Entropy solutions for nonlinear elliptic equations in L1

Pergamon Nonlinear Analysis. Theory. Methods & Applications, Vol. 32, No. 3, pp. 325-334, 1998

© 1998 Elsevier Science Ltd. All fights re~rvcd Printed in Great Britain

0362-546X/98 $19.00+0,00

PII: S0362-546X(96)00323-9

E N T R O P Y S O L U T I O N S F O R N O N L I N E A R E L L I P T I C

E Q U A T I O N S I N L 1

CHIARA LEONEI" and ALESSIO P O R R E T T A , t SISSA, Via Beirut 2-4, 34014 Trieste, Italy; and

:~ Dipartimento di Matematica "G. Castelnuovo", Universit~ di Roma I, P. le Aldo Moro 2, 00185 Roma, Italy

(Received 22 May 1996; received for publication 27 November 1997)

Key words and phrases: Entropy solutions, nonlinear elliptic equations in L ~.

1. INTRODUCTION AND ASSUMPTIONS

In this paper we prove results o f existence and continuous dependence for the following nonlinear Dirichlet problem

-d iv(a (x , u, Vu)) = f - div(F) in f~, (1.1)

u = 0 on Of~,

where f~ is a bounded open subset o f R N, N -> 2, and

f ~ LL(f~), F E LP'(f~) N. (1.2)

Moreover we assume that a: f~ × R × R N ~ R s is a Carathrodory function such that for every s in R, ~, r / in R N (~ :/: ~/), and for almost every x in f~,

a(x, s, ~) . ~ >- ~1~1", (1.3)

la(x, s, ~)l --- b(Isl)(k(x) + I~1"-1), (1.4)

(a(x, s, ~) - a(x, s, r/)). (~ - r/) > 0, (1.5)

where p > 1, ol is a positive real constant, k(x) belongs to LP'(~) (1/p + 1/p' = 1) and b: [0, +oo) ~ (0, +oo) is a continuous function. The model example of problem (1.1) under hypotheses (1.3), (1.4), (1.5) is the following boundary value problem:

-d iv (a (x , u)(1 + lUl)mlVul "-2 VU) = f - div(F) in ~ ,

u = 0 on 0~,

with 0 < ~ -< a(x, s) -< y, or, 7 ~ R +. Observe that, by the results o f [1], assumption (1.2) includes the case of a measure datum which is zero on the subsets o f zero p-capacity [i.e. the capacity defined with respect to Wol'P(fl)].

We note that, since no growth condition is required for b, it is not obvious that the term -d iv (a (x , u, Vu)) makes sense even as a distribution, that is to say a(x, u, Vu) belongs to Ll(f~) N. Thus, we consider a weaker formulation of (1.1) which allows us to overcome this difficulty. It is worth pointing out that similar questions can be found, for example, in [2], where the authors deal with an unbounded lower order term in divergence form. In the above paper, the obstacle is bypassed introducing the definition of renormalized solutions, while in [3] the same problem is treated using the notion o f entropy solution introduced in [4]. This is the point o f view that we will adopt.

325

326 C. LEONE and A. PORRETTA

Definition 1.1. A function u is an entropy solution of problem (1.1) if Tk(u) belongs to W~'P(~2) for every k > 0, and

In a(x, u, Vu) VT,(u - ~o) dx = IofTk(u - ~o) dx + In F VTk(u - ~o) dx (1.6)

for every ~0 in/'V~'P(f~) tq L~(f~) and every k > 0.

We recall that the gradient o fu which appears in (1.6) is defined as in Lemma 2.1 of [4], that is to say that Vu = VTk(u) on the set where lul < k; note however that, i fp > 2 - 1/N, then u belongs to a Sobolev space and this definition gives the usual notion of derivative in distributional sense. Let us remark that the use of Tk(u -- ~o) as test function gives a meaning to each term in (1.6), although a(x, u, Vu) might not be summable and Vu does not belong to LP'(f~)N; in fact the first integrand is zero on the set where lul > M = k + I1~011:(.), so that we can write

na(x, u, Vu) VTk(u - q~) dx = fo a(x, rn(u), VT~t(u)) VTk(TM(u) - ~o) dx,

to have, by (1.4), that

la(x, TM(U), VT~(u))I -< max b(lsl)(k(x ) + IVTM(u)] p-I) E LP'(f~). [0,M]

In Section 3 we will prove the following theorem of existence, which extends the results of [1,4-61.

Henceforward, we will denote by Mq(fl) the Marcinkiewicz space of order q, with 0 < q < +oo.

THEOREM 1.1. Let 1 < p < N, and letfbelong to LI(~) and F in LP'(f~) N. Assume that a(x, s, ~) satisfies hypotheses (1.3), (1.4) and (1.5). Then there exists an entropy solution u of problem (1.1); furthermore, u E M(N(p-I))/(N-P)(~'~) and IVul ~ M(N(P-1))/(N-1)(~'~); i fp > 2 - 1/N, then u belongs to w~'q(f~) for every q < (N(p - 1))/(N - 1).

Let us point out that there is an extensive literature concerning problem (1.1) under hypotheses (1.2), (1.3), (1.5) but with (1.4) replaced by the usual growth condition

]a(x, s, ~)] <- fl(k(x) + Is] p-l + I~]P-l), (1.7)

with fl > 0 and k(x) ~ LP'(f~). In the linear case, i.e. p = 2, and a(x, s, ~) = M(x)~, with M(x) a bounded coercive matrix,

the existence of a solution u of (1.1) belonging to the Sobolev space wd'q(f~) for every q < N/(N - 1) has been proved by Stampacchia in [7] by a duality argument, even in the more general case in whichf is a Radon measure. However, since this method cannot be used if the operator is nonlinear, a solution in the sense of distributions of problem (1.1) [with assumption (1.7)] has been obtained in [5] and [8] by approximation. That is to say, they take a sequence of regular functions {f,} converging to f i n L~(fl), solve the Dirichlet problems

-div(a(x, u,, Vu,)) =fn in f/,

un = 0 o n 0t'~,

and prove, if p > 2 - 1/N, that u, converges, up to subsequences, in wd'q(f~) for every q < (N(p - 1))/(N - 1) to a weak solution u of (1.1) (the restriction onp is due to the fact that

Entropy solutions for nonlinear elliptic equations 327

(N(p - 1))/(N - 1) > 1 if and only i fp > 2 - 1/N). This convergence cannot be obtained in a smaller Sobolev space; on the other hand, if we denote by Tk(s) = max(-k, min(k, s)) the mmcation at levels +k, it has been proved by Lions and Murat in [6] (see also [9] for the linear case, [10] for a simple proof in a particular framework and [11]) that Tk(u,) converges strongly to Tk(U) in W~'P(f~). In Section 2 we will provide a different proof of this result, which is also the main tool we need to obtain Theorem 1.1. Our proof of the convergence of truncations is inspired by the definition of entropy solution and it can be easily applied to obtain a stability result for problem (1.1); in fact, we will prove in Section 4 the following theorem, also obtained in [6] when a(x, s, ~) does not depend on s.

THEOREM 1.2. Let {f,} C LL(f~) and {F~} C LP'(~) N be such that

f~ ~ f weakly in L l(f~)

F~ ~ F strongly in LP'(f~) N

and let u, be entropy solutions with dataf~ - div(F,). Then there exist a subsequence, that we will still denote by {u,}, and a function u, which is an entropy solution relative to f - div(F), such that, for every k > 0:

lim Tk(u,) = Tk(u) strongly in Wol'P(~). e ~ 0

We observe that this result of continuous dependence completes the treatment of problem (1.1) in L~(~) which has been given in [4] in the framework of entropy solutions.

2. STRONG CONVERGENCE OF TRUNCATIONS

In this section we prove the strong convergence of truncations in the energy space, under assumption (1.7) instead of (1.4), so that it will be actually a new proof of the same result of [6]. We remark that this convergence plays a crucial role in the theory of elliptic equations with L l data and can be applied in several similar contexts. In addition, the method we employ will allow us to get easily the existence result of Theorem 1.1.

In the following, we will set ,6 = ( N ( p - 1) ) / (N-1 ) , whose Sobolev exponent is p* = (N(p - 1))/(N - p); note that bothp and/3" are positive real numbers, and that,6 > 1 if and only i fp > 2 - 1/N.

THEOREM 2.1. Assume that 1 < p < N and that a(x, s, ~) satisfies hypotheses (1.3), (1.5), (1.7). Let {u,} be a sequence in Wo~'P(f~) such that

with

-div(a(x, u,, Vu,)) = f , - div(F,) in f~ V n ~ N,

f , ~ f weakly in L l(~)

F, ~ F strongly in LP'(~) ~.

Then there exist a subsequence, that we will still denote by {u,}, and a function u which satisfies u E M'*(f~), IVul ~ MP(f~), such that, for every k > 0:

Tk(u,) ~ Tk(u) strongly in W~'P(f~).


Proof. The assumption on u. means that

If we take o = Tk(u.), applying in the last term Young's inequality and in the first one Assumption (1.3), we get:

and then

fnWTk(u.)lP dx <~ k v k >~ 1. (2.2) Cl

By means of (2.2), we can apply Lemmas 4.1 and 4.2 of [4] which imply that {u,} is bounded in MP'(~), and { [Vu.[ } is bounded in MP(92). Moreover, reasoning as in the proof of Theorem 6.1 of [4], it follows from (2.2) that {u,} is a Cauchy sequence in measure. As a consequence, there exist a function u and a subsequence, still denoted by {u,}, such that:

u, --+ u a.e. in ~. (2.3)

Putting together (2.2) and (2.3) it is possible to conclude that

Tk(u,) ~ Tk(u) weakly in Wol'P(~) for every k > 0, (2.4)

Tk(u,) --+ Tk(u) strongly in LP(f2) and a.e. in ~, for every k > 0.

Thus we have proved that Tk(u) belongs to W~'P(~) for every k > 0. Furthermore, by the weak lower semicontinuity of the norm in W~'P(~), estimate (2.2) still holds for u; applying again Lemmas 4.1 and 4.2 of [4] we find that u belongs to MP'(~) and [Vul belongs to MP(~).

Now we prove that Tk(u,) strongly converges to Tk(u) in W0~'P(f2), for every k > 0. Let h > k and let us take w, = T2k(u, - Th(u,) + Tk(u,) -- Tk(u)) as test function in (2.1); then if we set M = 4k + h, it is easy to see that Vw, = 0 where lu,[ > M; therefore, since Assumption (1.3) implies that a(x, s, 0) = 0, we can write:

faa(x, TM(un),VTM(u~))Vwndx=Infnwndx+I FnVw~dx.

Splitting the integral in the left-hand side on the sets where [u,[ -< k and where [u,[ > k we get [remember that a(x, s, ~) . ~ >- 0]:

n a(x, Tu(u,), VTM(u,)) VTz~(u, - Th(u,) + Tk(u,) -- Tk(u)) dx

Jn a(x, T~(u.), VTk(u,)) V(Tk(u,) - Tk(u)) dx

( la(x, TM(u,), 7TM(u,))I [7Tk(u)l dx, 1

J{ lu.I > ~}


and then from the equation it follows:

f (a(x, Tk(u.), VTk(u.)) a(x, Tk(u.), VTk(u))) V(Tk(u.) Tk(u)) dx l l

I la(x, rM(u.), VTM(u.))I IVTk(u)l dx 3~ lu.I > k}

+ J f . T2k(u. -- Th(u.) + Tk(u.) -- Tk(u)) dx

+ Jn F. VT2k(u. - Th(u.) + Tk(u.) - Tk(u)) dx

- Ia a(x, Tk(u.), 7Tk(u)) 7(Tk(u.) -- T~(u)) ~ . (2.5)

Now, if we take o. = Tzk(Un - Th(Un)) in (2.1), we can proceed as in the beginning of the proof in order to have:

In lVT2k(u. - Th(Un))l p c2(2k 1), dx < +

where c2 is a positive constant that does not depend on h. Since

T2k(u. - Th(u.)) ~ Tzk(u -- Th(u)) weakly in WI'p(fl),

we get

falVT2k(u - dx <- + Th(U))l p c2(2k 1),

from which we can deduce

f~ lFl lVT2k(u - Th(u))l dx <-- c3 f{ IFlp' I

where c3 depends on k but not on h. Therefore, by the absolute continuity of the integral we get:

t '

[ F VT2k(u - Th(U)) lira dx 0. h ~ + ~ 3n

Since the Lebesgue theorem implies also

lim [ f T z k ( u - Th(U))dx = O, h ~ + o o J.

we can fix a positive real number h, sufficiently large to have

InfT2k(u - Th~(u)) dx + f n F VT2k(u - Th.(u)) dx <-- e. (2.6)

Now we take h = h, in formula (2.5) (and then M = M, = 4k + h,), and observe that la(x, TM(u,), VTm(u,)) I is bounded in LP'(fl) while Z{lu.I > k~ IVTk(u)I converges strongly to zero

330 c. LEONE and A. PORRETTA

in LP(fl) as n tends to infinity; this allows us to write that

I la(x, TM(un), VTM(un))l IVTk(u)l dx = 0. (2.7) lim n--, +oo j { lu.i > k}

Furthermore, it is easy to see that, as n tends to infinity,

T2k(u, - Th(U,) + Tk(u,) -- T~(u)) ~ T2k(u - Th(U)) weakly in Wd'P([2),

so that, passing to the limit in (2.5), by means of (2.4), (2.6) and (2.7) we deduce:

lim® Ja(a(x, Tk(un), VT~(un)) - a(x, Tk(u.), VTk(u))) V(Tk(un) - Tk(u)) dx

<-- f fT2,(u - Tho(u)) dx + fo F VT2,(u - Th.(u)) dx <-- t.

that is to say. since e is arbitrarily small.

j (a(x, Tk(un), VTk(un)) - a(x, Tk(un), VTk(u))) V(Tk(un) - Tk(u)) dx = lim 0. n--* q -~ 30

Now we can apply Lemma 5 of [12] (which is actually a refinement o f a lemma due to Leray and Lions (see [13]) in case one assumes Assumption (1.3) instead of a milder coercivity) to conclude that:

Tk(un) ~ Tk(u) strongly in WI'P(~) for every k > 0. •

Remark 2.1. It is not difficult to see that the function u found in Theorem 2.1 is an entropy solution, in the sense of Definition 1.1, of the Dirichlet problem (1.1) under Assumption (1.7) instead of (1.4). Moreover, we point out that u is also a renormalized solution of the same boundary value problem, according to the definition of [2], that is to say that u satisfies:

I a(x,u, Vu)V(h(u) )dx=ffh(u) dx+IrV(h( ) )dx for every h in Ccl(R) and for every q~ in ~(f~).

Remark 2.2. The result of Theorem 2.1 does not extend to the case of a general {fn} converging .-weakly (in the sense of Radon measures) to a measure/z. For a counterexample, see [14]. On the other hand, it is proved in [15] that for every measure/z the same result can be obtained if {f.} is chosen in a suitable way.

3. PROOF OF THEOREM 1.1

We are able now to prove the main result of this paper.

Proof o f Theorem 1.1. We introduce the functions an(x, s, ~) = a(x, Tn(s), ~) as an approximation ofa(x, s, ~) as n tends to infinity. If we consider also two sequences of regular functions {fn} and {Fn} strongly converging respectively to f in L~(fl) and to F in L¢(fl) ~, by the classical results of Leray and Lions (see [13]) there exists a sequence {un} in Wd'P(fl) of solutions of the Dirichlet problems:

{ -div(a.(x, u., Vun)) = fn - div(Fn) in f~, (3. 1 )

u . = 0 on Of~.


Since a,(x, s, ~). ~ >-o~l~l p, i f we take Tk(u,) as test function in (3.1) and apply Young 's inequality in the right-hand side we have:

that yields, as before,

Ilrk(u.)ll~g~<~) --< kc, V k >- 1.

Thanks to this inequality we can proceed as in the Proof of Theorem 2.1. So first we deduce that there exists a function u such that, up to a subsequence,

u . ~ u a . e . in ~ ,

Tk(u,) ~ Tk(u) weakly in W0~"(f2), (3.2)

Tk(u,) --* Tk(u) strongly in L'(f~) and a.e. in ~ ,

and moreover u ~ M(N(p-I))/(N-P)(["~) and [Vu[ E M(N(P-1))/¢N-O([~). Now we take, as in Theorem 2.1, w. = Tzk(u.- Th(U.)+ Tk(u.)- Tk(u)) in the weak

formulation of (3.1); recalling that Vw. = 0 where [u.[ > M = 4k + h the following equality holds:

fn a.(x, u., Vu.) Vw. dx = fn a.(x, TM(u.), VT,~(u.)) Vw. dx.

Since we will let n tend to infinity, we can choose n greater than M in order to have

a.(x, Tu(u.), VTu(u.)) = a(x, TM(u.), VTu(u.)).

In conclusion, for n > M = 4k + h, with h > k, we have:

Ina(x, TM(u.), VTM(u.)) Vw. dx = f f .w. dx + InF. Vw. dx.

Thanks to this equality we are in the position to rewrite the Proof of Theorem 2.1 (remember that ]a(x, TM(u.), VTM(u.))I < maxl0,M 1 b(IsD(k(x) + IVTM(u.)l p-I) which is bounded in LP'(fl)), and again we get

Tk(u.) ~ Tk(u) strongly in W0~'P(f~), V k > 0.

We point out that this convergence implies that, for every fixed k > 0,

a(x, Tk(u.), VTk(u.)) ~ a(x, Tk(u), VTk(u)) in LP'(~) u. (3.3)

Finally, to show that u is an entropy solution, let us take v. = Tk(u. - ~o) as test function in (3.1) with ~0 in W~'P(f~) N L=(fl); we note that, i f L = k + IIq~llL-(n) and n > L, we have

fna.(x, u., Vu.) VT,(u. - q') dx = fna(x, TAu.), VTL(u.)) VT,(u. - ~o) dx,

so that (3.1) with this test function can be rewritten as


Hence we can pass to the limit as n tends to infinity, using (3.2) and (3.3) and therefore conclude:

fn a(x, u, Vu) V Tk(u - ~) dx = InfTk(u - ~o) dx + In F V Tk(u - ~o) dx,

for every ~0 in W~'P(f~) fq L~(f~) and every k > 0. •

We remark that it is not obvious that problem (1.1) can be formulated in the usual weak sense even i f f = 0, unless F is regular enough to have u in L®(f~) (it suffices in fact that F is in Lq(['~), q > N / ( p - 1), see [7]). On the other hand, if f = 0, it is easily seen that the approximating sequence {u,} of the previous theorem is bounded in W01'P(~), so that the solution u enjoys the same regularity. We can state this remark in a separate theorem.

THEOREM 3.1. Let 1 < p < N, f = 0 and let F be in LP'(f~) N. Assume that a(x, s, ~) satisfies hypotheses (1.3), (1.4) and (1.5). Then there exists an entropy solution u in Wol'P(f~) of problem (1.1).

We observe also that with very few modifications the Proof of Theorem 1.1 applies if we allow in equation (1.1) the presence of a lower order term in divergence form without any growth condition, like in [3], or in [2] where f = 0. To be more precise, let ~: R --, R N be a continuous function, and consider the boundary value problem

-div(a(x, u, Vu)) = f - div(F + ~(u)) in f], (3.4)

u = 0 o n 0 f ~ .

Note that the entropy formulation is well posed in this case too. We can use the same scheme of approximation as in Theorem 1.1, introducing the functions ~.(s) = dP(T.(s)) and defining u. as the solutions of the problems:

-div(a.(x, u., Vu.)) = f . - div(F. + ~.(u.)) in fl,

u . = 0 o n Of~.

Then it should be remarked that, by the divergence theorem (see [2] for more details), we have

fa ~.(u.) 7T,(u.) dx = 0 n E N, V

so that the estimates on the sequence {u,} still hold true. As far as the convergence of truncations is concerned, it suffices to observe that, for n large,

f dP.(u.) 7Tz,(u. - T,(u.) + T,(u.) - Tk(u)) dx

In ~(T4k+h(U.)) VT2k(u. - Th(Un) a t- Tk(u.) -- Tk(u)) dx,

which yields:

lira 3n Jo


The remainder of the proof applies easily, therefore we can state the following theorem, which is a generalization of the result in [3].

THEOREM 3.2. Let 1 < p < N, f i n L~(f~) and F in LP'(f~)/v. Assume that • E C°(R, R u ) and that hypotheses (1.3), (1.4) and (1.5) hold true. Then there exists an entropy solution u of problem (3.4). Moreover u E M(N(p-I))/(N-P)(~) and ]Vu[ ~ M(N(P-1))/(N-I)(['~); if 2 - 1/N< p < N, then u belongs to the Sobolev space W~'q(f~) for every q < (N(p - 1 ) ) / ( N - 1).

4. A STABILITY RESULT FOR ENTROPY SOLUTIONS

In this section we prove the result of continuous dependence for entropy solutions stated in Section 1. We remark that in this framework the "natural" continuous dependence concerns not the solutions themselves, but their truncations.

Proof of Theorem 1.2. Since the u, 's are entropy solutions, we have, for every k > 0 and every q~ in WI"p(f~) fq L~(~),

Ina(x ,u , ,Vu~)VTk(u,- (o)dx=fno~Tk(u~-{o)dx + fnF , VTk(u,-~o)dx. (4.1)

Taking ~0 = 0 we can reason as in Theorem 2.1 and get:

[[Tk(u~)[lP~.~n) <-- cok V k --- 1, (4.2)

which is the same as estimate (2.2). As we have already seen, it implies that there exists a subsequence, still denoted by {u~}, and a function u such that u ~M~utP-t)~/~u-P)(f2), IVul ~ M<u~P-')~'("-'~(f~) and

u, ---, u a.e. in fl ,

Tk(u,) ~ Tk(u) weakly in Wd'P(f~), (4.3)

Tk(u,) ~ Tk(u) strongly in LP(f~) and a.e. in f~.

In particular, we have that Tk(u) belongs to Wol'P(f~) for every k > 0. Let us now take ~0, = Th(u,) - Tj(u,) + Tj(u), h >j, and k = 2j in (4.1) in order to have:

fn[a(x, u~, ~Tu,) FJ VT2j(u, Th(U,) Ty(u,) Tj(u)) + dx i i

= Jnf, r2 j (u , - rh(u~) + ~(u~) - r A u ) ) dx.

Henceforth, thanks to (4.3), we can repeat the Proof of Theorem 1.1 with u~ instead of Un ; thus we find, in the same way, that for every j > 0

Tj(u~) ~ Tj(u) strongly in W~'P(f~).

Moreover, this convergence allows us to deduce that Vu~ converges to Vu almost everywhere in and then

a(x, Tj(u,), 7Tj(u,)) ~ a(x, Tj(u), VTj(u)) strongly in LP'(f~) N, V j > 0;

therefore, if we observe that

In a(x, u,, Vu,) VT~(u~ - ~o) dx = In a(x, TL(u,), VTL(u~)) VTk(u~ - ~o) dx,

334 c. LEONE and A. PORRETTA

with L = k + I[@[IL'¢.), we earl pass to the l imit in (4.1) as r tends to zero and get:

for every ~ in wl 'p(f~) A L~(I2) and every k > 0; this means that u is an entropy solution

relat ive to the datum f - d iv(F) .

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