Boccardo, L.

Brezis, H.

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Some remarks on a class of elliptic equationswith degenerate coercivityc© Lucio Boccardo, Ha¨ım Brezis1. Some elliptic equations with degenerate coercivity.In the paper [3], existence and regularity results of the following elliptic problem arestudied: −div(a(x, u)∇u) = f in Ω,u = 0 on ∂Ω,(0.1)where Ω is a bounded, open subset of IRN , with N > 2, and a(x, s) : Ω × IR → IR isa Carathe´odory function (which is measurable with respect to x for every s ∈ IR, andcontinuous with respect to s for almost every x ∈ Ω) satisfying the following conditions:α(1 + |s|)θ ≤ a(x, s) ≤ β , (0.2)for some real number θ such that0 ≤ θ ≤ 1 , (0.3)for almost every x ∈ Ω, for every s ∈ IR, where α and β are positive constants. Thedatum f belongs to Lm(Ω), for some m ≥ 1.The main difficulty in dealing with problem (0.1) is the fact that, because of assump-tion (0.2), the differential operator A(u) = −div(a(x, u)∇u), even if well defined betweenH10 (Ω) and its dual H−1(Ω), is not coercive on H10 (Ω) (when u is,1(1+|u|)θ goes to zero: foran explicit example see [6]).This implies that the classical methods used in order to prove the existence of asolution for problem (0.1) cannot be applied even if the datum f is regular.In this note, two new and short proofs of the existence theorems will be presented.The shortness depends on the use of previous results of [7] and [5]; by contrast the originalones of [3] are selfcontained.We will recall here the existence and regularity results proved in [3], then we will givenew proofs of the first two theorems.The first result concerns the existence of bounded solutions, and coincides with theclassical boundedness results for uniformly elliptic operators (see [7]). The main tool ofthe proof will be an L∞(Ω) a priori estimate, which then implies the H10 (Ω) estimate,since if u is bounded then the operator A is uniformly elliptic.Theorem 0.1 Let f be a function in Lm(Ω), with m > N2. Assume (0.2) and (0.3). Thenthere exists a weak solution of (0.1) u in H10 (Ω) ∩ L∞(Ω).The next result deals with data f which give unbounded solutions in H10 (Ω).Theorem 0.2 Let 0 < θ < 1 and f be a function in Lm(Ω), with m such that2NN + 2− θ(N − 2) ≤ m <N2. (0.4)Assume (0.2) and 0 < θ < 1. Then there exists a function u in H10 (Ω) ∩ Lm∗∗(1−θ)(Ω),which is weak solution of (0.1), where m∗∗ = (m∗)∗ = mNN−2m .Remark that, since 2NN+2−θ(N−2) ≤ m and 0 < θ < 1, then m ≥ 2NN+2 , so that f belongsto the dual of H10 (Ω).We recall that Example 1.5 of [3] shows that the result of Theorem 0.2 is sharp.Remark that if θ = 0, the result of the preceding theorems coincides with the classicalregularity results for uniformly elliptic equations (see [7] and [5]).We refer to [6] for a uniqueness result for (0.1).If we decrease the summability of f , we find solutions which do not in general belongany more to H10 (Ω), even if our assumptions on f (f ∈ Lm(Ω), m ≥ 2NN+2) implies thatf ∈ H−1(Ω).We recall now other results of [3]Theorem 0.3 Let 0 < θ < 1 and f be a function in Lm(Ω), with m such thatNN + 1− θ(N − 1) < m <2NN + 2− θ(N − 2) . (0.5)Then there exists a function u in W 1,q0 (Ω), withq =Nm(1− θ)N −m(1 + θ) < 2 , (0.6)which solves (0.1) in the sense of distributions, that is,∫Ωa(x, u)∇u · ∇ϕ =∫Ωf ϕ , ∀ϕ ∈ C∞0 (Ω) . (0.7)Moreover, the truncation Tk(u) belongs to H10 (Ω) for every k > 0, whereTk(s) = max{−k,min{k, s}} . (0.8)Up to now, we have obtained solutions belonging to some Sobolev space. If we weakenthe summability hypotheses on f , then the gradient of u (and even u itself) may no longerbe in L1(Ω). However, it is possible to give a meaning to solution for problem (0.1), usingthe concept of entropy solutions which has been introduced in [2].Definition 0.4 Let f be a function in L1(Ω). A measurable function u is an entropysolution of (0.1) if Tk(u) belongs to H10 (Ω) for every k > 0 and if∫Ωa(x, u)∇u · ∇Tk(u− ϕ) ≤∫Ωf Tk(u− ϕ) , (0.9)for every k > 0 and for every ϕ ∈ H10 (Ω) ∩ L∞(Ω).We observe that every term in (0.9) is meaningful. This is clear for the right handside, while for the left hand side we have∫Ωa(x, u)∇u · ∇Tk(u− ϕ) =∫Ωa(x, u)∇TM(u) · ∇Tk(u− ϕ) ,where M = k + ‖ϕ‖L∞(Ω). Then if f is a function in Lm(Ω), with1 ≤ m ≤ max{NN + 1− θ(N − 1) , 1}, (0.10)we refer to [3] for the existence of an entropy solution u of (0.1).Remark that if 0 ≤ θ < 1N−1 , then (0.10) becomes m = 1 and q =N(1−θ)N−1−θ which isgreater than 1. If in particular θ = 0, this is the same result obtained in [4] for ellipticequations with L1(Ω) (or measure) data.If 1N−1 ≤ θ < 1, then the upper bound on m is NN+1−θ(N−1) , which is the lower boundon m given by Theorem 0.3.2. Proofs.The proofs of the existence results will be obtained by approximation.Let f be a function in Lm(Ω), with m as in the statements of Theorems 0.1, 0.2 and0.3. Let {fn} be a sequence of functions such thatfn ∈ L 2NN+2 (Ω) , fn → f strongly in Lm(Ω), (0.11)and such that ‖fn‖Lm(Ω) ≤ ‖f‖Lm(Ω) , ∀n ∈ IN . (0.12)For instance, fn = Tn(f).Let us define the following sequence of problems:−div(a(x, Tn(un))∇un) = fn in Ω,un = 0 on ∂Ω.(0.13)The existence of weak solutions un in H10 (Ω) of the Dirichlet problem (0.13) is classical,since the differential operator in (0.13) is uniformly elliptic.Lemma 0.5 Assume the same hypotheses of Theorem 1.1. Let f be in Lm(Ω) and letun be a solution of (0.13) with fn = f for every n ∈ IN . Then the norms of un in L∞(Ω)and in H10 (Ω) are bounded by a constant which depends on θ, m, N , α, |Ω| and on thenorm of f in Lm(Ω).Proof. Let us start with the estimate in L∞(Ω). Define, for s in IR and for k > 0,Gk(s) = (|s| − k)+ sgn(s) = s− Tk(s) ,andH(s) =∫ s01(1 + |t|)θ dt.For k > 0, if we take Gk(H(un)) as test function in (0.13) and use assumption (0.2),we obtainα∫{x∈Ω:|H(un(x))|>k}|∇un|2(1 + |un|)2θ ≤∫{x∈Ω:|H(un(x))|>k}f Gk(H(un)).That isα∫Ak|∇(H(un))|2 ≤∫Akf Gk(H(un)) (0.14)where we have setAk = {x ∈ Ω : |H(un(x))| > k} .The inequality (0.14) is exactly the starting point of Stampacchia’s L∞-regularity proof(see [7]), so that there exists a constant c1 such that‖H(un)‖L∞(Ω) ≤ c1. (0.15)The properties of the functions h and H ( lims→∞H(s) = +∞) yield a bound for un in L∞(Ω)from (0.15):‖un‖L∞(Ω) ≤ c2.The estimate in H10 (Ω) is now very easy. Taking un as test function in (0.13), one obtainsα(1 + c2)θ∫Ω|∇un|2 ≤∫Ωf un ,and the right hand side is bounded since f belongs, at least, to L1(Ω).The next result will be used in the proof of Theorem 0.2.Lemma 0.6 Assume the same hypotheses as in Theorem 1.2. Let f belong to L∞(Ω),and let u ∈ H10 (Ω) ∩ L∞(Ω) be a solution of (0.1) (which exists by Theorem 1.1). Thenthe norms of u in Lm∗∗(1−θ)(Ω) and in H10 (Ω) are bounded by constants depending onlyon θ, m, N , α, |Ω| and the norm of f in Lm(Ω).Proof. Multiplying (0.1) by γ(u), whereγ(t) = ((1 + |t|)p − 1)sign(t), p = (1− θ)N(m− 1)N − 2m ,integrating on Ω and using the standard Sobolev imbedding H10 (Ω) ⊂ L2∗(Ω), yields‖u‖L(p+1−θ)2∗/2(Ω)≤ c3, (0.16)and ∫Ω|∇u|2(1 + |u|)p−1−θ ≤ c4. (0.17)It is convenient to observe that pm′ = (p + 1 − θ)2∗/2. So far, we have not used (0.4).Note that clearly(p+ 1− θ)2∗/2 = m∗∗(1− θ)and that (0.4) is equivalent to p − 1 − θ ≥ 0. Thus, if (0.4) holds, then (0.16) implies abound for ∇u in L2(Ω). Here the constants c3 and c4 depend only on θ, m, N , α, |Ω| andthe norm of f in Lm(Ω).Proof of Theorems 0.1 and 0.2The estimates for un in H10 (Ω) imply that there exist a subsequence {unj} and afunction u ∈ H10 (Ω), such that unj converges weakly in H10 (Ω) to u. The coefficienta(x, unj) converges to a(x, u) in any Lp(Ω). Thus it is possible to pass to the limit in(0.13) and to obtain the existence of a weak solution u.Remark 0.7 We point out that Lemma 0.5 can be proved under the slightly more generalassumptionh(s) ≤ a(x, s) ≤ β, (0.18)where the real function h(s) is continuous, decreasing, strictly positive and such that itsprimitiveH(s) =∫ s0h(t) dt (0.19)is unbounded.Thus it is possible, for instance, to study also problems where h(s) = 1(e+|s|) ln(e+|s|) .3. Lower order terms.The presence of lower order terms in the Dirichlet problem (0.1) can change the exis-tence results. For instance, consider the following boundary value problem−div(a(x, u)∇u) + u = f in Ω,u = 0 on ∂Ω,(0.20)where a(x, s) still satisfies the following inequalityα(1 + |s|)θ ≤ a(x, s) ≤ β , (0.21)for some real number θ > 0 (for almost every x ∈ Ω, for every s ∈ IR, α, β > 0) and fbelongs to Lm(Ω), for some m ≥ 1.Let {fn} be the sequence of functionsfn = Tn(f) (0.22)Define the following sequence of problems:−div(a(x, Tn(un))∇un) + un = fn in Ω,un = 0 on ∂Ω.(0.23)It is classical thatLemma 0.8 ‖un‖Lm(Ω) ≤ ‖fn‖Lm(Ω).Now we can prove thatLemma 0.9 Assumem ≥ θ + 2. (0.24)Then the sequence {un} is bounded in H10 (Ω).Proof. The use of[(1 + |un|)1+θ − 1]sign(un)]as test function in (0.23) implies that∫Ω|∇un|2 ≤ c5{1 + ‖fn‖Lm(Ω)}‖|un|(1+θ)‖Lm′(Ω).Remark that (1 + θ)m′ ≤ m if and only if m ≥ 2 + θ.Then, if f belongs to Lm(Ω) and if (0.24) holds, the existence of solutions follows asin the proofs of Theorems 1.1 and 1.2.References[1] A. Alvino, V. Ferone, G. Trombetti, A priori estimates for a class of non uniformlyelliptic equations, to appear on Atti Sem. Mat. Univ. Modena.[2] P. Benilan, L. Boccardo, T. Galloue¨t, R. Gariepy, M. Pierre, J.L. Vazquez, An L1theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola Norm.Sup. Pisa Cl. Sci., 22 (1995), 240–273.[3] L. Boccardo, A. Dall’Aglio, L. Orsina, Existence and regularity results for someelliptic equations with degenerate coercivity , to appear on Atti Sem. Mat. Univ.Modena.[4] L. Boccardo, T. Galloue¨t, Nonlinear elliptic equations with right-hand side measures,Comm. Partial Differential Equations 17 (1992), 641–655.[5] L. Boccardo, D. Giachetti, Ls-regularity of solutions of some nonlinear elliptic prob-lems, preprint.[6] A. Porretta, Uniqueness and homogenization for a class of non coercive operators indivergence form, to appear on Atti Sem. Mat. Univ. Modena.[7] G. Stampacchia, Le proble`me de Dirichlet pour les e´quations elliptiques du secondordre a` coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189–258.Lucio BoccardoDipartimento di Matematica, Universita’ di Roma 1Piazza A. Moro 2, 00185 RomaTel: +39.06.49.91.32.02; fax: +39.06.44.701.007E-mail: boccardo@gprme.sci.uniromal.ilAnalyse NumeriqueUniversite P.et M.Curie4, pl. Jussieu, 75252 ParisCedex 05, FranceE-mail: brezis@ccr.jussie.fr

Some remarks on a class of elliptic equations with degenerate coercivity

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Some remarks on a class of elliptic equations with degenerate coercivity

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)