The thermistor obstacle problem with periodic data

The Obstacle Thermistor Problem with Periodic Data

MAURIZIOBADII(*)

ABSTRACT- The object of this paper is the study of an obstacle thermistor problem with a nonlocal term. Using the classical Lax-Milgram theorem, a result of Lions and a fixed point argument we prove existence of weak periodic solutions. Fi- nally, by means of some a-priori estimates in Campanato's spaces, we obtain the regularity of these solutions.

1. Introduction.

Let V be a bounded open set ofRⁿ,nF1, with smooth boundary@V.

For givenv>0, we setQ:ˆVP,S:ˆ@VPandP:ˆ R

vZdenotes the period interval [0;v] so the functions defined inQandS arev-time pe- riodic. In this paper we consider the following parabolic-elliptic system arising from an obstacle evolution thermistor problem

u_t 4u‡ Z

V

G(x;y)u(y;t)dyFs(u)jrWj² inQ;

…1:1†

u(x;t)ˆ0 onS;

…1:2†

div(s(u)rW)ˆ0 inQ;

…1:3†

W(x;t)ˆW₀(x;t) onS:

…1:4†

HereVis the conductor and the unknownsuandWare the temperature and the electrical potential inR V, respectively. The nonlocal term

V G(x;y)u(y;t)dy, withG(x;y)F0, describes heat losses to the surrounding (*) Indirizzo dell'A.: Dipartimento di Matematica ``G. Castelnuovo'', UniversitaÁ di Roma ``La Sapienza'', P. A. Moro 2, 00185 Roma, Italy.

E-mail: badii@mat.uniroma1.it

gas whiles(u) denotes the temperature dependent electrical conductivity.

Finally, the terms(u)jrWj²represents the Joule heating inV. We note that ifW2L²(P;W^1;2(V))\L¹(Q), then (1.3) implies

s(u)jrWj²ˆdiv(s(u)WrW);

in the sense of distribution.

Thus, we shall study the problem u_t 4u‡

Z

V

G(x;y)u(y;t)dyFdiv(s(u)WrW) inQ;

…1:5†

u(x;t)ˆ0 onS …1:6†

div(s(u)rW)ˆ0 inQ;

…1:7†

W(x;t)ˆW₀(x;t); onS:

…1:8†

To solve (1.5)-(1.8) we follow the approach of [1], [2] and introduce the penalized problem

u_nt 4u_n‡ Z

V

G(x;y)u_n(y;t)dy

!

I_n(u_n)ˆdiv(s(u_n)W_nrW_n) inQ;

…1:9†

u_n(x;t)ˆ0 onS;

…1:10†

div(s(u_n)rW_n)ˆ0 inQ;

…1:11†

W_n(x;t)ˆW₀(x;t) on S;

…1:12†

where it is assumed thatI_nsatisfies

Hn) In2C(R), 0GIn(s)G1, for all s2R , In(s)ˆ0 if sG0 and I_n(s)!H(s) inL², whereH(s) denotes the Heaviside function.

The plan of the paper is as follows: in Section 2 we give some notations and preliminary results. In Section 3, we consider a linearized version of the elliptic problem (1.7)-(1.8) (see (3.5)-(3.6)) and prove existence and uniqueness of the weak periodic solutionWby means of the classical Lax- Milgram's theorem. Next, in Section 4, we use a result due to Lions [5, Theorem 6.1] to obtain existence and uniqueness of periodic solutionsu_n for the penalized problem (1.9)-(1.10). In Section 5, we deduce some uni- form estimates for (u_n;W_n) and utilize a fixed point argument to obtain the

existence of weak periodic solutions (u;W) of (1.5)-(1.8) by lettingn! 1.

Finally, in Section 6, we establish the HoÈlder regularity for (u;W). This will be done by proving a-priori estimates in Campanato's spaces as in [6], [7].

2. Preliminaries and auxiliary results.

Let us introduce an useful space of v-periodic functions in which we look for solutions to our problem.

We consider the Hilbert spaceV₀:ˆL²(P;W₀^1;2(V)) endowed with the norm

kvk_V0:ˆ Z

Q

jrv(x;t)j²dxdt

!₁₌₂ …2:1†

and its dualV:ˆL²(P;W ^1;2(V)). The duality pairing betweenV0andV shall be denoted byh:i. The spaceV0is the closure with respect to the norm (2.1) ofC¹₀ (Q) , the space of periodic functions vanishing nearS. We recall some notations concerning the Campanato spaces.

NOTATIONS. Let Qt0;t1:ˆV(t0,t1], 0Gt05t1. A point (x;t)2Qt0;t1

shall be denoted byz. LetBr(x0) be the ball centered atx0 with radiusr and let Qr(z0) be the cylinder Br(x0)(t0 r², t0]. Moreover, define Q_r[z₀,r]:ˆQ_r(z₀)\Q_t0;t1 and V[x₀,r]:ˆB_r(x₀)\V:

For 0Gm5n, the Campanato space L^2;m(V):ˆ

(

u2L²(V) : sup

x02V;r>0

r ^m Z

V[x0;r]

u²(x)dx

!₁₌₂ 5‡ 1

)

is a Banach space with the equivalent norm kuk_2;m;V[x0;r]ˆ sup

x02V;r>0

r ^m Z

V[x0;r]

u²(x)dx

!₁₌₂ : (see [3]).

For 0Gm5n‡2, the Campanato space L^2;m(Q_t0;t1):ˆ

(

u2L²(Q_t0;t1): sup

z02Qt0;t1;r>0 r ^m Z

Qr[z0;r]

u²(x;t)dxdt

!₁₌₂ 5‡1

) :

is a Banach space with the equivalent norm

kuk_2;m;Qt

0;t1 ˆ sup

z02Qt0;t1;r>0r ^m Z

Qr[z0;r]

u²(x;t)dxdt

!₁₌₂ :

Furthermore, we need the following result

PROPOSITIONA ([7]). The spaceL^2;n‡2‡2m(Q_t0;t1) is isomorphic topologi- cally and algebraically toC^m;m=2(Q_t0;t1), form2(0;1).

We shall study problem (1.5)-(1.8) under the following assumptions:

Hs)s2C(R), 05sGs(s)Gsfor anys2R;

H_G)G2C(R²),G(x;y)F0, we setGb :ˆsup_R2G(x;y);

H₀) W₀ is an v-periodic bounded function with an extension eW₀ to Q which satisfieseW₀2L¹(P;W^1;1(V)).

Moreover, we define the convex set

K:ˆ fv2L²(P;W₀^1;2(V));vF0 a:e:in Qg:

The notion of weak periodic solution for our problem is the fol- lowing

DEFINITION2.1. A pair of functions (u;W) are a weak periodic solution to (1.5)-(1.8) if the following conditions hold

u2K ; u_t 2L²(P;W ^1;2(V)) andW W₀2V₀;

…2:2†

Z

Q

ut(v u)dxdt‡ Z

Q

rur(v u)dxdt‡ Z

Q

Z

V

G(x;y)u(y;t)(v u)dydxdt

F Z

Q

s(u)WrWr(v u)dxdt; for anyv2K

and Z

Q

s(u)rWrjdxdtˆ0; for anyj2V₀:

3. The elliptic problem.

In this section we study problem (1.7)-(1.8). Set v(x;t):ˆ

:ˆW(x;t) W₀(x;t). Fixed w2 L²(Q) with wF0, we solve the problem div(s(w)rv)ˆ div(s(w)rW₀) inQ;

…3:1†

v(x;t)ˆ0 onS;

…3:2†

We state the weak formulation of solution to (3.1)-(3.2).

DEFINITION3.1. A function v2V₀ is called a weak periodic solution of (3.1)-(3.2) if

Z

Q

s(w)rvrjdxdtˆ Z

Q

s(w)rW₀rjdxdt; for allj2V₀: …3:3†

The existence and uniqueness of the weak solutionvshall be obtained by the classical Lax-Milgram theorem.

In agreement with this result, we define the bilinear form a:V0V0!R

by setting

a(v;j):ˆ

Z

Q

s(w)rvrjdxdt; for anyj2V0

We have

PROPOSITION3.2. If we assume H_s), then i) ais continuous;

ii) ais coercive.

PROOF. Condition i) is a consequence of HoÈlder's inequality. In fact

ja(v;j)jGs Z

Q

jrvj²dxdt

!₁₌₂ kjk_V0: Gskvk_V0kjk_V0:

ii) The coercivity ofafollows from a(v;v)ˆ

Z

Q

s(w)jrvj²dxdtFskvk²_V0

which implies

a(v;v)

kvk_V0Fskvk_V0! 1; askvk_V0! 1: p In this way if H₀) holds, problem (3.3) becomes equivalent to the problem

a(v;j)ˆ hG;ji …3:4†

whereG2V is the linear functional defined as follows hG;ji:ˆ

Z

Q

s(w)rW₀rjdxdt;8j2V0:

Now, we can state the main result of this section

THEOREM 3.3. Let H_s) and H₀) be satisfied. There exists a unique weak periodic solution to(3.4).

PROOF. By the theorem of Lax-Milgram we easily conclude the ex- istence and uniqueness of the weak periodic solution. p

Thus, forw2L²(Q),wF0 we have solved the problem div(s(w)rW)ˆ0 inQ;

…3:5†

W(x;t)ˆW₀(x;t) onS:

…3:6†

4. The parabolic penalized problem.

We consider the problem u_nt 4u_n‡

Z

V

G(x;y)w(y;t)dy

!

I_n(w)ˆdiv(s(w)WrW) inQ;

…4:1†

un(x;t)ˆ0 onS:

…4:2†

DEFINITION4.1. A weak periodic solution of (4.1)-(4.2) corresponding to w, is a functionun2V0such that

…4:3†

Z

Q

u_ntzdxdt‡ Z

Q

ru_nrzdxdt‡ Z

Q

Z

V

G(x;y)w(y;t)dy

!

I_n(w)zdxdt

ˆ Z

Q

s(w)WrWrzdxdt; for anyz2V₀:

The existence and uniqueness of the periodic solution follows from a Lion's result (see [5, Theorem 6.1]).

For anyu_n 2V₀ we define the mappingB:V₀!V as follows hBun;zi:ˆ

Z

Q

runrzdxdt; for any z2V0:

The set

D:ˆ fv2L²(P;W₀^1;2(V)): v_t2L²(P;W ^1;2(V))g is dense inV0, because of the density ofC¹(Q)DinV0.

Let

L:D!V

be the closed skew-adjoint (i.e.Lˆ L) linear operator defined by hLun;zi:ˆ

Z

Q

untzdxdt;8z2V0:

It is known that this operator is maximal monotone (see [4, Lemma 1.1, p. 318]).

The properties ofBare contained in the following result.

PROPOSITION4.2. Assume Hs), then j) Bis continuous;

jj) Bis coercive.

PROOF. j) The continuity ofBfollows from jhBu_n;zijGku_nk_V0kzk_V0:

jj) By

hBun;uni ˆ Z

Q

jrunj²dxdtˆ kunk²_V0

one has

hBu_n;u_ni

kunk_V0 ˆ ku_nk_V0! 1; asku_nk_V0! 1: p Finally, letM_n2Vbe the linear functional defined by

hMn;zi:ˆ

Z

Q

Z

V

G(x;y)w(y;t)dy

!

In(w)zdxdt Z

Q

s(w)WrWrzdxdt;

then, problem (4.3) can be considered in the framework of the abstract problems of the form

Lu_n‡Bu_nˆM_n …4:4†

to which we apply [5, Theorem 6.1] to establish the existence and unique- ness of the weak periodic solution.

5. Afixed point argument.

The periodicity of solutions to (1.5)-(1.8) shall be proved utilizing the Schauder fixed point theorem for a suitable operator equation. To this aim, let

U:S ! S be the mapping defined by

U(w)ˆun

and

S:ˆ fv2L²(Q):kvk_L2(Q)Mg;

where u_n is the unique weak periodic solution of (4.1)-(4.2) corre- sponding to w. The mapping U is well-defined. In order to prove its continuity we will prove some crucial estimates and convergences useful to utilize the Schauder fixed point theorem. Let w_k2L²(Q) be a non- negative sequence such thatw_k!wands(w_k)!s(w) strongly inL²(Q) as k! 1: We denote by u_nk and W_k, respectively, the weak periodic

solutions of …5:1†

Z

Q

unktzdxdt‡ Z

Q

runkrzdxdt‡ Z

Q

Z

V

G(x;y)wk(y;t)dy

!

In(wk)zdxdt

ˆ Z

Q

s(w_k)W_krW_krzdxdt; for any z2V₀

and Z

Q

s(wk)rW_krjdxdtˆ0; for anyj2V0: …5:2†

An estimate onrW_kis achieved choosingW_k W₀as a test function in (5.2).

In fact, s

Z

Q

jrW_kj²dxdtG Z

Q

s(w_k)jrW_kj²dxdtˆ Z

Q

s(w_k)rW_krW₀dxdt

Gs Z

Q

jrW_kj²dxdt

!₁₌₂ Z

Q

jrW₀j²dxdt

!₁₌₂

hence,

Z

Q

jrW_kj²dxdtG s s

2krW₀k²_L2(Q): …5:3†

Moreover, by the weak maximum principle we derive that kW_kk_L1(Q)GkeW₀k_L1(Q):

…5:4†

Combining (5.3) and (5.4), one has the usual energy estimate Z

Q

jW_k(x;t)j²dxdt‡ Z

Q

jrW_k(x;t)j²dxdtGC:

…5:5†

In what follows,Cstands for a generic positive constant independent ofk andn.

Thanks to (5.3), we deduce thatW_kis uniformly bounded in theVnorm, with respect tok. Therefore

W_k*WinV askgoes to infinity:

Furthermore, one has

LEMMA 5.1. The sequence rW_k converges strongly to rW in L²(P;(L²(V))ⁿ):

PROOF. TakingW_k Was a test function in (5.2), we get Z

Q

s(wk)jr(W_k W)j²dxdtˆ Z

Q

s(wk)r(W W_k)rWdxdt

from which it follows that s

Z

Q

jr(W_k W)j²dxdtG Z

Q

s(w_k)r(W W_k)rWdxdt:

The weak convergence of s(w_k)r(W W_k) to zero ask! 1 leads to the

conclusion. p

Choosingu_nkas a test function in (5.1), we have Z

Q

jru_nkj²dxdt‡ Z

Q

Z

V

G(x;y)w_k(y;t)dy

!

In(w_k)u_nk(x;t)dxdt

ˆ Z

Q

s(w_k)W_krW_kru_nkdxdt:

Applying Young's and PoincareÂ's inequalities, one obtains 1

2 Z

Q

jru_nkj²dxdt

Ge 2 Z

Q

ju_nkj²dxdt‡Gb² 2e Z

Q

Z

V

w_k(y;t)dy

!₂ dxdt

‡(skeW₀k_L1(Q))² 2

s s

2krW₀k²_L2(Q)

GeC 2

Z

Q

jru_nkj²dxdt‡(GjVj)b ² 2e

Z^v

0

Z

V

jw_k(y;t)j²dydt

‡(skeW₀k_L1(Q))² 2

s s

2krW₀k²_L2(Q)

GeC 2

Z

Q

jrunkj²dxdt‡(GjVj)b ² 2e [e‡

Z^v

0

Z

V

jw(y;t)j²dydt

‡(skeW₀k_L1(Q))² 2

s s

₂

krW₀k²_L2(Q):

Thus, Z

Q

jru_nk(x;t)j²dxdtGC

and we infer the classical energy estimate Z

Q

ju_nk(x;t)j²dxdt‡ Z

Q

jru_nk(x;t)j²dxdtGC:

…5:6†

By virtue of (5.5), (5.6) and (5.1),u_nktis uniformly bounded in theVnorm.

Thereforeu_nkbelongs to a bounded set ofDi.e.

ku_nkk_DGC:

Thus, we can select a subsequences, still denoted byu_nk, such that u_nk*u_n; inDask! 1:

A well known result of [4, Lemma 5.1] guarantees that the sequenceu_nkis precompact inL²(Q), then

unk!un in L²(Q) and a:e:inQ:

LEMMA5.2. The operatorUis continuous.

PROOF. From the preceding result, we have unk!un in L²(Q) and a:e:inQ:

u_nkt*u_ntin L²(P;W ^1;2(V)) ru_nk*ru_n inL²(P;(L²(V))ⁿ)

rW_k ! rWinL²(P;(L²(V))ⁿ) W_k!Win V and a:e:inQ

w_k!winL²(Q) s(w_k)!s(w) inL²(Q) In(wk)!In(w) inL²(Q):

These convergences enable us to conclude thatU(wk)ˆunkconverges

strongly toU(w)ˆun inL²(Q). p

LEMMA5.3. There exists a constantM>0 such that kU(w)k_L2(Q)GM; for allw2L²(Q):

PROOF. The assertion is obtained as above. p SinceU(L²(Q))Dand the embeddingD,!L²(Q) is compact,Uis a

compact operator fromL²(Q) into itself. p

Our main result, is given in the next statement.

THEOREM5.4. If H_s) - H₀) are fulfilled, there exists at least one weak periodic solution to (1.5)-(1.8).

PROOF. As a consequence of Lemmas 5.2 and 5.3 the mapping Uis both continuous and compact. Hence, by the Schauder fixed point theo- rem,Uhas a fixed points which corresponds to a weak periodic solutions

to (1.9)-(1.12). p

As far as previously proved, we can conclude that u_nt*u_t inL²(P;W ^1;2(V)) u_n!uinL²(Q) and a:e:in Q:

run*ruin L²(P;(L²(V))ⁿ) rW_n! rWin L²(P;(L²(V))ⁿ) W_n!W inV and a:e:inQ

wn!winL²(Q) s(wn)!s(w) inL²(Q)

I_n(u_n)!rweak-* inL¹(Q); with 0GrG1:

Theseunare nonnegative. In fact, forzˆu_n, we have …5:7†

Z

Q

untu_ndxdt‡ Z

Q

runru_ndxdt‡ Z

Q

Z

V

G(x;y)un(y;t)In(un)u_n(x;t)dydxdt

ˆ Z

Q

s(un)jrWnj²u_n dxdt;8z2V0

from which it follows that Z

Q

jrunj²dxdtˆ Z

Q

s(un)jrW_nj²u_n dxdt:

This implies the nonnegativity ofu_n. Sinceu_nF0, so isu:

Passing to the limit in …5:8†

Z

Q

untzdxdt‡ Z

Q

runrzdxdt‡ Z

Q

Z

V

G(x;y)un(y;t)In(un)z(x;t)dydxdt

ˆ Z

Q

s(un)W_nrW_nrzdxdt

we have …5:9†

Z

Q

u_tzdxdt‡ Z

Q

rurzdxdt‡ Z

Q

Z

V

G(x;y)u(y;t)r(x;t)z(x;t)dydxdt

ˆ Z

Q

s(u)WrWzdxdt;8 z2V0: We observe that r(x0;t0) ˆ1 if u(x0;t0)>0 thus r(x;t)(v u)G(v u) for all vF0:If we replace z withv uin (5.9) with v2K, then (2.2) is

satisfied. p

6. Regularity.

This part of paper is devoted to the regularity of weak periodic solu- tions (u;W) to problem (1.5)-(1.8). We will prove the HoÈlder continuity of (u;W) by means of some a priori estimates in the Campanato spaces.

LEMMA 6.1. Let W_n be the weak periodic solution of (1.11)-(1.12). If 0Gm₀5n 2‡2d₀,d₀2(0;1) one has

krW_n(t)k_2;m0;VGc(kW₀(t)k_2;m0;V‡ krW₀(t)k_{2;(m0 2)}^‡_;V);

…6:1†

for a.e:t2P:

PROOF. From a result of [6] we get

krW_n(t)k_2;m0;VGc(kW₀(t)k_2;m0;V‡ krW₀(t)k_{2;(m0 2)}^‡_;V‡ kW_n(t)k_H1(V))

a.e. t. SinceW₀(t),rW₀(t)2L¹(V) for a.e: tand the fact that L¹(V) is a multiplier forL^2;m1(V) form₁5n, by the inclusionL^{2;(m0 2)‡}(V),!L²(V) and (5.3) we infer that

krW_n(t)k_L2(V)G s

s krW₀(t)k_{2;(m0 2)}^‡_;V for a.e.t, which implies

kW_n(t)k_H1(V)GckW₀(t)k_2;m0;V‡ s

s krW₀(t)k_{2;(m0 2)}^‡_;V:

This completes the proof of Lemma 6.1. p

PROPOSITION 6.2. The weak periodic solutions (u;W) of (1.5)-(1.8) belong to the spaceC^a;a=2(Q).

PROOF. From (6.1), one has

krW_n(t)k_2;m0;VGC

for a.e.t. The definition of Campanato's spaces gives us r ^(m0‡2)

Z

Q[z0;r]

jrW_n(x;t)j²dxdtGC …6:2†

for allz₀2Qandr>0:

Being s(u_n)W_n 2L¹(Q) and L¹(Q) a multiplier of L^2;m(Q) if 0GmGn‡2d₀, then s(u_n)W_nrW_n2 L^2;m(Q) and the inequality (6.2) im- plies that

krW_nk²_2;mGC:

By the comparison principle, we have that the nonnegative weak pe- riodic solutioncof problem

c_t 4cˆdiv(s(u_n)W_nrW_n) inQ;

…6:3†

cˆ0 onS;

is a supersolution for (5.2), so that

0Gu_nGc:

…6:4†

According to a result in [7, Theorem 1], we get krck_2;mGC

for 05mGn‡2aandcbelongs toL^2;m‡2(Q) (see [7, Lemma 2.6]). Because of Proposition A, one has thatc2C^a;a=2(Q) and consequently

kck_L1(Q)GC:

The boundedness ofcimplies thatun2L¹(Q):Theorem 1 in [7] allows to conclude that

…6:5† kru_nk_2;mGc(ks(u_n)W_nrW_nk_2;m

‡kI_n(u_n) Z

V

G(x;y)u_n(y;t)dy

!

k_{2;(m 2)}^‡‡ ku_nk_V)

and the embedding ofL¹(Q),!L^2;m(Q) yields Z

V

I_n(u_n)(G(x;y)u_n(y;t)dy)

2;(m 2)^‡Gc Z

V

G(x;y)u_n(y;t)dy

L¹(Q)GC:

Thus, if 05mGn‡2aby (6.5), we obtain kru_nk_2;mGC

andu_n2 L^2;m‡2(Q), (see [7, Lemma 2.6]). Since we showed thatu_n !uand W_n !Wa.e. inQwithu_n,W_ninL¹(Q), we conclude thatu,W2 L^2;m(Q) for mˆn‡2‡2a. Finally, by Proposition Au,W2C^a;a=2(Q) foraˆm n

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Manoscritto pervenuto in redazione il 27 settembre 2007.