Brooke L. Hollingsworth and R. E. Showalter Department of Mathematics The University of Texas at Austin Austin, TX USA

Transcription

1 SEMILINEAR DEGENERATE PARABOLIC SYSTEMS AND DISTRIBUTED CAPACITANCE MODELS Brooke L. Hollingsworth an R. E. Showalter Department of Mathematics The University of Texas at Austin Austin, TX 7872 USA Abstract. A two-scale microstructure moel of current flow in a meium with continuously istribute capacitance is extene to inclue nonlinearities in the conuctance across the interface between the local capacitors an the global conucting meium. The resulting egenerate system of partial ifferential equations is shown to be in the form of a semilinear parabolic evolution equation in Hilbert space. It is shown irectly that such an equation is equivalent to a subgraient flow an, hence, isplays the appropriate parabolic regularizing effects. Various limiting cases are ientifie an the corresponing convergence results obtaine by letting selecte parameters ten to infinity.. Introuction. As integrate circuits become smaller, istribute capacitors receive corresponingly more attention; they have become the big components in integrate circuits. An example of istribute capacitance is the tantulum capacitor. A porous slug presse out of tantalum power is sintere to make the metal particles cohere an then anoize to prouce a film of tantalum oxie, which serves as the ielectric of the capacitor. Next the slug is immerse in a solution of manganese nitrate an heate; this process leaves eposits of semiconucting manganese ioxie in its pores. The manganese ioxie serves as one electroe, while the unerlying tantalum serves as the other [24]. Due to the intricate fine scale of its geometry, the real moel for a capacitor of this type is too singular to be useful. Thus we use This material is base upon work supporte by a grant from the National Science Founation. Typeset by AMS-TEX

2 2 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER the istribute capacitance moel, in which a microcapacitor is ientifie with each point of a larger omain, as a continuous approximation to the actual situation. These microstructure moels contain the fine scale geometry of the microcapacitors as well as the current flux across the intricate interface by which they are connecte to the global fiel. Another moel for istribute capacitance is the layere meium equation. This moel represents a continuous approximation to a meium consisting of alternating thin layers of conuctive an ielectric materials an is given by the equation.) t zcx, z) z u) z G V x, z) z u) x G H x, z) x u) = F x, z, t). Instea of the thin layers of ielectic an conucting materials moele by the layere meium equation, we shall consier small horizontally aligne microcapacitors istribute continuously throughout the conucting meium. We will show that the layere meium equation can be obtaine as a singular limit of the microstructure moel by approximating the istribute capacitors by single points of charge storage. The actual system of equations for the istribute capacitance moel will be given in the next section. It will be shown that the system has the abstract form.2) Bu) + Au) f t where the linear operator B is non-negative an symmetric but egenerate an the nonlinear operator A is monotone. The operator B is necessarily egenerate, since it arises as a consequence of the charging capacitor an thus is nonzero in the local cell equations but zero in the global fiel equations of the system. The layere meium equation.) is also of this form. In orer to show that the Cauchy problem for the evolution equation.2) is well-pose, we will use some techniques of convex analysis. For etails, see [4] an [9]. Let V be a Banach space, an let ϕ : V, + ] be convex, proper, an lower-semi-continuous. Then w V, the ual space, is a subgraient of ϕ at u V

3 SEMILINEAR DEGENERATE PARABOLIC SYSTEMS 3 if wv u) ϕv) ϕu) for all v V. The set of all subgraients of ϕ at u is enote by ϕu). The subgraient is a generalize notion of the erivative, comparable to a irectional erivative. We regar ϕ as a multivalue operator from V to V ; it is easily shown to be monotone. Equations of the general form of.2) have been stuie for a long time by a variety of methos. These equations are of interest not only for the sake of generalization but also because they arise naturally in a vast variety of applications. The case of linear B may have egenerate behaviour ue to a possibly spatially epenent) coefficient that vanishes somewhere. That situation is in essence the type encountere here. The earliest general treatment of semilinear an egenerate evolution equations of the form of.2) in an abstract setting occurs in the work [22]. There the nonlinear B is monotone an continuous an A is structure after a family of linear elliptic operators; both are permitte to be time epenent. In a similar but simpler setting, results were obtaine in [2] by a backwar ifference approximation technique, an such results were subsequently obtaine irectly from a characterization of maximal monotone operators in [5]. Thereafter this situation was shown in [2] to be attainable irectly as an application of nonlinear semigroups in Hilbert space; the solution was stronger but require smoother ata initially. The parabolic case with A being a subgraient coul be hanle this way if B were invertible, but this oes not cover the egenerate case. See [7,,23].) For a review of work prior to 976 an many examples, we refer to Chapter 3 of the book [8]. Perturbation an continuous epenence results were emonstrate in [7,8] an [25], an aitional extensions were subsequently evelope in the series of papers [2,3,4,5,6]. Our purpose here is to evelop a nonlinear microstructure moel for istribute capacitance in a conucting meium, to formulate it in an abstract form of a system of egenerate semilinear parabolic equations, an to show that this system is wellpose. We will fin that the operator A of.2) that arises from our istribute

4 4 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER capacitance moel is the subgraient of a convex function, i.e., there exists a convex function ϕ such that A = ϕ. Then we will show in the abstract setting that there exists another convex function Φ such that.2) is equivalent to an explicit equation of the stanar form.3) t w + Φw) g, so this problem is parabolic an has the corresponing regularizing effects on the ata. See [4,6]. Our plan is as follows. In Section 2 we erive the equations in their variational form involving the subgraient of a convex function. In Section 3 we evelop existence an uniqueness results for the Cauchy problem for.2); we then prove results in Section 4 concerning limiting cases. In Section 5 we apply these results to the istribute capacitance moel. 2. The Distribute Capacitance Moel. In this section we escribe the istribute microstructure moel. Let Ω be a boune omain in R 3 ; this omain represents the conucting material in which the capacitance is to be istribute. For each point x Ω, there is given a boune cylinrical omain of the form Ω x = S x [ h 2, h 2 ], where S x is a cross section in R 2 an h > is the thickness. Each Ω x represents the generic horizontally oriente microcapacitor embee in the conuctor Ω in the vicinity of the point x. Each x Ω represents a point in real space, an y S x is the local variable in the small scale. Let ux, t) be the voltage istribution in the global region Ω. We will represent the voltage ifference across the capacitor Ω x by Ux, y, t); specifically, at any time t an point y S x, Ux, y, t) is equal to the voltage at the point y, h 2 ) on the top minus the voltage at the bottom point y, h 2 ). Using the approximation ux, x 2, x 3 + h 2 ) ux, x 2, x 3 h 2 ) h u, we have the voltage ifference across the interface between the microcapacitor an the surrouning conucting meium given by h u U. In each cell Ω x, the horizontal voltage graient inuces a current ensity given by Ohm s law as K y U, where K is the horizontal conuctance of the capacitor

5 SEMILINEAR DEGENERATE PARABOLIC SYSTEMS 5 surfaces. We use the subscript y on the graient symbol to inicate that the graient is taken with respect to the local variable y. The graient symbol with no subscript will be use to inicate a graient taken with respect to the global variable x. The capacitor charges in time at a rate of Cx, y)u t, where Cx, y) is the istribute capacitance. A vertical current input to Ω x from the surrouning conuctor is inuce by the voltage rop h u by ψ G h u U; we will assume that this current is given U), where ψ G is the monotone nonlinear conuctance function obtaine as the erivative or subgraient of a convex function ψ G. The principle of conservation of charge then yiels 2.) t Cx, y)u) y K y U) ψ G h u ) U F, y S x where the function F enotes any aitional istribute current sources. Similarly, the surrouning conuctor inuces a current of magnitue h ψ g h u U) across the bounary of the capacitor in the horizontal normal irection n, where ψ g is the subgraient of a convex function ψ g. Thus we have 2.2) K U n h ψ g h u ) γu, s S x At the global level, the istribute current arises from two sources: the global voltage, ux, t), an the normal current exiting the microcapacitor at the point x. This current is given by j = k u e 3 K U S x S x n s + ψ G h u ) ) U y S x where k is a positive efinite an symmetric matrix representing the conuctance of the surrouning material. This matrix will reflect the fact that the current will flow more easily in the horizontal irections than in the vertical one. Conservation of charge on the global scale gives j =. Thus we have 2.3) k u ψ G h u ) U y + K U ) S x S x S x n s,, x Ω

6 6 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER We will assume a groune bounary at the global level, 2.4) ux) =, x Ω, although any one of the usual bounary constraints can be hanle similarly. Finally, we nee to specify initial values for the charge istribution U, 2.5) Cx, y)ux, y, ) = Cx, y)u x, y), x Ω, y S x. The system given by 2.) 2.4) is our istribute RC network moel for istribute capacitance. This system of partial ifferential equations is of mixe egenerate parabolic-elliptic type. It consists of a family of iffusion equations, given by 2.), each of which escribes the conuction an storage of charge on the local scale of an iniviual capacitor at a specific site on the global conuction meium, an the single elliptic equation 2.3) which governs the interconnection by conservation of charge on the global scale of the conuctor. This moel contains the geometry of the iniviual capacitors an the current flux across the intricate interface by which they are connecte to the global current fiel. Notice that the total charge rate of the capacitor is given by CU y = F + ψ G h u ) ) U y + h ψ g t S x S x S x = F y + J S, S x Ω x in which the function 2.6) J = ψ G h u U ψ g h u γu is the current flux across the bounary Ω x. ) /2 at ȳ = y, ± h ), y S x 2 ) at ȳ = y, y 3 ), y S x h u ) γu y In this evelopment we have permitte the current flux in 2.6) to be riven by a nonlinear conuctance. Our hypotheses below require that the convex functions

7 SEMILINEAR DEGENERATE PARABOLIC SYSTEMS 7 have at most quaratic growth, so the nonlinear terms in 2.6) will be linearly boune. These restrictions are only technical an convenient; one can easily inclue more general nonlinearities an also quasilinear moels arising from nonlinear conuctances in 2.) an 2.3). In orer to obtain the variational formulation of the istribute capacitance moel, we specify the spaces to be use. Let L 2 Ω) be the Lebesgue space of equivalence classes of functions that are square-integrable on Ω, an let H Ω) be the Sobolev space consisting of those functions in L 2 Ω) having each of their partial erivatives also in L 2 Ω). Denote by C Ω) the space of infinitely ifferentiable functions with support containe in Ω; the space H Ω) is the closure in H Ω) of C Ω). For more information on these spaces, see []. Let Q Ω R 2 be a measurable set in R 5, an set S x = {y R 2 : x, y) Q}; this provies an explicit contruction of the measurable family of cells mentione above. By zero-extension, we can ientify L 2 Q) as a subspace of L 2 Ω R 2 ) an each L 2 S x ) as a subspace of L 2 R 2 ). Thus we have the ientification L 2 Q) = {U L 2 Ω, L 2 R 2 ) : Ux) L 2 S x ) for a.e. x Ω}. We will enote this space by L 2 Ω, L 2 S x )), with the inner prouct { } U, Θ) L2 Ω,L 2 S x )) = Ux, y)θx, y)y x. Ω x S x Ω This is a continuous irect sum of Hilbert spaces, since a function that is in L 2 Ω, L 2 S x )) takes values in a ifferent Hilbert space at each point x Ω. We efine Sobolev spaces in a similar manner. Define { L 2 Ω, H S x )) U L 2 Ω, L 2 S x )) : Ux) H S x ) a.e. x Ω, } an Ux) 2 H S x ) x < ; this irect sum is a Hilbert space. Ω The state space for our problem will be the prouct H L 2 Ω) L 2 Ω, L 2 S x )), an the energy space will be V H Ω) L 2 Ω, H S x )). We will enote an element of these prouct spaces by ũ = [u, U]. In orer to efine trace maps on these spaces, we require that each S x is a boune

8 8 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER omain in R 2 which lies locally on one sie of its bounary, S x, an that S x be a smooth curve in R 2. Let γ x : H S x ) L 2 S x ) be the trace map from each cell to its bounary. We assume these maps are uniformly boune so that we may efine the istribute trace γ : L 2 Ω, H S x )) L 2 Ω, L 2 S x )) by γu)x, s) = γ x U)s); in this case, γ is boune an linear. These efinitions enable us to state precisely the weak formulation of our system. Suppose [u, U] is an appropriately smooth solution of 2.) 2.4), an let [θ, Θ] V be corresponing test functions. Multiply 2.) by Θ an integrate over S x. Using Green s Theorem an 2.2), we obtain { 2.7) Ω x S x t Cx, y)u)θ + K y U y Θ ψ G h u ) U γθ s = S x Ω x S x ψ g h u ) U S x F Θ y. } Θ y Technically, an equation like this is an abuse of notation since ψ G may be multivalue. When an equation like this is use, we mean that there exists a representative from the multivalue operator such that equality hols. We will use this notational convenience hereafter.) Similarly, multiply 2.3) by θ an integrate over Ω to obtain { 2.8) k u θ + ψ G h u ) U y + K U ) θ Ω S x S x S x n s Finally, use 2.2) an a the integral of 2.7) over Ω to 2.8) to obtain { k u θ + 2.9) Ω Ω x S x t Cx, y)u)θ + K y U y Θ + ψ G h u ) U h θ ) ) Θ y + ψ g h u ) γu h θ ) } γθ s x S x S x = F Θ y x. Ω x S x Ω } x =. In Section 3 we shall efine a generalize solution of 2.) 2.4) to be a pair of appropriate functions u an U such that 2.9) hols for all corresponing test

9 SEMILINEAR DEGENERATE PARABOLIC SYSTEMS 9 functions θ an Θ. Conversely, a generalize solution ũ = [u, U] V of 2.9) can be shown to satisfy 2.) 2.4). We have shown that the variational form of the system 2.) 2.4) can be written succinctly as 2.) t Bũt)) + Aũt) ft) in V Bu) = Bu, where B : H H an A : V V are the operators given by Bũ θ) = Cx, y)uθ y x, an, Ω Ω x S x { Aũ θ) = k u θ + [K Ω Ω x y U y Θ + ψ G h u ) U h θ ) ] Θ y S x + ψ g h u ) γu h θ ) } γθ s x. S x S x We assume that Cx, y) is a boune, nonnegative function, that k is a positive efinite an symmetric matrix, an that K is a positive constant. We also assume that the convex functions ψ G an ψ g satisfies the conitions i) ψ) = = minψ), an are lower semi-continuous, an that each ii) there exists c > such that ψs) c + s 2 ), for s R. Uner these assumptions, it follows that the operator A can be written as the subgraient of the convex function ϕ : V R given by { ϕũ) = Ω 2 k u) u + K Ω x S x + h u ) } γu s x. S x S x ψ g We state this as the following. 2 y U 2 + ψ G h u Proposition 2.. The subgraient of ϕ, ϕ : V V, is given by where A is given as above. ϕũ), θ = Aũ θ), The proof follows by stanar methos of convex analysis [4,9]. ) ) U y

10 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER 3. The Abstract Cauchy Problem. Let V be a separable reflexive Banach space, ense an continuously embee in a Hilbert space H. Let B be a continuous linear operator from V to V ; we will assume that B is positive an self-ajoint. Define the semi-norme space W b to be the completion of V with respect to the seminorm inuce by the semiscalar prouct u, v) Wb = Buv). Then the ual space W b is a Hilbert space, an B is a strict homomorphism from W b into W b. Let ϕ : V [, ] be a proper, convex, an lower semi-continuous function. We consier the egenerate Cauchy problem 3.) But)) + ϕut)) ft) for a.e. t [, T ] t Bu) = Bu. A solution of 3.) is a function u C[, T ], W b ) such that u is absolutely continuous on [δ, T ] for all δ >, an 3.) hols in W b for almost every t [, T ]. We will show that 3.) is equivalent to an evolution equation in H in the explicit form wt) + Cwt)) gt) t in which the operator C is the subgraient of a convex function, i.e., C = Φ. Stanar results on maximal monotone operators in Hilbert space will then apply irectly to yiel existence an uniqueness results. We will use the square root of the operator B []. Define the Hilbert space V b to be the completion of V with respect to the scalar prouct u, v) Vb = u, v) H + Buv). Since u Wb u Vb for all u V, we have V b W b, an V b is ense in W b. Also, the space V is ense an continuously embee in V b, which is ense an continuously embee in H. Therefore, by extension, we can regar B as a continuous linear operator from V b to V b. Since the bilinear form u, v) Wb = Buv) is ensely efine, close, an symmetric on V b, there exists a positive, self-ajoint, close

11 SEMILINEAR DEGENERATE PARABOLIC SYSTEMS linear operator B : omb) V b H such that Buv) = Bu, v) H for u omb), v V b. Thus, B is obtaine from B by restricting the range to H V b ; the omain of B is ense in V b. Next one constructs the positive, self-ajoint, continuous, linear operator B /2 : omb /2 ) = V b H such that B /2 B /2 = B, an B /2 u, B /2 v) H = Buv) = u, v) Wb for u, v V b. It is the operator B /2 which will be of primary interest to us. Since B /2 LV b, H), we can efine the Banach space ajoint, B /2 LH, V b ), by B /2 wv) = w, B /2 v) H. It follows that B /2 B /2 = B on V b. For v V b, B /2 v H = v Wb, so B /2 is continuous on V b with the W b seminorm. Thus B /2 has a unique continuous extension which we also enote by B /2 ) to a strict homomorphism from W b to H, an so the ajoint B /2 : H W b is continuous an onto. Since V b is ense in W b, the ientity B /2 B /2 = B also extens to W b. These properties of B /2 an B /2 permit us to reformulate 3.) in an explicit form. Proposition 3.. Let g B /2 f. Then the equation 3.2) t wt)) + B /2 ϕb /2 wt)) gt) for a.e. t [, T ] is equivalent to 3.) in the following sense: If w C[, T ]; H) is a solution to 3.2), then there exists u B /2 w such that u C[, T ]; W b ) is a solution of 3.). If u C[, T ]; W b ) is a solution to 3.), set w = B /2 u. Then w C[, T ]; H), is a solution to 3.2).

12 2 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER Next we show that the explicit operator B /2 ϕb /2 is the subgraient of a convex function. Recall from [9, p.7] that the polar function of ϕ, ϕ : V, ], efine by ϕ f) = sup{fv) ϕv)}, v V is proper, convex, an lower semi-continuous; the relationship ϕ = ϕ) hols in the sense of multifunctions; an ϕ ) = ϕ. We will use the chain rule for subgraients, [9, p.27], on the composition ϕ B /2. The following coercivity conition on the function ϕ will guarantee the require continuity: there exist constants c > an k > such that if v V with v V k, then ϕv) c v V. Lemma 3.. If ϕ satisfies the coercivity conition, then ϕ is continuous at some point of the range of B /2. Proof. We will show that ϕ is continuous at. boune on the neighborhoo N = {f V : f V It suffices to show that ϕ is c}, where c is the constant from the coercivity estimate. Define V = {v V : v V < k}, an V 2 = {v V : v V k}, an let f be an element of N. Since ϕv), sup {fv) ϕv)} sup f V v V ) k f V c k. v V v V Using the coercivity conition, sup {fv) ϕv)} sup { f V v V c v V }. v V 2 v V 2 Hence ϕ f) c k on the neighborhoo N. Proposition 3.2. The function Φ ϕ B /2 ) : H [, ] is proper, convex, an lower semi-continuous with Φ = B /2 ϕb /2. Proof. Consier the composite function ϕ B /2 : H, ]. Using Lemma 3., an application of the chain rule yiels ϕ B /2 ) = B /2 ϕ B /2.

13 SEMILINEAR DEGENERATE PARABOLIC SYSTEMS 3 But the subgraient of the polar is given by ϕ B /2 ) ) = ϕ B /2 )) = B /2 ϕ ) B /2 = B /2 ϕb /2. Finally, note that Φ is proper, convex an lower semi-continuous, an rgφ) rgϕ ) ) = rgϕ) [, ]. Thus we can write 3.2) in the form 3.3) t wt) + Φwt)) B /2 ft) in H. We now relate the initial conitions for 3.) to those appropriate for 3.3). Lemma 3.2. B /2 omϕ) W b) omφ) H. Proof. We first show that B /2 omϕ)) omφ). Suppose that h = B /2 u for some u omϕ). Then for every g H, ϕu) = ϕ u) = sup f V {fu) ϕ f)} B /2 gu) ϕ B /2 g), an so ϕ B /2 ) h) = ϕ B /2 ) B /2 u) = sup{b /2 u, g) H ϕ B /2 )g)} g H sup{b /2 u, g) H + ϕu) B /2 gu)} g H = ϕu). Since u omϕ), we have h omϕ B /2 ) ). Thus B /2 omϕ)) H omφ) H. Next, we show that B /2 om ϕ W b) = B /2 omϕ)) H ; this will complete the proof. Let {u n } be a sequence in omϕ) such that u n u in W b, an suppose that h = B /2 u, i.e., h B /2 om ϕ W b). Set h n = B /2 u n. Then, since h n h H = B /2 u n u) H = u n u Wb,

14 4 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER h n h in H, an so h B /2 omϕ)) H. The converse follows similarly. Finally, we note that if f L 2, T ; W b ), then there exists g L 2, T ; H) such that g B /2 f. We have thus complete all the steps necessary to reuce the Cauchy problem 3.) to the form 3.4) w t) + Φwt) gt) in H for a.e. t [, T ] w) = w, for which there is a complete theory [4, p.3]. Theorem 3. Existence). Let u omϕ) W b, f L 2, T ; W b ), an B an ϕ be given as above. Then there exists a solution u C[, T ]; W b ) to 3.5) t But)) + ϕut)) ft) in W b, for a.e. t [, T ] Bu) = Bu, an ut) om ϕ) for a.e. t [, T ]. Proof. By Lemma 3.2, if u omϕ) W b then w B /2 u B /2 omϕ) W b) omφ) H. Also, there exists g L 2, T ; H) such that g B /2 f). Choosing Φ = ϕ B /2 ) an g as above, we obtain the existence of a unique solution w C[, T ]; H) of 3.4). Proposition 3. an Proposition 3.2 show that there exists a function u C[, T ]; W b ) satisfying 3.5) an we have Bu) = B /2 B /2 u) = B /2 w) = B /2 w = B /2 B /2 u = Bu. Even though the solution to the explicit equation is unique, the choice of ut) as an element of the set B /2 wt) coul introuce nonuniqueness. To insure uniqueness, we impose an aitional conition.

15 SEMILINEAR DEGENERATE PARABOLIC SYSTEMS 5 Theorem 3.2 Uniqueness). In the situation of Theorem 3., if B+ ϕ is strictly monotone, then the solution ut) is unique. Thus we have sufficient conitions for existence an uniqueness for the Cauchy problem 3.5). Continuous epenence on the ata u an f can be shown using stanar methos. Also see [25]. In the next section, we will nee the solution u to be slightly more regular. In anticipation of this, we have the following result. Theorem 3.3. If, in aition to the previous hypotheses, u omϕ), then the solution u satisfies for t [, T ]. t 2 s Bus))us)s = ut) 2 W b u) 2 W b Proof. From the proof of Lemma 3.2, we have B /2 omϕ)) omφ). Thus w = B /2 u omφ), an so we have w t is absolutely continuous on [, T ] an L2, T ; H). It follows that wt) 2 H t wt) 2 H) = t B/2 ut) 2 H) = 2 But)), ut). t W b,w b Note that the solution of 3.5) satisfies ut) om ϕ) V an the equation hols in V b V at a.e. t, T ], even though u) is given in omϕ) W b. This is the parabolic regularizing effect. 4. Limiting Cases. Suppose that ϕ = ϕ + ϕ, with ϕ an ϕ proper, convex, an continuous from V into [, ], an set ϕ ɛ = ϕ + ɛ ϕ with < ɛ. In this section we will consier the limiting problem obtaine by replacing ϕ with ϕ ɛ in 3.) an letting ɛ. We assume that V = {v V : ϕ v) = } is a linear subspace of V. Let V = L 2, T ; V ), H = L 2, T ; H), an V = L 2, T ; V ); let H be the closure in H of V an H = L 2, T ; H ). Similarly, let W be the closure in W b of V an

16 6 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER W = L 2, T ; W ). In aition, we assume that ϕ an hence ϕ ɛ ) is V -coercive; this implies that ϕ ɛ satisfies the coercivity conition of Section 3. We also assume that B + ϕ an hence B + ϕ ɛ ) is strictly monotone. Let f L 2, T ; W b ) an u ɛ omϕ ɛ ) for each ɛ, ). From Theorem 3. an Theorem 3.2, there exists a unique solution u ɛ C[, T ]; W b ) of 4.) Similarly, let f = f W t Bu ɛ) + ϕ ɛ u ɛ ) f in W b for a.e. t [, T ] Bu ɛ ) = Bu ɛ. an let u omϕ ). Then another application of Theorem 3. an Theorem 3.2 gives a unique solution u C[, T ]; W ) of 4.2) t Bu ) + ϕ u ) f in W for a.e. t [, T ] Bu ) = Bu. Our goal is to show that, with appropriate hypotheses on the initial conitions, u ɛ u in V. Attaining this will require several lemmas. Lemma 4.. If {u ɛ} is boune in W b, then {u ɛ } is boune in V. Proof. Apply 4.) to u ɛ an integrate to obtain: t t ϕ ɛ u ɛ ) s + s Bu ɛ)u ɛ s Theorem 3.3 yiels 2 t ϕ u ɛ ) s + 2 ɛ t f W b t t ϕ u ɛ ) s + u ɛ t) 2 W b f, u ɛ s. ) /2 u ɛ 2 W b s + u ɛ 2 W b. A Gronwall-type inequality then shows that u ɛ C[,T ];Wb ) is boune. Since ϕ ɛ is V -coercive, this implies that {u ɛ } is also boune in V. Thus there exists a subsequence, which we will also enote by {u ɛ }, such that u ɛ u for some u V. From the proof of Lemma 4. we see that T ɛ ϕ u ɛ ) t is boune, so Fatou s Lemma an the weak lower semi-continuity of ϕ imply that u V.

17 SEMILINEAR DEGENERATE PARABOLIC SYSTEMS 7 Lemma 4.2. If { ɛ ϕ u ɛ)} is boune, then { t Bu ɛ} is boune in W b. Proof. Let w ɛ t) B /2 u ɛ t) an Φ ɛ ϕ ɛ B /2 ), so that w ɛ + Φ ɛ w ɛ ) B /2 f in H, for a.e. t [, T ]. Taking the scalar prouct in H with w ɛ an integrating yiels w ɛ 2 H + Φ ɛ w ɛ T )) g H w ɛ H + Φ ɛ w ɛ )). Since inf v V {ϕ ɛ v)}, we also have inf w H {Φ ɛ w)}, so Φ ɛ w ɛ T )) is positive. Also, from the proof of Lemma 3.2, we see that Φ ɛ w ɛ )) ϕ ɛ u ɛ), which is boune since { ɛ ϕ u ɛ)} is boune, so we have w ɛ H boune. Since B /2 is an isomorphism from H to W b, the result follows. Thus we can choose a further subsequence of {u ɛ } such that t Bu ɛ t Bu in W b an Bu ɛt ) Bu T ) in W b. Lemma 4.3. Assume that Bu ɛ Bu in W b. Then the equation hols in W for a.e. t [, T ]. t Bu ) + ϕ u ) f Proof. For every v V an almost every t [, T ], 4.) yiels f t Bu ɛ), v u ɛ ϕ ɛ v) ϕ ɛ u ɛ ). Restricting this to v V, applying Theorem 3.3, an noting that ϕ ɛ ϕ gives 4.3) T f, v u ɛ t Bu ɛ), v t + 2 u ɛt ) 2 W b 2 u ɛ 2 W b T ϕ v) t T ϕ u ɛ ) t. Using weak lower semi-continuity, we have T ϕ u ) t lim inf ɛ T ϕ u ɛ ) t an Bu T ) 2 W b lim inf ɛ Bu ɛ T ) 2 W b. Since Bu ɛ Bu in W b an B is an

18 8 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER isomorphism from W b to W b, u ɛ u in W b. Taking the lim inf of 4.3) thus gives T f t Bu ), v u t for every v V. Thus we obtain T ϕ v) ϕ u ) t f t Bu ) ϕ u ) in V, hence, in W for a.e. t [, T ]. Theorem 4.. Let u ɛ an u be the generalize solutions to 4.) an 4.2), respectively. If { ɛ ϕ u ɛ)} is boune an Bu ɛ Bu in W b, then u ɛ u in V. Proof. We have shown in the previous lemmas that a subsequence of {u ɛ } converges weakly in V to u V satisfying t Bu ) + ϕ u ) f in W for a.e. t [, T ]. To show that u is a solution to 4.2), we assert that Bu ) = Bu. Using the continuous embeing of H, T ; H) into C[, T ], H), we have the estimate B /2 u ɛ t) H c B /2 u ɛ H + t ) B/2 u ɛ ) H for a.e. t [, T ]. This implies that the map from H, T ; H) to H which takes B /2 u ɛ, t B/2 u ɛ )) to B /2 u ɛ ) is strongly continuous; it is also linear an hence weakly continuous. The same reasoning shows that the operator B /2 : V H is weakly continuous as well, so B /2 u ɛ B /2 u in H. Lemma 4.2 shows that the erivatives also converge weakly in H, an so we have B /2 u ɛ ) B /2 u ). It follows that Bu ɛ ) Bu ) since B /2 is weakly continuous. But Bu ɛ ) = Bu ɛ Bu, an so, since weak limits are unique, Bu ) = Bu. Thus u is a generalize solution of 4.2), an, since the solution is unique, it follows that u = u an the original sequence satisfies u ɛ u.

19 5. Examples. SEMILINEAR DEGENERATE PARABOLIC SYSTEMS 9 We will apply the preceeing results to the istribute capacitance moel to obtain existence an uniqueness of a generalize solution an characterize three limiting problems. As in Section 2, efine the spaces V = H Ω) L 2 Ω, H S x )) an H = L 2 Ω) L 2 Ω, L 2 S x )). We have shown that, with B : V V given by Bũ θ) = an ϕ : V [, ) efine by ϕũ) = Ω + S x Ω Cx, y)uθ y x, ũ = [u, U], θ = [θ, Θ], Ω x S x { 2 k u) u + K Ω x S x 2 y U 2 + ψ G h u ) } γu s x, ũ = [u, U], S x ψ g h u )) U y the istribute capacitance moel can be written in the abstract form 5.) t Bũ) + ϕũ) f in V Bũ) = Bũ. The generalize solution that we obtain for this equation will be in the sense of that efine in Section 3. Notice that the space W b is a set of pairs of functions [u, U] whose secon component U satisfies C 2 U L 2 Ω, L 2 S x )), an W b is the set of pairs of functionals of the form [, C 2 f] with f L 2 Ω, L 2 S x )). A precise statement of our notion of solution is foun in the existence theorem below. We will use two versions of Poincaré s inequality. Lemma 5.. Assume that G is an open set in R n with sup{ x : x, x 2,..., x n ) G} = D <. If v H G), then G v 2 x 4D 2 G v 2 x. Lemma 5.2. Assume in aition that G lies locally on one sie of its bounary, G, an that G is a smooth surface in R n. If v H G), then G v 2 x 2D γvs) 2 s + 4D 2 v 2 x. G G

20 2 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER Recall that each S x satisfies the hypotheses for Lemma 5.2), an furthermore these sets are uniformly boune, say S x for all x Ω. Also, we have the estimate S x S x S x 2 π, since the bounary of minimum length enclosing a given area is a circle. Theorem 5. Existence for the Distribute Capacitance Moel). Let the measurable set Q in Ω R 2 an the corresponing sets S x, x Ω, be given as in Section 2. Let the positive efinite matrix k, the constant K >, the non-negative function C L Q), an the convex functions ψ G an ψ g be given as in Section 2. Suppose that there exists a number a > for which either ψ G s) as 2 for all s R or ψ g s) as 2 for all s R. Let T > an assume the measurable functions F : Q, T ) R an U : Q R are given with C 2 F L 2, T ), L 2 Q)) an C /2 U L 2 Q). Then there exist measurable functions u : Ω, T ) R an U : Q, T ) R for which ut) H Ω) an Ut) L 2 Ω, H S x )) for a.e. t, T ), C /2 U C[, T ], L 2 Q)) an is locally absolutely continuous, 2.9) hols at a.e. t, T ) for every θ H Ω), Θ L 2 Ω, H S x )), an lim t C /2 Ut) = C /2 U in L 2 Q). Proof. This is a irect application of Theorem 3.; we only nee to show that ϕ satisfies the coercivity conition of Section 3.3. We write ϕ in four positive parts as ϕ = ϕ + ϕ 2 + ϕ 3 + ϕ 4. Let k be the coercivity constant for the matrix k. That is, k u) u k u 2 for all u H Ω). From Lemma 5., we have 2 ϕ u) c u 2 H Ω).

21 SEMILINEAR DEGENERATE PARABOLIC SYSTEMS 2 Assume now that ψ G s) as 2 for all s R. Then 2 ϕ ũ) + ϕ 3 ũ) { 2 k u + a h u ) ) } 2 2h u U + U 2 y x Ω 4 Ω x S x { 2 k u + a h u ) 2 h ɛ u ) ) } 2 h Ω 4 Ω x S x ɛ 2 U 2 + U 2 y x { ) k u 2 = + ha aɛ2 + a hɛ ) } Ω 4 Ω x S 2 U 2 y x x If ɛ is chosen so that h < ɛ < h + k 4a then this is boune below by c 2 Ω Ω x S x U 2 y x. Thus for each ũ = [u, U] in V we have { ) } K ϕũ) c u 2 H Ω) + Ω x 2 y U 2 + c 2 U 2 y x c ũ 2 V. Ω S x If instea we assume that ψ g s) as 2 for all s R, then, as above, we have by another appropriate choice of ɛ 2 ϕ ũ) + ϕ 4 ũ) c 2 Ω Ω x γu))2 s x. Using Lemma 5.2 an the fact that S x for all x Ω, we obtain { ϕũ) c u 2 H Ω) + K Ω Ω x S x 2 y U 2 y + c } 2 γu)) 2 s x S x S x c ũ 2 V. Note that the solution is smooth enough for the equation to hol in W b whereas only the minimal requirements are aske of the initial function U. In orer to insure that the solution is unique, it is sufficient for B + ϕ : V V to be strictly monotone. Theorem 5.2 Uniqueness for the Distribute Capacitance Moel). If, in aition to the hypotheses of Theorem 5., either ψ G or ψ g is strictly monotone, or if S x Cx, y) y > for a.e. x Ω, then the solution to 5.) is unique.

22 22 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER Proof. Assume that ũ an w are elements of V, an that Bũ w)ũ w) + ϕũ ϕ w, ũ w =. Since this expression is a sum of positive terms, we have 5.2) ϕ j ũ ϕ j w, ũ w = for each j {, 2, 3, 4}. When j =, this gives Ω u w) 2 x =, which implies that u = w in H Ω). Also the case j = 2 implies that Ux, y) W x, y) = U W )x), i.e., this ifference oes not epen on y. If ψ G is strictly monotone, then 5.2) with j = 3 implies that U = W an thus ũ = w in V. The same follows from Bũ w)ũ w) = if instea we assume the above conition on Cx, y). Alternatively, assume that ψ g is strictly monotone. Then 5.2) with j = 4 shows that γu) = γw ), an hence U = W, so in this case also ũ = w in V. Notice that we coul have equivalently assume that either ψ G or ψ g is strictly convex, since a convex function ψ is strictly convex if an only if its subgraient ψ is strictly monotone. Finally, apply the results of Section 4 to the istribute capacitance moel. Specifically, we will multiply ψ g, ψ G, or K by ɛ an use Theorem 4. to characterize each of the three corresponing limiting problems. For the first case set ϕ ũ) = Ω ψ g h u ) γu s x, S x S x an let ϕ = ϕ ϕ. We originally assume that ψ g ) = ; here we will also nee to assume that ψ g s) = only if s =. This is guarantee if ψ g s) as 2 as in Theorem 5..) In this case, 4.) is the weak form of the istribute capacitance moel with ψ g replace by ɛ ψ g, an Theorem 4. shows that, with appropriate initial conitions, its solution converges to that of 4.2). Using the above assumption on ψ g, we see that V kerϕ ) = { ũ V : h u } = γu in L 2 Ω, L 2 S x )).

23 SEMILINEAR DEGENERATE PARABOLIC SYSTEMS 23 From calculations similar to those in Section 2.4, we see that 4.2) is a weak form of the system t Cx, y)u) K yu ψ G h u ) U F, h u = γu, s S x k u) ψ G h u ) U) y, S x S x ux, t) =, x Ω, t [, T ]. y S x x Ω That is, 2.2) is replace by the Dirichlet conition above. This limiting problem is the matche moel in which the istribute voltage ifferences on the capacitor bounaries, γu, are in perfect contact with the global voltage graient, h u. Next we consier the case where ψ G is replace by ɛ ψ G; we set ϕ ũ) = Ω ψ G h u ) U y s. Ω x S x As above, we assume aitionally that ψ G s) = only if s =. Then V kerϕ ) = an we fin that 4.2) is a generalize form of k u + e 3 S x S x { ũ V : h u } U = in L 2 Ω, L 2 S x )) Cx, y) h u )) ) y = ) F x, y) y t S x S x This is a egenerate form of the layere meium equation.), an Theorem 4. shows that the solution of 4.) converges to its solution as ɛ. Finally, we consier the case where K. Define Then we have ϕ ũ) = Ω Ω x S x K 2 yu 2 y x.,. V = {ũ V : Ux, y) = vx) for some vx) L 2 Ω)}V = H Ω) L 2 Ω).

24 24 BROOKE L. HOLLINGSWORTH AND R. E. SHOWALTER As above, we fin that 4.2) is the weak formulation of the system = k u) ψ G h u ) ) v + s h ψ g h u ))) v, S x Cx, y) y )v ψ G h u ) v t S x S x x e ) h s ψ g h u )) v = F y. S x S x S x S x Again, Theorem 4. guarantees that the solution of 4.) converges to the solution of this system as ɛ i.e., as K ). In summary, we have escribe a PDE moel of current flow in a meium with continuously istribute capacitance an nonlinear connections between the local capacitors an the global conucting meium. Clearly one coul obtain corresponing results for more general situations. For example, one coul permit monotone nonlinearities in the conuctance in both the capacitors an in the meium an resolve as above the corresponing quasilinear system. Aitionally, one coul supplement 2.) an 2.3) with terms representing losses ue to leakage of current between capacitor plates or connections, an one coul permit the convex functions to be more general, specifically, to inclue unilateral constraints such as arise in ioe nonlinearities. Our techniques apply to such moels after some technical work to bring them to the form of the semilinear egenerate evolution.2). Finally, our main result, Theorem 3., shows that.2) is really parabolic when the operator A is a subgraient. The improvement over earlier work is to show that a very strong solution is obtaine when one begins the evolution with very general ata. References. R.A. Aams, Sobolev Spaces, Acaemic Press, New York, C. Baros an H. Brezis, Sur une classe e problemes evolution nonlineaires, J. Differential Equations ), M.-P. Bosse an R.E. Showalter, Homogenization of the layere meium equation, Appl. Anal ), H. Brezis, Monotonicity methos in Hilbert spaces an applications to nonlinear partial ifferential equations, Contributions to Nonlinear Functional Analysis E.H. Zarantonello, e.), Acaemic Press, New York, 97, pp. 56.

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