Regularity for entropy solutions of parabolic p-Laplacian type equations

Publicacions Matem`atiques, Vol 43 (1999), 665–683. REGULARITY FOR ENTROPY SOLUTIONS OF PARABOLIC p-LAPLACIAN TYPE EQUATIONS S. Segura de Le´ on and J. Toledo Abstract In this note we give some summability results for entropy solutions of the nonlinear parabolic equation ut−div ap(x, ∇u)=f in ]0,T[×Ω with initial datum in L1(Ω) and assuming Dirichlet’s boundary condition, where ap(., .) is a Carath´eodory function satisfying the classical Leray-Lions hypotheses, f∈L1(]0,T[×Ω) and Ω is a domain in RN. We find spaces of type Lr(0,T;Mq(Ω)) containing the entropy solution and its gradient. We also include some summability results when f= 0 and the p-Laplacian equation is considered. 1. Introduction Let Ω be a domain in RN(bounded or not) and let 1 <p<N. Let ap:Ω×RN→RNbe a Carath´eodory function satisfying the classical Leray-Lions conditions in such a way that div ap(x, ∇u(x)) defines an operator from W1,p 0(Ω) onto W−1,p(Ω) (see [10]). The model example of such function is ap(x, ξ)=|ξ|p−2ξwhich determines the p-Laplacian operator ∆pu= div(|∇u|p−2∇u). 1991 Mathematics subject classifications: 35K65, 47H20. This research has been supported by the Spanish DGICYT, Proyecto PB94-0960. 666 S. Segura, J. Toledo We are interested in the regularity of entropy solutions (see Definition 2.4 below) of the following parabolic problem: (P)      ut−div ap(x, ∇u)=f, in QT:=]0,T[×Ω, u=0,on ST:=]0,T[×∂Ω, u(0) = u0,in Ω; where u0∈L1(Ω) and f∈L1(QT). A distributional solution of this problem was found in [9]. The notion of entropy solution of (P) is used in [1] and [11] to obtain uniqueness. (An equivalent concept, in terms of renormalized solution, can be found in [7], [13], [14] and [16].) We remark that in [1] it is only considered the case f= 0, although the proof of the general case can be easily obtained following the same steps and taking into account the arguments of [6, Chapter 4]. On the other hand, the hypotheses in [11] are p>2−1 N+1 and Ω bounded. In both articles some summability results on the entropy solution uand its gradient are given; more precisely, u∈M (N+1)p−N N(QT) and |∇u|∈M p−N N+1 (QT) (see [1]), and |∇u|∈∩ q<p−N N+1 Lq(QT) (see [11]or[9]). These summability results are obtained jointly for time and space variables; thus, as a consequence of these papers, it is not possible to get optimal regularity when both variables are separately considered. The purpose of this note is to give a precise summability result of the entropy solution and its gradient with respect to space and time. This was studied in [8] for weak solutions of (P) in the framework of Lebesgue and Sobolev spaces when p≥2, u0= 0 and Ω is bounded. The main result of [8] states that there exists a weak solution uof (P) such that u∈Lr(0,T;W1,q 0(Ω)) for p/2≤q<N(p−1)/(N−1) and 1≤r<q((N+1)p−2N)/((N+1)q−N), and for 1 ≤q<p/2 and r=p. Some remarks on their assumptions are as follows. Firstly, it is possible to get a similar result if 2 −1/(N+1)<p<2, as it is observed in [8, Remark 1.7]. Moreover, analogous results also hold for other initial data u0= 0, as it is pointed out in [12]. Nevertheless, their hypothesis of Ω bounded cannot be removed if their arguments are to be followed, in fact it is needed from the very begining when they obtain the a priori estimates in [8, Lemma 2.2]. We want to show in this paper what the situation looks like in the framework of Marcinkiewicz spaces (see Definition 2.1 below) since we think they should be more suitable for L1-data (see [3] for the elliptic case). We obtain an improvement of the results of [8] since our arguments work for small p>1 and unbounded domains. Regularity for entropy solutions 667 Theorem 1. Let Ωbe a bounded domain and let ube the entropy solution of problem (P). (1) If 2N N+1 <p<N, then u∈Lr(0,T;Mq(Ω)) for 1<q<N(p−1) N−p and r<(N+1)p−2N N q q−1, |∇u|∈Lr(0,T;Mq(Ω)) for p 2<q<N(p−1) N−1 and r<q(N+1)p−2N (N+1)q−N, for q=p 2and r<p, and for q<p 2and r=p. (2) If 1<p≤2N N+1 , then |∇u|∈Lp(0,T;Mq(Ω)) for q<p 2. Remark 1.1. When Ω is bounded, one can replace Mq(Ω) by Lq(Ω) due to the relations between Marcinkiewicz and Lqspaces in bounded domains, and to the fact that qsatisfies strict inequalities. Theorem 2. Let Ωbe an unbounded domain and let ube the entropy solution of problem (P). (1) If 2N N+1 <p<N, then u∈Lr(0,T;Mq(Ω)) for 1<q<N(p−1) N−p and r<(N+1)p−2N N q q−1, |∇u|∈Lr(0,T;Mq(Ω)) for p 2<q<N(p−1) N−1 and r<q(N+1)p−2N (N+1)q−N. 668 S. Segura, J. Toledo (2) If 1<p< 2N N+1 , then u∈Lr(0,T;Mq(Ω)) for N(p−1) N−p<q<1 and r<(N+1)p−2N N q q−1, |∇u|∈Lr(0,T;Mq(Ω)) for N(p−1) N−1<q<p 2 and r<q(N+1)p−2N (N+1)q−N. Remark 1.2. For p= 2, the bound on ris exactly the same one that can be computed for the fundamental solution of the Heat equation. We improve these regularity results when f= 0 and the p-Laplacian operator is considered. Theorem 3. Let Ωbe a bounded or unbounded domain, 1<p<N and let ube the entropy solution of problem (P) with the p-Laplacian operator and f=0, then u∈Lr(0,T;MN(p−1) N−p(Ω)) for r<p−1. |∇u|∈Lr(0,T;MN(p−1) N−1(Ω)) for r<p−1. |∇u|∈Lr(0,T;Mp 2(Ω)) for r<p. This paper is divided into five sections. The next one is on preliminaries: we include the definitions of Marcinkiewicz spaces and entropy solutions of problem (P) and the first estimates on the solution and its derivative. Section 3 is devoted to prove Theorem 1 (Theorems 3.1 and 3.2), while in section 4 we prove Theorem 2 (Theorems 4.1 and 4.2). Finally, in section 5, we prove Theorem 3, and we also give other regularity results when f= 0 and the initial datum u0lies in Ms(Ω) ∩L1(Ω) or Ls(Ω) ∩L1(Ω), 1 <s<2. Regularity for entropy solutions 669 2. Preliminaries Throughout this note the Lebesgue measure on Ω will be denoted by µN. Definition 2.1. For 0 <q<∞, the set of all measurable functions u: Ω→Rsuch that the functional [u]q:= supk>0kµN{|u|>k}1/q is finite is called a Marcinkiewicz space and is denoted by Mq(Ω). It is straightforward that, for bounded Ω, we have Mq(Ω) ⊂M r(Ω) for r<q. The connection between Marcinkiewicz and Lebesgue spaces is easy: Lq(Ω) ⊂M q(Ω) ⊂Lr loc(Ω) for r<q(see, for instance, [17]); let us point out that Marcinkiewicz spaces are also known as weak-Lebesgue spaces. When q>1, the Marcinkiewicz space Mq(Ω) is a Banach space with the norm defined by uq= supt>0t1−q qt 0u∗(τ)dτ, where u∗(τ)= infk>0:µN{|u|>k}≤τdefines the non-increasing rearrangement of u(see, for instance, [17, Definition 1.8.6]). As a consequence of this definition one has that K|u|≤µN{K}q−1 quqfor any K⊂Ω with finite measure. Note that, endowed with this norm, if u, v ∈M q(Ω), then t 0u∗(τ)dτ ≤t 0v∗(τ)dτ for all t>0 implies uq≤vq; that is, Mq(Ω) is a normal space in the sense of [5, Definition 2.8]. Definition 2.2. If r,q∈]0,+∞[, we will say that a measurable function u:]0,T[×Ω→Rbelongs to Lr(0,T;Mq(Ω)) if T 0[u(t)]r qdt is finite. The following result is well known; a proof may be seen, for instance, in [2, Lemma 1.3]. Lemma 2.3. Let I⊂Rbe an interval and denote by Λthe set of all measurable functions λ:[0,T]→I.Letf:[0,T]×I→]0,+∞[ be a function such that for each λ∈Λthe function t→f(t, λ(t)) is measurable. Then T 0supk∈If(t, k)dt = supλ∈ΛT 0f(t, λ(t)) dt. Before introducing the concept of entropy solution, we will define the following functional spaces (see [3]). Given k>0, define the truncature operator by Tk(ζ)=(ζ∧k)∨(−k), whose primitive is Jk(ξ)=ξ 0Tk(s)ds. The space of all measurable functions u:Ω→Rsuch that Tku∈ W1,1 loc (Ω) and ∇Tku∈Lp(Ω) for all k>0 is denoted by T1,p(Ω) while T1,p 0(Ω) denotes the space of all functions u∈T1,p(Ω) such that for every k>0 there exists a sequence (φn)inC∞ 0(Ω) satisfying ∇φn→∇Tkuin 670 S. Segura, J. Toledo Lp(Ω) and φn→Tkuin L1 loc(Ω). If u∈T1,p(Ω), a derivative ∇ucan be defined as the unique measurable function such that ∇Tku=∇u·χ{|u|<k} for all k>0 (see [3, Lemma 2.1]). Definition 2.4. We will say that a function u∈C[0,T]; L1(Ω)is an entropy solution of (P) if u(t)∈T1,p 0(Ω) for almost all t,∇Tk(u)∈ Lp(QT) for all k>0 and t 0Ω ϕTk(u−ϕ)+t 0Ω ap(x, ∇u),∇Tk(u−ϕ) ≤Ω Jku0−ϕ(0)−Ω Jku(t)−ϕ(t)+t 0Ω fTk(u−ϕ) for all k>0, t∈[0,T] and ϕ∈L∞(QT)∩Lp0,T;W1,p 0(Ω)∩ W1,10,T;L1(Ω). Taking ϕ= 0 in the formulation of entropy solution it follows that 1 kT 0Ω ap(x, ∇u),∇Tku ≤Ω Jk(u0) k+T 0Ω fTku k≤Ω |u0|+T 0Ω |f| for all k>0. Consequently, (2.1) T 0Ω |∇Tku(t)|p k≤M, ∀k>0, being M=Ω|u0|+T 0Ω|f|. (Recall that one of the Leray-Lions assumptions asserts that there exists α>0 satisfying α|ξ|p≤ap(x, ξ),ξ; we consider α= 1 since there is not loss of generality in doing so.) From now on uwill denote the entropy solution of (P). We will finish this section giving the basic estimates in order to prove our regularity results. We will use the fact that u∈C([0,T],L 1(Ω)) and the estimate given in (2.1). Proposition 2.5. For every δ>0there is g∈L1(0,T)such that Ω |∇Tku(t)|p k=kδg(t),if t∈[0,T]and k≥1; k−δg(t),if t∈[0,T]and k≤1. Regularity for entropy solutions 671 Proof: We just prove the first assertion, the second one can be proved similarly. Consider a measurable function λ:[0,T]→[1,+∞[ and the sets Aj:= {t∈[0,T]:2 j≤λ(t)<2j+1},j≥0, and compute to get the following inequalities T 0Ω |∇Tλ(t)u(t)|p λ(t)1+δ≤ ∞  j=0 AjΩ |∇T2j+1 u(t)|p 2j(1+δ) = ∞  j=0 2 2jδ AjΩ |∇T2j+1 u(t)|p 2j+1 ≤2M ∞  j=0 1 2jδ . By Lemma 2.3, T 0 sup k≥1Ω |∇Tku|p k1+δ<∞, and now Fubini’s theorem allows us to take g(t)≥supk≥1Ω |∇Tku(t)|p k1+δ. Observe that there is no loss of generality in taking g(t)≥MN−p Nas we will do in our following estimates. Proposition 2.6. Let δ∈]0,p−1[. Then there exists a constant C>0 such that for almost all t∈[0,T], sup k≥1 kµN{|u(t)|>k}N−p N(p−1−δ)≤(Cg(t)) 1 p−1−δ (1) sup k≤1 kµN{|u(t)|>k}N−p N(p−1+δ)≤(Cg(t)) 1 p−1+δ.(2) Proof: Proposition 2.5 and Sobolev’s inequality imply two facts: on the one hand, for all k≥1 and almost all t, µN{|u(t)|≥k}≤Ω |Tku(t)|p∗ kp∗ ≤C kp∗Ω |∇Tku(t)|pN N−p ≤Cg(t)N N−p kN(p−1−δ)/(N−p), and, on the other hand, for all k≤1 and almost all t, µN{|u(t)|≥k}≤Ω |Tku(t)|p∗ kp∗ ≤C kp∗Ω |∇Tku(t)|pN N−p ≤Cg(t)N N−p kN(p−1+δ)/(N−p), from where the result follows. 672 S. Segura, J. Toledo Proposition 2.7. Let δ∈]0,p−1[. Then there exists a constant C>0 such that for almost all t∈[0,T], sup h≥g(t)1/pM−1/p hµN{|∇u(t)|>h}2+δ p≤(Cg(t))1 p.(1) sup h≥g(t)−1/(N−p) hµN{|∇u(t)|>h}N−1−δ N(p−1−δ)≤(Cg(t)) 1 p−1−δ.(2) sup h≤g(t)1/pM−1/p hµN{|∇u(t)|>h}2−δ p≤(Cg(t))1 p.(3) sup h≤g(t)−1/(N−p) hµN{|∇u(t)|>h}N−1+δ N(p−1+δ)≤(Cg(t)) 1 p−1+δ.(4) Proof: A similar argument to the one in [3, Lemma 4.2] can be applied. We just prove (1) and (2), since the other assertions can be proved in an analogous manner. Before showing (1), let us point out that the operator associated with the elliptic term −div ap(x, ∇u) + the Dirichlet boundary condition, is accretive in L1(Ω) (see [3, Theorem 7.1] or [1]); thus, u(t)1≤u01+ f1for all t∈[0,T] and so (2.2) µN{|u(t)|≥k}≤1 kΩ |u(t)|≤1 kΩ |u0|+T 0Ω |f|=M k. This fact and Proposition 2.5 imply the following inequalities. µN{|∇u(t)|>h}≤µN{|∇u(t)−∇Tku(t)|>h/2}+µN{|∇Tku(t)|>h/2} ≤µN{|u(t)|≥k}+2pΩ |∇Tku(t)|p hp≤M k+2pk1+δg(t) hp. By taking k=hpM g(t)1/(2+δ)≥1, we get µN{|∇u(t)|>h}≤(Ch−pg(t))1/(2+δ), and (1) follows. To show (2), apply Proposition 2.6 (1) to obtain kp−1−δµN{|u(t)|≥ k}N−p N≤Cg(t) for all k≥1. Then, using Proposition 2.5, µN{|∇u(t)|>h}≤µN{|u(t)|≥k}+2 pΩ |∇Tku(t)|p hp ≤Cg(t)N N−p kN(p−1−δ)/(N−p)+Ck1+δg(t) hp. Taking k=g(t)1/(N−1−δ)h(N−p)/(N−1−δ)≥1, (2) is obtained. Regularity for entropy solutions 673 3. Case Ωbounded Theorem 3.1. If 2N N+1 <p<N, then u∈Lr(0,T;Mq(Ω)) for 1<q<N(p−1) N−p and r<(N+1)p−2N N q q−1. Proof: Let 2N N+1 <p<Nand qbe as above and take δ>0 such that p>N(2+δ) N+1 and 1 <q<N(p−1−δ) N−p. It will be enough to show that u∈Lr0,T;Mq(Ω)for r=p(N+1)−N(2+δ) N q q−1. First observe that the above inequalities imply that r>p−1−δ; thus, for k≥1, we obtain that krµN{|u(t)|≥k}r/q =kp−1−δµN{|u(t)|≥k}N−p N·kr−p+1+δµN{|u(t)|≥k}r−p+1+δ. On the one hand, applying Proposition 2.6, kp−1−δµN{|u(t)|≥k}N−p N≤ Cg(t) and on the other hand, by (2.2), kµN{|u(t)|≥k}≤M, so that sup k≥1 krµN{|u(t)|≥k}r/q ≤CMr−p+1+δg(t), gbeing an integrable function. Since we also have, for 0 <k≤1, that krµN{|u(t)|≥k}r/q ≤ µN{Ω}r/q; the function defined by [u(t)]r qis integrable and consequently u∈Lr(0,T;Mq(Ω)). Theorem 3.2. (1) If 2N N+1 <p<N, then |∇u|∈Lr(0,T;Mq(Ω)) for p 2<q<N(p−1) N−1 and r<q(N+1)p−2N (N+1)q−N, for q=p 2and r<p, and for q<p 2and r=p. (2) If 1<p≤2N N+1 , then |∇u|∈Lp(0,T;Mq(Ω)) for q<p 2. 680 S. Segura, J. Toledo Theorem 5.2. Let ube the entropy solution of (P) and let 1<s<2. (1) If u0∈Ls(Ω) ∩L1(Ω), then u∈Lr(0,T;Mq(Ω)), where r≤ q(N+s)p−2N N(q−s)and s<q≤N(p+s−2) N−p, and |∇u|∈Lr(0,T;Mq(Ω)), where r≤q(N+s)p−2N (N+s)q−sN and qlies between N(p+s−2) N+s−2and sp 2. (2) If u0∈M s(Ω) ∩L1(Ω), then u∈Lr(0,T;Mq(Ω)), where r< q(N+s)p−2N N(q−s)and s<q<N(p+s−2) N−p, and |∇u|∈Lr(0,T;Mq(Ω)), for q strictly between N(p+s−2) N+s−2and sp 2, and r<q(N+s)p−2N (N+s)q−sN . To finish this paper, we prove that the above estimates are sharp in the framework of Marcinkiewicz spaces. We will show that, in the case p= 2 and u0∈M s(Ω) ∩L1(Ω), we have u∈Lr(0,T;Mq(Ω)), with r<q 2s N(q−s)and s<q< Ns N−p, but u/∈Lr(0,T;Mq(Ω)), with r=q2s N(q−s)and s<q< Ns N−p. Example 5.3. Let us consider Ω = RNand the problem of finding u∈C[0,T],L 1(Ω)such that ut−∆u=0,in ]0,T[×RN; u(0,.)=u0,in RN; where u0(x)= 1 |x|αχB(0,1)(x), with N/2<α<N. Denoting s=N/α, it is straighforward that 1 <s<2 and u0∈M s(RN)∩L1(RN), but u0/∈Ls(RN). We know that the solution of this linear problem is given by u(t, x)= 1 (4πt)N/2RN e−|x−y|2 4tu0(y)dy. Changing variables: x=ηtt/2and y=ξtt/2, we obtain T 0 [u(t)]r qdt =T 0 sup k>0 krµ{x∈RN:|u(t, x)|>k}r/q dt =T 0 sup k>0 krRN χ1/(4π)N/2RNe −|x−y|2 4tu0(y)dy>ktN/2(x)dxr/q dt =T 0 tNr 2qsup k>0 krRN χ1/(4π)N/2RNe −|η−ξ|2 4u0(ξt1/2)dξ>k(η)dηr/q dt =T 0 tNr 2qsup k>0 krµ   η∈RN:1 (4π)N/2B(0,t−1/2) e−|η−ξ|2 4 |ξ|αdξ >ktα/2   r/q dt. Regularity for entropy solutions 681 By denoting now h=kt1/2and g(t, η)= 1 (4π)N/2B(0,t−1/2) e −|η−ξ|2 4 |ξ|αdξ, we get T 0 [u(t)]r qdt =T 0 tr 2(N q−α)sup h>0 hrµ{η∈RN:g(t, η)>h}r/q dt. Define g1(η)= 1 (4π)N/2B(0,T −1/2) e−|η−ξ|2 4 |ξ|αdξ, g2(η)= 1 (4π)N/2RN e−|η−ξ|2 4 |ξ|αdξ and ci= suph>0hrµ{η∈RN:gi(η)>h}r/q,i=1,2 (observe that both constants are finite). Since g1(η)≤g(t, η)≤g2(η) for all t∈[0,T], it follows that c2T 0 tr 2(N q−α)dt ≤T 0 [u(t)]r qdt ≤c1T 0 tr 2(N q−α)dt. Thus, if r<q 2s N(q−s), then r 2N q−α>−1 and so T 0[u(t)]r qdt < +∞; however, if r=q2s N(q−s), then r 2N q−α=−1 and consequently T 0[u(t)]r qdt =+∞. Therefore, in the first case, the solution ubelongs to Lr(0,T;Mq(RN)), but it is not so in the second one. References 1. F. Andreu, J. M. Maz´ on, S. Segura de Le´ on and J. Toledo, Existence and uniqueness for a degenerate parabolic equation with L1data, Trans. Amer. Math. Soc. 351 (1999), 285–306. 2. J. Appell and P. P. Zabrejko,“Nonlinear Superposition Operators,” Cambridge Univ. Press, Cambridge, 1990. 3. Ph. B´ enilan, L. Boccardo, Th. Gallou¨ et, R. Gariepy, M. Pierre and J. L. V´ azquez,AnL1theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), 241–273. 682 S. Segura, J. Toledo 4. Ph. B´ enilan and M. G. Crandall, Regularizing effects of homogeneous evolution equations, in “Contributions to Analysis and Geometry,” (D. N. Clark et al., ed.), John Hopkins Univ. Press, Baltimore, Md., 1981, pp. 23–39. 5. Ph. B´ enilan and M. G. Crandall, Completely accretive operators, in “Semigroup Theory and Evolution Equations,” (Ph. Cl´ement, E. Mitidieri and B. de Pagter, ed.), Marcel Dekker Inc., New York, 1991, pp. 41–75. 6. Ph. B´ enilan, M. G. Crandall and A. Pazy,“Evolution Equations Governed by Accretive Operators,” forthcoming book. 7. D. Blanchard and F. Murat, Renormalized solutions of non linear parabolic problems with L1data: Existence and uniqueness, Proc. Roy. Soc. Edinbourgh Sect. A 127(6) (1997), 1137–1152. 8. L. Boccardo, A. Dall’Aglio, Th. Gallou¨ et and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal. 147 (1997), 237–258. 9. L. Boccardo and Th. Gallou¨ et, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149–169. 10. J. L. Lions,“Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non Lin´eaires,” Dunod, Gauthier-Villars, Paris, 1969. 11. A. Prignet, Existence and uniqueness of entropy solutions of parabolic problems with L1data, Nonlinear Anal. 28 (1997), 1943–1954. 12. A. Porretta, Regularity for entropy solutions of a class of parabolic equations with non regular initial datum, Dynam. Systems Appl. 7(1998), 53–71. 13. J. M. Rakotoson, Generalized solutions in a type of sets for problems with measures as data, Differential Integral Equations 6(1993), 27–36. 14. J. M. Rakotoson, Uniqueness of renormalized solutions in a T-set for the L1-data problem and the link between various formulations, Indiana Univ. Math. J. 43(2) (1994), 285–293. 15. S. Segura de Le´ on, Estimates for solutions of non-linear parabolic equations, Boll. Un. Mat. Ital. B (7) 11 (1997), 987–996. 16. X. Xu, On the initial-boundary-value problem for ut−div(|∇u|p−2 ∇u)=0,Arch. Rational Mech. Anal. 127 (1994), 319–335. Regularity for entropy solutions 683 17. W. P. Ziemer,“Weakly Differentiable Functions,” Springer-Verlag, New York, 1989. Departament d’An`alisi Matem`atica Universitat de Val`encia Dr. Moliner 50 46100 Burjassot Val`encia SPAIN e-mail: [email protected] e-mail: Jose.T[email protected] Primera versi´o rebuda el 22 d’octubre de 1998, darrera versi´o rebuda el 6 de setembre de 1999