On certain doubly non-uniformly and singular non-uniformly elliptic systems
Abstract
We consider the steady state of the thermistor problem consisting of a coupled set of nonlinear elliptic equations governing the temperature and the electric potential. We study the existence of weak solutions under two kind of assumptions. The first one considers the case in which the two diffusion coefficients are not bounded below far from zero, arising to a doubly non-uniformly elliptic system. In the second one, we assume in addition that the thermal conductivity blows up for a finite value of the temperature, arising to a singular and non-uniformly coupled system.
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On certain doubly non-uniformly and singular non-uniformly elliptic systems M.T. Gonz)alez Montesinosa, F. Orteg)on Gallegob aDepartamento de Matematicas, Facultad de Ciencias Economicas y Empresariales, Universidad de Cadiz, 11002 Cadiz, Spain b CASEM, Departamento de Matematicas, Universidad de Cadiz, 11510 P uerto Real, Spain Abstract We consider the steady state of the thermistor problem consisting of a coupled set of nonlinear elliptic equations governing the temperature and the electric potential. We study the existence of weak solutions under two kind of assumptions. The 4rst one considers the case in which the two di5usion coe6cients are not b ounded b elow far from zero, arising to a doub ly non-uniformly elliptic system. In the second one, we assume in addition that the thermal conductivity blows up for a 4nite value of the temperature, arising to a singular and non-uniformly coupled system. Keywords: Non-uniformly and singular elliptic systems; Nonlinear elliptic equations; Thermistor problem; Sobolev spaces 1. Introduction The heat produced b y an electrical current passing through a conductor device is governed b y the so-called thermistor prob lem. This prob lem consists of a system of nonlinear parab olic–elliptic system describ ing the temperature, u, and the electric potential ’ [2,8]. Here, we consider the steady-state case, resulting in a coupled non-linear elliptic system. Let J b e the current density, Q the heat >ux and E = −∇’ the electric 4eld; then b y Ohm’s and Fourier’s law we have J = (u)E; Q = −a(u)∇u;
where a(u) and (u) are, respectively, the thermal and electric conductivities. Also, from the usual conservation laws ∇· J =0, ∇· Q = E · J we obtain −∇ · (a(u)∇u)=(u)|∇’|2in ; ∇·((u)∇’)=0 in ; u=0 on @; ’=’0on @; (1) where is an open, bounded and smooth enough set in RN,N¿1. We note that, the right-hand side of the equation for the temperature may be written as ∇·((u)’∇’) thanks to the equation veri4ed by ’; this is true, for instance, if ’∈H1() (this aspect is also discussed in Remark 2). The steady-state thermistor problem has been studied by several authors along the last two decades. Among them, we refer to Cimatti [4–7] and Cimatti and Prodi [9]. In these papers, the authors have obtained some existence results of weak solutions in both, two and three dimensions, using the so-called Diesselhorst transformation, and under the conditions u=u0on @, and u0being a constant value, or u=u0¿um¿0 on @, together with the hypothesis 0 ¡a m6a(u), or a(u)=a0constant, or even under the Wiedemann–Franz law (that is, a(s)=Ls(s), L¿0 a constant value) with metallic conduction, and certain assumptions on (u). We notice that in all these papers is assumed that a(s)¿a0¿0, for all s. In the present work we show two existence results of weak solution to the steady-state thermistor problem under the general assumption that both a(s) and (s) are not bounded below far from zero (Theorems 1and 2). In this way, system (1) becomes doubly non-uniformly elliptic; in general, we cannot expect the regularity ’∈H1()∩ L∞(), or that ubelongs to some Sobolev space. We point out that the technique used here in the analysis of the non-singular case is not based on the derivation of L∞-estimates for the temperature. On the other hand, our setting for the singular and non-uniformly elliptic problem leads us to derive L∞-estimates for the temperature, and this implies more regularity for both, the electric potential ’and the temperature uitself. 2. Setting of the non-uniformly elliptic problem We consider the steady-state thermistor problem in divergence form, namely −∇ · (a(u)∇u)=∇·((u)’∇’)in; ∇·((u)∇’)=0 in ; u=0 on @; ’=’0on @; (2)
together with the following hypotheses on data: (H.1) ∈C(R) and 0 ¡(s)6J, for all s∈R. (H.2) a∈C(R)∩L∞(R), +∞ 0a(s)ds=+∞, and A(r)=r 0a(s)dsis a strictly increasing function. (H.3) ’0∈H1(). (H.4) There exist an integer M¿1 and a function :[M; +∞)→Rsuch that (s)¿0, for all s¿M,is non-increasing and (s)¿(s)¿0. (H.5) Let p∈(2N=(N+2);2) if N¿2, p∈(1;2) if N= 1 and p=2−p, then +∞ M ds (s)p=pA(s−1) Jq=2¡+∞with Jq=2 ∗if N¿3; Jq∈[2;+∞)ifN=2; Jq∈[1;+∞)ifN=1: (3) Remark 1. If we consider that and ’0are smooth enough, then ’will be smooth too, for instance |∇’|∈L∞(), and then we could obtain that u∈L∞(); in particular, this implies (u)¿(uL∞()), that is, (u) becomes uniformly elliptic; thus, we have again that and ’0are smooth, starting over the cycle. However, this does not correspond with our setting, since both aand are non-uniformly elliptic. The main result of this section now follows. Theorem 1. Under assumptions (H.1)–(H.5), problem −LA(u)=∇·((u)’∇’)inD(); ∇·((u)∇’)=0 in ; u=0 on @; ’=’0on @; (4) has a weak solution (u; ’)in the following sense: ∀q¡ N N−1if N ¿2;q=2 if N =1;A(u)∈W1;q 0();(5) ’−’0∈W1;p 0();(u)1=2∇’∈L2()N;(6) ∇A(u)∇=− (u)’∇’∇for all ∈D();(7) (u)∇’∇=0 for all ∈H1 0():(8) Furthermore,the term ∇·((u)’∇’)is a Radon measure and u¿0almost everywhere in . The proof of this theorem is developed along the next paragraphs.
2.1. Approximate problems Let n ∈ N and introduce the functions an(s)= a(s)+1=n, n(s)= (s)+1=n, then we set the approximate prob lem given as follows: −∇ · (an(un)∇un)=n(un)|∇’n|2in ; ∇·(n(un)∇’n)=0 in ; un=0 on @; ’n=Tn(’0)on@; (9) where Tn(s) is the truncation function at height n, that is Tn(s) = min(|s|;n) sign s: (10) By virtue of the classical existence results [2], problem (9) has a solution such that un∈H1 0(), ’n−Tn(’0)∈H1 0()∩L∞(). 2.2. Estimates and passing to the limit Since n(un)∇’n∇= 0 for all ∈H1 0();(11) taking =’n−Tn(’0) yields n(un)|∇’n|2= n(un)∇’n∇Tn(’0) 6 n(un)|∇’n|21=2 n(un)|∇’0|21=2 ; hence n(un)|∇’n|26˜ |∇’0|26˜’0H1()=C(˜; ’0)=C1;(12) therefore, (fn)=(n(un)|∇’n|2) is bounded in L1(). Let vn=An(un), An(r)=r 0an(s)dsand consider the elliptic problem −Lvn=fnin ; vn=0 on @: From Boccardo–GallouOet estimates [3,11], we deduce that (vn) is bounded in W1;q 0();for all q¡ N N−1 if N¿2;q=2 if N=1:(13)
As a result, there exist a subsequence (still denoted in the same way) and v∈W1;q 0() such that vn*v in W1;q 0()-weakly:(14) Since the embeddings W1;q 0(),→Lr(), for all r¡N=(N−2) if N¿2, or W1;q 0()= H1 0(),→C(J )ifN= 1, are compacts, we may also assume that vn→vin Lr()-strongly if N¿2;(15) vn→vin C(J )-strongly if N=1;(16) vn→va:e:in : (17) Moreover, since fn¿0in, then vn¿0in. Since Anis strictly increasing, we also have un¿0in. Now, we show that (A(un))⊂H1 0() is bounded in W1;q 0(). Indeed, |∇A(un)|=|a(un)∇un|6|an(un)∇un|=|∇An(un)|=|∇vn| and by virtue of (13), (A(un)) is also bounded in W1;q 0(); then there exist a subsequence (denoted in the same way) and z∈W1;q 0() such that A(un)*z in W1;q 0()-weakly;(18) A(un)→zin Lr()-strongly for all r¡ N N−2if N¿2;(19) A(un)→zin C(J )-strongly if N=1;(20) A(un)→za:e:in (21) But, since Ais one-to-one, from (21) we deduce un→A−1(z)=ua:e:in (22) with u¿0 a.e. in . Thanks to the de4nition of n, together with (22), we obtain n(un)→(u)a:e:in : (23) Also, by virtue of (H.1), (n(un)) is bounded in L∞(), and taking into account (23), we have n(un)→(u)inL∞()-weakly- ∗:(24) Now, we seek for estimates to the sequence (’n) in some Sobolev space W1;p(), with 1 ¡p¡2. By virtue of (H.5), 2=pis the conjugate exponent of 2=p. Applying Young’s inequality and taking into account (12), we obtain |∇’n|p6 n(un)−p=pp=2 n(un)|∇’n|2p=2 6Cp=2 1 n(un)−p=pp=2 :
Let us show the following estimate: n(un)−p=p 6C2:(25) From 0 ¡(s)6n(s)6˜, for all s∈R, it yields ˜−p=p 6n(s)−p=p 6(s)−p=pfor all s∈R; hence n(un)−p=p 6 (un)−p=p 6{|un|6M} (un)−p=p+{un¿M} (un)−p=p: Thanks to (H.1), −1is bounded on compact sets of R, in particular, there exists a constant value CM¿0 such that min|s|6M(s)=CM, and this implies that (un)−p=p#{|un|6M}6C−p=p M, and {|un|6M} (un)−p=p 6C−p=p M||=C(M; p; p;)=C3: On the other hand, by virtue of (H.4), we deduce {un¿M} (un)−p=p 6{un¿M} (un)−p=p 6 i¿M{i6un¡i+1} (un)−p=p 6 i¿M{i6un¡i+1} (i+1) −p=p 6 i¿M (i+1) −p=p|{un¿i}| (26) In order to derive some estimate to |{un¿i}|, we 4rst study |{vn=An(un)¿i}|.To do so, we take Ti(vn) as a test function in the equation of un; then ∇vn∇Ti(vn)= n(un)|∇’n|2Ti(vn)6C1i; the left-hand side can be written as ∇vn∇Ti(vn)=|∇Ti(vn)|2=Ii;n. By Sobolev’s inequality we have Ii;n ¿C |Ti(vn)|Jq2=Jq ¿C{vn¿i} |Ti(vn)|Jq2=Jq =C{vn¿i} iJq2=Jq =Ci2|{vn¿i}|2=Jq; where Jq=2 ∗=2N=(N−2) and C=C(; N), if N¿3, Jq∈[2;+∞) and C=C(; Jq), if N62. Consequently, |{vn¿i}|2=Jq6C1i i2C=C1 iC ;
which yields, |{vn¿i}| 6(C1=iC)Jq=2=C4=i Jq=2. Since un¿0in,An(un)¿A(un)in ,{A(un)¿i}⊂{vn=An(un)¿i}and |{A(un)¿i}| 6|{vn¿i}| 6C4 iJq=2; hence |{un¿A−1(i)}| 6C4 iJq=2; this can be expressed as |{un¿l}| 6C4 A(l)Jq=2: Therefore, thanks to (3) in (H.5) and (26), we have {un¿M} (un)−p=p 6 i¿M (i+1) −p=pC4 A(i)Jq=2 6C4+∞ M−1 ds (s+1) p=pA(s)Jq=2=C5: This shows (25) and we deduce that |∇’n|p6Cp=2 1Cp=2 2=C6;(27) which means that, ’n−Tn(’0) is bounded in W1;p 0(). We then take a subsequence (still denoted in the same way) and ’∈W1;p() such that ’n*’ in W1;p()-weakly;(28) ’n→’in LJr()-strongly for all Jr¡p ∗if N¿2;(29) ’n→’in C(J )-strongly if N=1;(30) ’n→’a:e:in : (31) From (H.5), p¿2N=(N+ 2) which implies that p∗=Np=(N−p)¿2. In particular ’n→’in L2()-strongly:(32) Thanks to (12) the sequence (n(un)1=2∇’n) is bounded in L2()N; thus there exist a subsequence (still denoted in the same way) and &∈L2()Nsuch that n(un)1=2∇’n*& in L2()N-weakly:(33) From (24), (28) and (33) it is deduced that &=(u)1=2∇’∈L2()N. Moreover, taking into account (H.1), (24) and (33), we also have n(un)∇’n*(u)∇’in L2()N-weakly:(34)
Going b ack to (11) and taking = ’n, with ∈ D(). Then 0= n(un)∇’n∇(’n)= n(un)|∇’n|2+ n(un)∇’n’n∇ = n(un)|∇’n|2− ∇·(n(un)’n∇’n) and so, n(un)|∇’n|2=∇·(n(un)’n∇’n)enD():(35) From the equality n(un)’n∇’n∇= n(un)1=2’nn(un)1=2∇’n∇ and by virtue of (23), (32) and (33), passing to the limit, it yields (u)1=2’(u)1=2∇’∇= (u)’∇’∇for all ∈D(); so, n(un)|∇’n|2=∇·(n(un)’n∇’n)→∇·((u)’∇’)enD(). Since n(un)|∇’n|2¿0 is bounded in L1(), we conclude that ∇·((u)’∇’) is a positive Radon measure. This ends up the proof of Theorem 1. Remark 2. It is interesting to know if the equality ∇·((u)’∇’)=(u)|∇’|2holds in our setting. There are cases where this holds true (for instance in N= 1). We also know some situations where we 4nd that this equality is true: •In the regular case, that is, if ’∈H1(). •In the non-regular case the situation is more complicated. Indeed, according to Theorem 1, we have ’∈W1;p();(u)−1∈L1=(r−1)();r=2 p: Then, we may show that the equality still holds true in the following cases: ◦If (u) is a weight of the Muckenhoupt class [13]; this means that, for all x∈RN, 1 |BR(x)|BR(x) (u) 1 |BR(x)|BR(x) (u)−1=(r−1)r−1 6C: ◦Or if the linear problem ∈W1;p 0();(u)1=2 ∈L2()N; (u)∇ ∇= 0 for all ∈H1 0(); has only the trivial solution =0 (note that in this linear problem, we cannot take = in the variational formulation).
In the general case and with the regularity deduced here for uand ’,wedonot know if this equality still holds [12]). 3. Analysis of a singular and non-uniformly elliptic problem The case described in the previous section does not lead to L∞-estimates on the temperature u. The situation that we are presenting now considers the case of a singular thermal conductivity, that is, a(s) blows up for a 4nite value s=+¿0. Under certain hypotheses on data, we show that the temperature remains bounded in . Speci4cally, it will be shown that 0 6u(x)¡+, almost everywhere in . We consider the thermistor problem (1). All along this section, we will assume the following hypotheses: (H:6) ∈C(R) and 0 ¡(s) for all s∈R. (H:7) a∈C(−∞;+), +¿0, a(0)=0, a(s)¿0 for all s∈(0;+), a(s)¿0 for all s¡+, and + 0a(s)ds=+∞. (H:8) There exists n0¿1=+ such that a(s) is an increasing function in the interval (+−1=n0;+). (H:9) ’0∈H1(). Remark 3. Hypothesis (H.6) is very general. Indeed, like in the previous section we are not assuming to be uniformly elliptic. Besides, we do not make any assumption on the asymptotic behavior of (s) for great values of s. Remark 4. Hypotheses (H.7) and (H.8) yield lims→+−a(s)=+∞;soa(s) becomes singular for the 4nite value s=+. We have the following existence result: Theorem 2. Under assumptions (H.6)–(H.9), problem (1)has a weak solution (u; ’) in the following sense: u∈L∞();06u ¡ + almost everywhere in ; (36) ∀q¡ N N−1if N ¿2;q=2 if N =1; u∈W1;q loc ();A(u)∈W1;q 0();∇A(u)=a(u)∇u; (37) ’−’0∈H1 0();(38) a(u)∇u∇= (u)|∇’|2for all ∈D();(39) (u)∇’∇=0 for all ∈H1 0():(40)