity coe cient A(y) is positive definite and that it is uniformly bounded, i.e., it satisfies (3). Let g œ L ( ) be a given function. Then there eists a unique solution v œ H per( )/R to the following periodic boundary value problem ] div y A (y) Ò y v(y) = g(y) for y œ, () y æ v(y) is -periodic if and only if the source term g(y) in () satisfies the compatibility condition (3) ÈgÍ := g(y)dy =. Proof. We know that the the quotient space H per( ) has the equivalent norm ÎvÎ H per ( ) = ÎÒ yvî L ( ). Solving the periodic boundary value problem () is equivalent to solving the variational problem: Find a unique v œ H per( ) such that w œ H per( ), we have a(v, w) =l(w) with a(v, w) := A(y)Ò y v(y) Ò y w(y)dy and l(w) := g(y)w(y)dy.

7 TOPICS IN HOMOGENIZATION OF DIFFERENTIAL EQUATIONS 3 To employ the La-Milgram theorem, we need to show that the bilinear form a(, ) is coercive on H per( ). This is straightforward. Net, we need to show that the linear form l( ) is bounded. Compute the modulus of the linear form l(w) = - g(y)w(y)dy - = g(y)w(y) ÈwÍg(y)+ÈwÍg(y) dy - = -- - g(y)w(y) ÈwÍg(y)- dy ÆÎw ÈwÍÎ L ( ) ÎgÎ L ( ) Æ C ÎÒ y wî L ( ) ÎgÎ L ( ) where we have used the Hölder inequality and the Poincaré inequality. Note that the above computation only works under the assumption that g(y) is of zero mean. The eistence and uniqueness of v œ H per( )/R then follows from La-Milgram. Remark.4. This remark is about the solvability of the boundary value problems (7)-(8)-()-(). As Lemma.3 asserts, only detail that one needs to check is that the source terms in all these periodic boundary value problems are of zero average on the periodicity cell. The source term in the O( ) equation (7) is g(y). Hence the compatibility condition (3) is trivially satisfied. The result of Lemma.3 then implies that the unique solution to (7) is œ H per( )/R i.e., u (, y) u () which is what we deduced earlier via energy computations. The source term in the O( ) equation (8) is g(y) =div y A (y) Ò u (). As the variable plays the role of a parameter, we will not make eplicit the dependence of the source g on the variable. Let us now compute the average of g(y) over the periodic cell div y A (y) Ò u () dy = A (y) Ò u () n(y)d (y) =. ˆ Thus the compatibility condition (3) is satisfied. Hence the eistence of a unique solution u (, ) œ H per( )/R to the O( ) equation (8). A similar arguments work for the cell problems () whose source terms are g(y) =div y A (y) e i. Finally, to solve the periodic boundary value problem for u (, ) œ H per( )/R, let us make sure that the source term in () satisfies the compatibility condition. In fact, writing down the compatibility condition for () yields the

8 4 HARSHA HUTRIDURGA (4) homogenized equation for u (). The source term in the O() equation () is g(y) =div y A (y) Ò u (, y) +div A (y)! Ò u ()+Ò y u (, y) " + f() Integrating the above given g(y) over the periodicity cell yields g(y)dy = I + I + I 3 := div y A (y) Ò u (, y) dy + div A (y)! Ò u ()+Ò y u (, y) " dy + f()dy. Remark that, because A( ) and u (, ) are -periodic, we have I =. Furthermore, as f is independent of the y variable, I 3 = f(). From Lemma.3, we can solve for u (, ) uniquely, provided we satisfy the compatibility condition which translates here as I = I 3 which is nothing but div A (y) Ò u ()+Ò y u (, y) dy = f() for œ. Observe that substituting for u (, y) in the above equation in terms of the cell solutions Ê i (y) and the gradient of u (), i.e., using (9) results in a second order di erential equation for u (). Let us pick up our calculations from (4). Substituting for u (, y) using(9) results in div A (y) Ò u (, y)+ò y u (, y) dy A A = div A (y) Ò u (, y)+ = div A hom Ò u () dÿ k= Ò y Ê k (y) ˆu ˆ k () with the constant matri A hom called the homogenized matri which is given as A B dÿ A hom = A (y) + k Ò y Ê k (y) dy k= BB dy

9 TOPICS IN HOMOGENIZATION OF DIFFERENTIAL EQUATIONS 5 for any œ R d. Note in particular that the elements of the matri A hom are (5) A hom ] ij = A (y) e j + Ò y Ê j (y) e i dy for i, j œ{,...,d}. The calculations so far are summarised below. Proposition.5. The solution u () to the boundary value problem () is approimated as dÿ u ˆu () u ()+ Ê j () ˆ j j= where u () is called the homogenized approimation which solves the homogenized boundary value problem ] div A hom Òu () = f in, (6) u = on ˆ, with A hom being the constant homogenized conductivity whose elements are A hom ] ij = A (y) e j + Ò y Ê j (y) e i dy for i, j œ{,...,d}. The functions Ê j (y) are called the cell solutions which solve the cell problems ] div y A (y) Ò y Ê i (y) =div y A (y) e i for y œ, y æ Ê i (y) is -periodic for each i œ{,...,d}. Furthermore, the homogenized conductivity is uniformly positive definite, i.e., there eists a positive > such that (7) for all œ R d \{}. A hom Ø Proof. A lion share of the proof of Proposition.5 is already present in the calculations preceding the statement of the proposition. We are left, however, to show that the homogenized approimation u () satisfies zero Dirichlet data on the boundary and to show that A hom is uniformly positive definite. As we are postulating an asymptotic epansion (5), let us simply equate the epansion to zero on the boundary =u ()+u, + u, +... for œ ˆ. As the above equality should hold in the æ limit, at least formally, we obtain u () = for œ ˆ. In order to prove the positive definiteness of A hom,letusmultiplybyê i (y) thecell problem for Ê j (y) and integrate over the periodicity cell yielding A(y) Ò y Ê j (y)+e j Ò y Ê i (y)dy =.

10 6 HARSHA HUTRIDURGA Add the above zero to the epression (5) yielding an alternate epression for the elements of the homogenized conductivity (8) A hom ] ij = A (y) e j + Ò y Ê j (y) e i + Ò y Ê i (y) dy for i, j œ{,...,d}. Using the above alternate epression for the homogenized conductivity, let us compute the dot product A hom = A (y) + Ò y Ê (y) + Ò y Ê (y) dy where the function Ê (y) is defined as Ê (y) = dÿ Ê k (y) k. The coercivity assumption (3) on the conductivity coe cient A(y) implies A hom Ø + Ò y Ê (y) dy = Ò y Ê (y) dy + dy + Ò y Ê (y)dy Ø = as Ê periodic Hence Suppose now that which implies that i.e. k= A hom Ø. A hom = + Ò y Ê (y) = y œ Ê (y) =C y for some constant C. This says that Ê (y) is an a ne function in the y variable. This is a contradiction to the periodicity of Ê (y) eceptif =. Remark.6. If the conductivity coe cient A(y) is assumed to be symmetric, then the epression (8) for the homogenized conductivity implies that A hom is symmetric as well. Eercise.7. Using the method of two-scale asymptotic epansions, prove a result similar in flavour to Proposition.5 for the following periodic parabolic problem: fl ˆtu (t, ) div A Òu (t, ) = f(t, ) in (,T), _] (9) u (t, ) = on (,T) ˆ, _ u (,)=u in () in,

11 TOPICS IN HOMOGENIZATION OF DIFFERENTIAL EQUATIONS 7 where the coe cients fl( ) and A( ) are -periodic. Assume that the coe cient fl is bounded from above and away from zero, i.e., < c Æ fl( ) Æ c < Œ. Assume further that the coe cient A( ) satisfies the assumption (3). The bulk source f(t, ) and the initial datum u in () are nice functions and are as regular as the computations demand. Hint: Treat the time variable t as a parameter while performing the asymptotic epansions, i.e., postulate that the solution u (t, ) to the above initial boundary value problem can be written as u (t, ) =u t,, + u t,, + u t,, +... where the coe variable. cient functions u i (t,, y) are assumed to be -periodic in the y Eercise.8. Assume that the coe cients fl( ), A( ) satisfy the same assumptions as in Eercise.7. Assume further that the data f(t, ), u in () and u in () are nice functions and that they are as regular as the computations demand. Using the method of two-scale asymptotic epansions, prove a result similar in flavour to Proposition.5 for the following periodic hyperbolic problem: fl ˆttu (t, ) div A Òu (t, ) = f(t, ) in (,T), () _] _ u (t, ) = on (,T) ˆ, u (,)=u in () in, ˆtu (,)=u in () in... One dimensional problem. In this section, we will present some eplicit computations in a one dimensional setting. Let the strictly positive and bounded conductivity coe cient be denoted by a(y) which is -periodic. Then one dimensional analogue of the boundary value problem () is _] d 3 4 du a = f() in (, ), () d d _ u ( ) =u ( ) =, for an interval (, ) µ R. The one-dimensional analogue of Proposition.5 is stated here. Corollary.9. The cell solution in the one-dimensional setting is given by () Ê(y) =C y + a hom y d a( )

12 8 HARSHA HUTRIDURGA for a constant C. Furthermore, the constant a hom is the homogenized conductivity given by 3 4 dy (3) a hom = a(y) Proof. Let us write down the one-dimensional analogue of the cell problem (). _] d dê a (y) dy dy (y)+ = in(, ), _ Ê() = Ê(). Eplicitly integrating the above di erential equation yields for some constant C, 3 4 dê a (y) dy (y)+ = C = dê dy (y) = + C a(y) Integrating again would yield Ê(y) =C y + y C a( ) d. The constant of integration C is a dummy variable (for now). It can be chosen by normalization as we look for solutions in the quotient space H per(, )/R. The constant C, however, is determined by the periodic boundary condition in the cell problem. 3 4 dy C =. a(y) Now, let us write down the one dimensional analogue of the homogenized conductivity (5). 3 4 dê a hom = a (y) dy (y)+ dy which is nothing but C dy. Hence we have deduced that the one dimensional homogenized conductivity is nothing but the harmonic average over the periodicity cell, i.e., 3 4 dy a hom = a(y) This proves the corollary. Remark.. Note that the following strict inequality holds 3 4 dy < a(y)dy a(y) for any non-trivial function a(y). By, trivial, we mean functions that are identically constants in which case the harmonic average and the simple average coincide. This observation implies that the homogenized conductivity of a periodic composite is

13 TOPICS IN HOMOGENIZATION OF DIFFERENTIAL EQUATIONS 9 always less than the associated simple average (arithmetic average), at least in one dimension. Remark.. Let us reconsider the one dimensional illustrative Eample.. We chose the periodic conductivity to be a(y) = + cos(fiy). Computing the homogenized conductivity in this case yields 3 4 a hom = + cos(fiy) dy =. Hence the homoegnized equation for the Eample. becomes _] d u = in (, ), d _ u () = u () = which integrates eplicitly as u () = This is the parabolic profile which we found as a macroscopic approimation to the solution u () in Eample.. Eercise.. Compute the homogenized conductivity a hom for the following periodic conductivity which is piecewise constant: for y œ! ", 3 _] for y œ # 3 a(y) =, " 3 for y œ # _, " for y œ # 3 4, " Compute also its arithmetic average..3. Laminated geometries in higher dimensions. The one dimensional setting of section. is pretty special in the sense that given a periodic conductivity a(y), we can straightaway compute the homogenized conductivity a hom without having to go though the cell problem. In this section, we consider the eamples of periodic laminated structures in higher dimensions, i.e., scenarios where the conductivity coe cient depends on a single direction (i.e. varying only in one direction). We take A(y) =A(y ). In this section, we further make the assumption that the above conductivity coe - cient is isotropic, i.e., there eists a scalar function a(y ) such that (4) A(y) =a(y )Id. We will prove the following interesting result on the above chosen laminated structures.

14 HARSHA HUTRIDURGA Proposition.3. The homogenized conductivity associated with the isotropic conductivity coe cient (4) is the following diagonal matri 3 4 dy _] for i =, a(y ) A hom ] ii = _ a(y )dy for i =. Remark.4. It is to be noted that even though we chose an isotropic periodic conductivity coe cient to begin with, the homogenization procedure destroys the isotropy as can be noticed in Proposition.3. Proof of Proposition.3. Let us rewrite the cell problem () with the chosen isotropic conductivity (4) as ] div y a (y ) Ò y Ê i (y) =div y a (y ) e i for y œ, y æ Ê i (y) is -periodic for each i œ{,...,d}. In particular, for i =, we have _] div y a (y ) Ò y Ê (y) = ˆ a (y ) for y œ, ˆy _ y æ Ê (y) is -periodic Observe that the solution Ê (y) depends only on the y direction, i.e., Ê (y) = (y ) where (y ) solves the one dimensional cell problem _] d d a (y ) (y )+ = in(, ), dy dy _ () = (). Net, for i =, we have _] div y a (y ) Ò y Ê i (y) _ = ˆ ˆy i a (y ) = for y œ, y æ Ê i (y) is -periodic Observe that the solutions to the above (d ) cell problems are all in the zero equivalence class (see Lemma.3). Hence we have Ê i (y) for i =. The above information on the cell solutions help us get some insight into the homogenized coe cient which in this case is a diagonal matri. Let us compute A hom ] using the formula (5). A hom ] = d a(y ) (y )+ dy = dy 4 dy a(y )

15 TOPICS IN HOMOGENIZATION OF DIFFERENTIAL EQUATIONS where the last equality follows from Corollary.9. Net, let us compute A hom ] ii for i = usingtheformula(5). A hom ] ii = a(y ) Ò y Ê i (y)+e i e i dy = a(y )dy because the cell solutions Ê i (y) for i =. Eercise.5. Using Proposition.3, compute the homogenized conductivity A hom for the following laminar periodic conductivity: I d for y œ!, " A(y) = d for y œ #, " with strictly positive constants d,d. Eercise.6. Mimicking the proof of Proposition.3, obtain analytic epressions for the homogenized conductivity A hom in two dimensions when the periodic conductivity coe cient is laminar, i.e., A(y) =A(y ), but not necessarily isotropic as was the case in (4).

University of Alabama Department of Physics and Astronomy. PH 125 / LeClair Spring A Short Math Guide. Cartesian (x, y) Polar (r, θ)

University of Alabama Department of Physics and Astronomy PH 125 / LeClair Spring 2009 A Short Math Guide 1 Definition of coordinates Relationship between 2D cartesian (, y) and polar (r, θ) coordinates.