HARNACK S INEQUALITY FOR PARABOLIC DE GIORGI CLASSES IN METRIC SPACES

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1 Advances in Differential Equations Volume 17, Numbers ), HARNACK S INEQUALITY FOR PARABOLIC DE GIORGI CLASSES IN METRIC SPACES Juha Kinnunen Department of Mathematics, P.O Box FI Aalto University, Finland Niko Marola Department of Mathematics and Statistics, University of Helsinki P.O. Box 68, FI University of Helsinki Michele Miranda Jr. Dipartimento di Matematica, University of Ferrara via Machiavelli 35, Ferrara, Italy Fabio Paronetto Dipartimento di Matematica, Università degli Studi di Padova via Trieste 63, Padova, Italy Submitted by: Emmanuele DiBenedetto) Abstract. In this paper we study problems related to parabolic partial differential equations in metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. We give a definition of parabolic De Giorgi classes and compare this notion with that of parabolic quasiminimizers. The main result, after proving the local boundedness, is a scale- and location-invariant Harnack inequality for functions belonging to parabolic De Giorgi classes. In particular, the results hold true for parabolic quasiminimizers. 1. Introduction The purpose of this paper is to study problems related to the heat equation u t u = 0 in metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. We consider a notion of parabolic De Giorgi classes and parabolic quasiminimizers with quadratic structure conditions and study Accepted for publication: May AMS Subject Classifications: 30L99, 31E05, 35K05, 35K99, 49N

2 802 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto local regularity properties of functions belonging to these classes. More precisely, we show that functions in parabolic De Giorgi classes satisfy a scaleand location-invariant Harnack inequality; see Theorem 5.7. Some consequences of the parabolic Harnack inequality are the local Hölder continuity and the strong maximum principle for the parabolic De Giorgi classes. Our assumptions on the metric space are rather standard to allow a reasonable first-order calculus; the reader should consult, e.g., Björn and Björn [3] and Heinonen [20], and the references therein. Harnack-type inequalities play an important role in the regularity theory of solutions to both elliptic and parabolic partial differential equations as it implies local Hölder continuity for the solutions. A parabolic Harnack inequality is logically weaker than an elliptic one since the reproduction at each time of the same harmonic function is a solution of the heat equation. There is, however, a well-known fundamental difference between elliptic and parabolic Harnack estimates. Roughly speaking, in the elliptic case the information of a positive solution on a ball is controlled by the infimum on the same ball. In the parabolic case a delay in time is needed: the information of a positive solution at a point and at instant t 0 is controlled by a ball centered at the same point but at later time t 0 + t 1, where t 1 depends on the parabolic equation. Elliptic quasiminimizers were introduced by Giaquinta Giusti [14] and [15] as a tool for a unified treatment of variational integrals, elliptic equations and systems, and quasiregular mappings on R n. Let Ω R n be a nonempty open set. A function u W 1,p loc Ω) is a Q-quasiminimizer, Q 1, related to the power p in Ω if u p dx Q u φ) p dx suppφ) suppφ) for all φ W 1,p 0 Ω). Giaquinta and Giusti realized that De Giorgi s method [7] could be extended to quasiminimizers, obtaining, in particular, local Hölder continuity. DiBenedetto and Trudinger [11] proved the Harnack inequality for quasiminimizers. These results were extended to metric spaces by Kinnunen and Shanmugalingam [24]. Elliptic quasiminimizers enable the study of elliptic problems, such as the p-laplace equation and p-harmonic functions, in metric spaces under the doubling property of the measure and a Poincaré inequality. Compared with the theory of p-harmonic functions we have no partial differential equation, only the variational approach is available. There is also no comparison principle nor uniqueness for the Dirichlet problem for quasiminimizers. See, e.g., J. Björn [4], Kinnunen Martio [23],

3 Harnack s inequality for parabolic De Giorgi classes 803 Martio Sbordone [28] and the references in these papers for more on elliptic quasiminimizers. Following Giaquinta Giusti, Wieser [36] generalized the notion of quasiminimizers to the parabolic setting in Euclidean spaces. A function u : Ω 0, T ) R, u L 2 1,2 loc 0, T ; Wloc Ω)), is a parabolic Q-quasiminimizer, Q 1, for the heat equation thus related to the power 2) if u φ u 2 u φ) 2 dx dt+ dx dt Q dx dt suppφ) t suppφ) 2 suppφ) 2 for every smooth compactly supported function φ in Ω 0, T ). Parabolic quasiminimizers have also been studied by Zhou [37, 38], Gianazza Vespri [13], Marchi [27], and Wang [35]. The literature for parabolic quasiminimizers is very small compared to the elliptic case. One of the goals of this work is to introduce parabolic quasiminimizers in metric measure spaces. This opens up a possibility to develop a systematic theory for parabolic problems in this generality. The present paper is using the ideas of DiBenedetto [9] and is based on the lecture notes [12] of the course held by V. Vespri in Lecce. We would like to point out that the definition for the parabolic De Giorgi classes given by Gianazza and Vespri [13] is slightly different from ours, and it seems that our class is larger. Naturally, our abstract setting causes new difficulties. For example, Lemma 2.5 plays a crucial role in the proof of Harnack s inequality. In Euclidean spaces this abstract lemma dates back to DiBenedetto Gianazza-Vespri [10], but as the proof uses the linear structure of the ambient space a new proof in the metric setting was needed. Motivation for this work is to consider the Saloff-Coste Grigor yan theorem in metric measure spaces. Grigor yan [17] and Saloff-Coste [29] observed independently that the doubling property for the measure and the Poincaré inequality are sufficient and necessary conditions for a scale-invariant parabolic Harnack inequality for the heat equation on Riemannian manifolds. Sturm [32] generalized this result to the setting of local Dirichlet spaces; such an approach works also in fractal geometries, but always when a Dirichlet form is defined. For references, see for instance Barlow Bass Kumagai [1] and also the forthcoming paper by Barlow Grigor yan Kumagai [2]. In this paper we introduce a version of parabolic De Giorgi classes that include parabolic quasiminimizers and show the sufficiency of the Saloff- Coste Grigor yan theorem in metric measure spaces without using Dirichlet spaces nor the Cheeger differentiable structure [6]. Very recently a similar

4 804 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto question has been studied for degenerate parabolic quasilinear partial differential equations in the subelliptic case by Caponga Citti Rea [5]. Their motivating example is a class of subelliptic operators associated to a family of Hörmander vector fields and their Carnot Carathéodory distance. We show that the doubling property and the Poincaré inequality implies a scale- and location-invariant parabolic Harnack inequality for functions belonging to De Giorgi classes, and thus for parabolic quasiminimizers, in general metric measure spaces. It would be very interesting to know whether also necessity holds in this setting for De Giorgi classes or for parabolic quasiminimizers; using the results contained in Sturm [33, 34], it is possible to construct a regular Dirichlet form, and then a diffusion process, on every locally compact metric space, and this combined with Sturm [32] can be used to obtain the reverse implication. This is, however, a very abstract result and it is based on Γ-convergence of non-local Dirichlet forms, with no information on the limiting Dirichlet form. Such a geometric characterization via the doubling property of the measure and a Poincaré inequality is not available for an elliptic Harnack inequality; see Delmotte [8]. The paper is organized as follows. In Section 2 we recall the definition of Newton Sobolev spaces and prove some preliminary technical results; these results are general results on Sobolev functions and are of independent interest. In Section 3 we introduce the parabolic De Giorgi classes of order 2 and define parabolic quasiminimizers. In Section 4 we prove the local boundedness of elements in the De Giorgi classes, and finally, in Section 5 we prove a Harnack-type inequality. 2. Preliminaries In this section we briefly recall the basic definitions and collect some results needed in the sequel. For a more detailed treatment we refer, for instance, to the forthcoming monograph by A. and J. Björn [3] and the references therein. Standing assumptions in this paper are as follows. By the triplet X, d, µ) we will denote a complete metric space X, where d is the metric and µ a Borel measure on X. The measure µ is supposed to be doubling; i.e., there exists a constant c 1 such that 0 < µb 2r x)) cµb r x)) < 2.1) for every r > 0 and x X. Here B r x) = Bx, r) = {y X : dy, x) < r} is the open ball centered at x with radius r > 0. We want to mention in passing that to require the measure of every ball in X to be positive

5 Harnack s inequality for parabolic De Giorgi classes 805 and finite is anything but restrictive; it does not rule out any interesting measures. The doubling constant of µ is defined to be c d := inf{c 1, ) : 2.1) holds true}. The doubling condition implies that for any x X, we have µb R x)) R ) d N µb r x)) c = 2 N R ) N, 2.2) r r for all 0 < r R with N := log 2 c d. The exponent N serves as a counterpart of dimension related to the measure. Moreover, the product measure in the space X 0, T ), T > 0, is denoted by µ L 1, where L 1 is the one-dimensional Lebesgue measure. We follow Heinonen and Koskela [21] in introducing upper gradients as follows. A Borel function g : X [0, ] is said to be an upper gradient for an extended real-valued function u on X if for all rectifiable paths γ : [0, l γ ] X, we have uγ0)) uγl γ )) g ds. 2.3) If 2.3) holds for p almost every curve, we say that g is a p weak upper gradient of u; here by p almost every curve we mean that 2.3) fails only for a curve family Γ with zero p modulus. Recall that the p modulus of a curve family Γ is defined as { } Mod p Γ = inf ϱ p dµ : ϱ 0 is a Borel function, ϱ 1 for all γ Γ. X From the definition, it follows immediately that if g is a p weak upper gradient for u, then g is a p weak upper gradient also for u k, and k g for ku, for any k R. The p-weak upper gradients were introduced in Koskela MacManus [25]. They also showed that if g L p X) is a p weak upper gradient of u, then, for any ε > 0, one can find an upper gradient g ε of u such that g ε > g and g ε g L p X) < ε. Hence for most practical purposes it is enough to consider upper gradients instead of p weak upper gradients. If u has an upper gradient in L p X), then it has a unique minimal p weak upper gradient g u L p X) in the sense that for every p weak upper gradient g L p X) of u, g u g almost everywhere; see Corollary 3.7 in Shanmugalingam [31] and Haj lasz [19] for the case p = 1. Let Ω be an open subset of X and 1 p <. Following the definition of Shanmugalingam [30], we define for u L p Ω), u p N 1,p Ω) := u p L p Ω) + inf g p L p Ω), γ γ

6 806 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto where the infimum is taken over all upper gradients of u. The Newtonian space N 1,p Ω) is the quotient space N 1,p Ω) = { u L p Ω) : u N 1,p Ω) < } /, where u v if and only if u v N 1,p Ω) = 0. If u, v N 1,p X) and v = u µ almost everywhere, then u v. Moreover, if u N 1,p X), then u v if and only if u = v outside a set of zero p capacity. The space N 1,p Ω) is a Banach space see Shanmugalingam [30, Theorem 3.7] and it is easily verified that it has a lattice structure. A function u belongs to the local Newtonian space N 1,p loc Ω) if u N 1,p V ) for all bounded open sets V with V Ω, the latter space being defined by considering V as a metric space with the metric d and the measure µ restricted to it. Newtonian spaces share many properties of the classical Sobolev spaces. For example, if u, v N 1,p loc Ω), then g u = g v almost everywhere in {x Ω : ux) = vx)}; in particular, g min{u,c} = g u χ {u c} for c R. We shall also need a Newtonian space with zero boundary values; for the detailed definition and main properties we refer to Shanmugalingam [31, Definition 4.1]. For a measurable set E X, let N 1,p 0 E) = {f E : f N 1,p X) and f = 0 p a.e. on X \ E}. The notion of p a.e. is based on sets of null p capacity; the p capacity of a set E can be defined as Cap p E = inf { u p N 1,p X) : u N 1,p X), u 1 on E This space equipped with the norm inherited from N 1,p X) is a Banach space. We shall assume that X supports a weak 1, 2) Poincaré inequality; that is, there exist constants C 2 > 0 and Λ 1 such that for all balls B ρ X, all integrable functions u on X, and all upper gradients g of u, B ρ u u Bρ dµ C 2 ρ B Λρ g 2 dµ) 1/2, 2.4) where u B := u dµ := 1 u dµ. B µb) B It is noteworthy that by a result of Keith and Zhong [22] if a complete metric space is equipped with a doubling measure and supports a weak 1, 2) Poincaré inequality, then there exists ε > 0 such that the space admits a weak 1, p) Poincaré inequality for each p > 2 ε. We shall use this fact }.

7 Harnack s inequality for parabolic De Giorgi classes 807 in the proof of Lemma 5.6, which is crucial for the proof of a parabolic Harnack inequality. For more detailed references of Poincaré inequality, see Heinonen Koskela [21] and Haj lasz Koskela [18]. In particular, in the latter it has been shown that if a weak 1, 2) Poincaré inequality is assumed, then the Sobolev embedding theorem holds true and so a weak q, 2) Poincaré inequality holds for all q 2, where for a fixed exponent p we have defined p = { pn N p, p < N, +, p N. 2.5) In addition, we have that if u N 1,2 0 B ρ), B ρ Ω, then the following Sobolev-type inequality is valid: ) 1/q ) 1/2, u q dµ c ρ gu 2 dµ 1 q 2 ; 2.6) B ρ B ρ for a proof of this fact we refer to Kinnunen Shanmugalingam [24, Lemma 2.1]. The crucial fact here for us is that 2 > 2. We also point out that since u N 1,2 0 B ρ), then the balls in the previous inequality have the same radius. The fact that a weak 1, p) Poincaré inequality holds for p > 2 ε implies also the following Sobolev-type inequality: ) 1/q ) 1/p, u q dµ Cp ρ g p dµ 1 q p, 2.7) B ρ B ρ for any function u with zero boundary values and any g upper gradient of u. The constant c depends only on c d and on the constants in the weak 1, 2) Poincaré inequality. We also point out that requiring a Poincaré inequality implies in particular the existence of enough rectifiable curves; this also implies that the continuous embedding N 1,2 L 2, given by the identity map, is not onto. We now state and prove some results that are needed in the paper; these results are stated for functions in N 1,2, but can be easily generalized to any N 1,p, 1 p < + if we assumed instead a weak 1, p) Poincaré inequality. Theorem 2.1. Assume u N 1,2 0 B ρ), 0 < ρ < diamx)/3 any ρ > 0 in case X has infinite diameter); then there exists κ > 1 such that we have ) κ 1 u 2κ dµ c 2 ρ 2 u 2 dµ gu B ρ B ρ B 2 dµ. Λρ

8 808 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto Proof. Let κ = 2 2/2, where 2 is as in the Sobolev inequality 2.6). By Hölder s inequality and 2.6), we obtain the claim κ 1 u B 2κ dµ u dµ) ) 2/2 ρ B 2 u 2 dµ ρ B ρ c 2 ρ 2 ) κ 1 u B 2 dµ gu ρ B 2 dµ. Λρ By integrating the previous inequality in time, we obtain a parabolic Sobolev inequality. Proposition 2.2. Assume u C[s 1, s 2 ]; L 2 X)) L 2 s 1, s 2 ; N 1,2 0 B ρ)). Then there exists κ > 1 such that s2 ) κ 1 s2 s 1 B ρ u 2κ dµ dt c 2 ρ 2 sup t s 1,s 2 ) B ρ ux, t) 2 dµx) We shall also need the following De Giorgi-type lemma. s 1 B ρ g 2 u dµ dt. Lemma 2.3. Let p > 2 ε and 1 q p ; moreover, let k, l R with k < l, and u N 1,2 B ρ ). Then l k)µ{u k} B ρ ) 1/q µ{u > l} B ρ ) 1/q 2C p ρµb ρ ) 2/q 1/p {k<u<l} B Λρ g p u dµ) 1/p. Remark 2.4. The previous result holds in every open set Ω X, provided that 2.6) holds with Ω in place of B ρ. Proof. Denote A = {x B ρ : ux) k}; if µa) = 0, the result is immediate; otherwise, if µa) > 0, we define { min{u, l} k, if u > k, v := 0, if u k. We have that v v Bρ q dµ = B ρ and consequently B ρ\a v v Bρ q dµ + v Bρ q dµ v Bρ q µa) A v Bρ q 1 µa) B ρ v v Bρ q dµ. 2.8)

9 Harnack s inequality for parabolic De Giorgi classes 809 On the other hand, we see that v q dµ = l k) q dµ + B ρ {u>l} B ρ {k<u l} B ρ v q dµ l k) q µ{u > l} B ρ ), and using 2.8) we obtain ) 1/q ) 1/q v q dµ v v Bρ q dµ + v Bρ q µb ρ ) B ρ B ρ µbρ ) ) 1/q. 2 v v Bρ q dµ µa) B ρ By 2.7) and the doubling property, we finally conclude that l k)µ{u > l} B ρ ) 1/q 2C p ρ µb ρ) 2/q 1/p which is the required inequality. µa) 1/q B Λρ g p v dµ ) 1/q ) 1/p, 2.9) The following measure-theoretic lemma is a generalization of a result obtained in [10] to the metric space setting. Roughly speaking, the lemma states that if the set where u N 1,1 loc X) is bounded away from zero occupies a good piece of the ball B, then there exists at least one point and a neighborhood about this point such that u remains large in a large portion of the neighborhood. In other words, the set where u is positive clusters about at least one point of the ball B. Lemma 2.5. Let x 0 X, ρ 0 > ρ > 0 with µ B ρ x 0 )) = 0 and α, β > 0. Then, for every λ, δ 0, 1) there exists η 0, 1) such that for every u X) satisfying N 1,2 loc B ρ0 x 0 ) gu 2 dµ β µb ρ 0 x 0 )) ρ 2, 0 and µ{u > 1} B ρ x 0 )) αµb ρ x 0 )), there exists x B ρ x 0 ) with B ηρ x ) B ρ x 0 ) and µ{u > λ} B ηρ x )) > 1 δ)µb ηρ x )). Remark 2.6. The assumption µ B ρ x 0 )) = 0 is not restrictive, since this property holds except for at most countably many radii ρ > 0 and we can choose the appropriate radius ρ as we like. We also point out that the two previous lemmas can also be stated for functions of bounded variation instead of Sobolev functions, once a weak 1, 1) Poincaré inequality is assumed; the

10 810 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto proofs given here can be easily adapted to this case by using the notion of the perimeter. Proof. For every η < ρ 0 ρ)/2λρ), we may consider a finite family of disjoint balls {B ηρ x i )} i I, x i B ρ x 0 ) for every i I, B ηρ x i ) B ρ x 0 ), such that B ρ x 0 ) i I B 2ηρx i ) B ρ0 x 0 ). Observe that B 2Ληρ x i ) B ρ0 x 0 ) for every i I and by the doubling property, the balls B 2Ληρ x i ) have bounded overlap with bound independent of η. We denote and I + = I = By assumption, we get { i I : µ{u > 1} B 2ηρ x i )) > { i I : µ{u > 1} B 2ηρ x i )) α 2c d µb 2ηρ x i )) } α } µb 2ηρ x i )). 2c d αµb ρ x 0 )) µ{u > 1} B ρ x 0 )) µ{u > 1} B 2ηρ x i )) + α µb 2ηρ x i )) 2c d i I + i I µ{u > 1} B 2ηρ x i )) + α µb ηρ x i )) 2 i I + i I µ{u > 1} B 2ηρ x i )) + α 2 µb 1+η)ρx 0 )) i I + and consequently α µbρ x 0 )) µb 2 1+η)ρ x 0 ) \ B ρ x 0 )) ) µ{u > 1} B 2ηρ x i )). i I + Assume for the sake of contradiction that 2.10) µ{u > λ} B ηρ x i )) 1 δ)µb ηρ x i )), 2.11) for every i I + ; this clearly implies that µ{u λ} B ηρ x i )) µb ηρ x i )) The doubling condition on µ also implies that µ{u λ} B 2ηρ x i )) µb 2ηρ x i )) δ. δ c d.

11 Harnack s inequality for parabolic De Giorgi classes 811 By Lemma 2.3 with q = 2, k = λ, and l = 1, we obtain that δ µ{u > 1} B 2ηρ x i )) µ{u λ} B 2ηρx i )) µ{u > 1} B 2ηρ x i )) c d µb 2ηρ x i )) 16C2 2 η2 ρ 2 1 λ) 2 gu 2 dµ. 2.12) {λ<u<1} B 2Ληρ x i ) Summing up over I + and using the bounded overlapping property, from 2.10) we get α 2 1 δ λ)2 µbρ x 0 )) µb c 1+η)ρ x 0 ) \ B ρ x 0 ) ) d 16C2η 2 2 ρ 2 gu 2 dµ c η 2 ρ 2 i I + B ρ0 x 0 ) {λ u<1} B 2Ληρ x i ) g 2 u dµ c βµb ρ0 x 0 ))η 2, where the constant c is given by 16C2 2 multiplied by the overlapping constant. The conclusion follows by passing to the limit with η 0, since the condition µ B ρ x 0 )) = 0 implies that the left-hand side of the previous equation tends to α 2 1 δ λ)2 µb ρ x 0 )). c d We conclude with a result which will be needed later; for the proof we refer, for instance, to [16, Lemma 7.1]. Lemma 2.7. Let {y h } h=0 be a sequence of positive real numbers such that y h+1 cb h y 1+α h, where c > 0, b > 1, and α > 0. Then if y 0 c 1/α b 1/α2, we have lim h y h = Parabolic De Giorgi classes and quasiminimizers We consider a variational approach related to the heat equation see Definition 3.3) u u = 0 in Ω 0, T ) 3.1) t and provide a Harnack inequality for a class of functions in a metric measure space generalizing the known result for positive solutions of 3.1) in the Euclidean case. The following definition is essentially based on the approach of DiBenedetto Gianazza Vespri [10] and also of Wieser [36]; we refer also to the book of Lieberman [26] for a more detailed description.

12 812 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto Definition 3.1 Parabolic De Giorgi classes of order 2). Let Ω be a nonempty open subset of X and T > 0. A function u : Ω 0, T ) R belongs to the class DG + Ω, T, γ), if u C[0, T ]; L 2 loc Ω)) L2 1,2 loc 0, T ; Nloc Ω)), and for all k R the following energy estimate holds: s2 sup u k) 2 +x, t) dµ + gu k) 2 + dµ ds 3.2) t τ,s 2 ) B rx 0 ) τ B rx 0 ) α u k) 2 +x, s 1 ) dµx) B R x 0 ) + γ α θ ) 1 R r) 2 s2 s 1 B R x 0 ) u k) 2 + dµ ds, where x 0, t 0 ) Ω 0, T ), and θ > 0, 0 < r < R, α [0, 1], s 1, s 2 0, T ), and s 1 < s 2 are so that τ, t 0 [s 1, s 2 ], s 2 s 1 = θr 2, τ s 1 = θr r) 2, and B R x 0 ) t 0 θr 2, t 0 + θr 2 ) Ω 0, T ). The function u belongs to DG Ω, T, γ) if 3.2) holds with u k) + replaced by u k). The function u is said to belong to the parabolic De Giorgi class of order 2, denoted DGΩ, T, γ), if u DG + Ω, T, γ) DG Ω, T, γ). In what follows, the estimate 3.2) given in Definition 3.1 is referred to as energy estimate or Caccioppoli-type estimate. We also point out that our definition of parabolic De Giorgi classes of order 2 is slightly different from that given in the Euclidean case by Gianazza Vespri [13]; our classes seem to be larger, but it is not known to us whether they are equivalent. Denote KΩ 0, T )) = {K Ω 0, T ) : K compact} and consider the functional E : L 2 0, T ; N 1,2 Ω)) KΩ 0, T )) R, Ew, K) = 1 gw 2 dµ dt. 2 Definition 3.2 Parabolic quasiminimizer). Let Ω be an open subset of X. A function u L 2 1,2 loc 0, T ; Nloc Ω)) is said to be a parabolic Q-quasiminimizer, Q 1, related to the heat equation 3.1) if u φ dµ dt + Eu, suppφ)) QEu φ, suppφ)) 3.3) t suppφ) for every φ Lip c Ω 0, T )) = {f LipΩ 0, T )) : suppf) Ω 0, T )}. In the Euclidean case with Lebesgue measure it can be shown that u is a weak solution of 3.1) if and only if u is a 1-quasiminimizer for 3.1); see [36]. Hence 1-quasiminimizers can be seen as weak solutions of 3.1) in metric measure spaces. This motivates the following definition. K

13 Harnack s inequality for parabolic De Giorgi classes 813 Definition 3.3. A function u is a parabolic minimizer if u is a parabolic Q-quasiminimizer with Q = 1. We also point out that the class of Q-quasiminimizers is non-empty and non-trivial, since it contains the elliptic Q-quasiminimizers as defined in [14, 15] and as shown there, there exist many other examples as well. Remark 3.4. It is possible to prove, by using the Cheeger differentiable structure and the same proof contained in Wieser [36, Section 4], that a parabolic Q-quasiminimizer belongs to a suitable parabolic De Giorgi class. We are not able to prove this result directly without using the Cheeger differentiable structure; the main problem is that the map u g u is only sublinear and not linear, and linearity is a main tool used in the argument. 4. Parabolic De Giorgi classes and local boundedness We shall use the following notation: Q + ρ,θ x 0, t 0 ) = B ρ x 0 ) [t 0, t 0 + θρ 2 ), Q ρ,θ x 0, t 0 ) = B ρ x 0 ) t 0 θρ 2, t 0 ], Q ρ,θ x 0, t 0 ) = B ρ x 0 ) t 0 θρ 2, t 0 + θρ 2 ). When θ = 1 we shall simplify the notation by writing Q + ρ x 0, t 0 ) = Q + ρ,1 x 0, t 0 ), Q ρ x 0, t 0 ) = Q ρ,1 x 0, t 0 ), Q ρ x 0, t 0 ) = Q ρ,1 x 0, t 0 ). We shall show that functions belonging to DGΩ, T, γ) are locally bounded. Here we follow the analogous proof contained in [24] for the elliptic case. Consider r, R > 0 such that R/2 < r < R, s 1, s 2 0, T ) with 2s 2 s 1 ) = R 2, and σ s 1, s 2 ) such that σ < s 1 + s 2 )/2; fix x 0 X. We define level sets at scale ρ > 0 as follows: Ak; ρ; t 1, t 2 ) := {x, t) B ρ x 0 ) t 1, t 2 ) : ux, t) > k}. Let r := R + r)/2, i.e., R/2 < r < r < R, and η Lip c B r ) such that 0 η 1, η = 1 on B r, and g η 2/R r). Then v = u k) + η N 1,2 0 B r) and g v g u k)+ + 2u k) + /R r). We have u k) 2 + dµ dt 2 N u k) 2 +η 2 dµ dt B r σ,s 2 ) B r σ,s 2 ) 2 N µ L 1 B r σ, s 2 )) B r σ,s 2 ) u k) q + ηq dµ dt ) 2/q

14 814 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto ) q 2)/q µ L 1 Ak; r; σ, s 2 ) 2 N µ L 1 Ak; r; σ, s 2 )) ) q 2)/q 2/q. µ L 1 u k) q + B r σ, s 2 )) ηq dµ dt) B r σ,s 2 ) We now use Proposition 2.2 taking q = 2κ. We get u k) 2 + dµ dt 2 N 2 2/κ c 2/κ r 2/κ µ L 1 Ak; r; σ, s 2 )) B r σ,s 2 ) µ L 1 B r σ, s 2 )) κ 1)/κ sup u k) 2 +η dµ) ) 1/κ. 2 gu k) 2 + η t σ,s 2 ) B r dµ dt B r σ,s 2 ) ) κ 1)/κ By applying 3.2) with τ = σ, α = 0, and θ = 1/2, since κ > 1, we arrive at u k) 2 + dµ dt 2 N+2 c 2/κ r 2/κ µ L 1 Ak; r; σ, s 2 )) ) κ 1)/κ B r σ,s 2 ) s 2 σ) 1/κ µ L 1 B r σ, s 2 )) sup t σ,s 2 ) + ) κ 1)/κ u k) 2 +x, t) dµ 2 B r 8 R r) 2 s2 σ B r u k) 2 + dµ dt s2 σ ) 1/κ 2 N+2 c 2/κ r 2/κ µ L 1 Ak; r; σ, s 2 )) ) κ 1)/κ µbr ) s 2 σ) 1/κ µ L 1 B r σ, s 2 )) µb r ) 6γ + s2 22N+3 R r) 2 u k) B 2 + dµ dt. R s 1 B r g 2 u k) + dµ dt By the the choice of σ, we see that s 2 σ) 1 < 2s 2 s 1 ) 1, and consequently u k) 2 + dµ dt 4.1) B r σ,s 2 ) ) κ 1)/κ 2 N+4 3γ + 2 2N+2 )c 2/κ r 2/κ µ L 1 Ak; r; σ, s 2 )) µ L 1 B r σ, s 2 )) µb R) µb r ) s 2 σ) κ 1)/κ 1 R r) 2 u k) 2 + dµ dt. B R s 1,s 2 ) Consider h < k. Then k h) 2 µ L 1 Ak; r; σ, s 2 )) ) u h) 2 + dµ dt Ah; r;σ,s 2 ) u h) 2 + dµ dt = Ak; r;σ,s 2 ) s2 σ B r u h) 2 + dµ dt,

15 Harnack s inequality for parabolic De Giorgi classes 815 from which, using the doubling property 2.2), it follows that µ L 1 1 ) Ak; r; σ, s 2 )) k h) 2 µ L 1 B r σ, s 2 )) uh; r; σ, s 2 ) 2 4.2) where 2N+1 k h) 2 µ L 1 B r σ, s 2 )) ) uh; R; s 1, s 2 ) 2, 1/2. ul; ρ; t 1, t 2 ) := u l) 2 + dµ dt) B ρ t 1,t 2 ) By plugging 4.2) into 4.1) and arranging terms we arrive at uk; r; σ, s 2 ) c r1/κ s 2 σ) κ 1)/2κ k h) κ 1)/κ R r) uk; R; s 1, s 2 )uh; R; s 1, s 2 ) κ 1)/κ, 4.3) with c = 2 N+2+N+1)κ 1)/2κ) 3γ + 2 2N+2 ) 1/2 c 1/κ. Let us consider the following sequences: for n N, k 0 R, and fixed d we define k n := k 0 + d 1 1 ) 2 n k 0 + d, r n := R 2 + R 2 n+1 R 2, and σ n := s 1 + s 2 R2 2 4 n+1 s 1 + s 2. 2 This is possible since 2s 2 s 1 ) = R 2. The following technical result will be useful for us. Lemma 4.1. Let u 0 := uk 0 ; R; s 1, s 2 ), u n := uk n ; r n ; σ n, s 2 ), θ := κ 1 κ, a := 1 + θ θ = 1 + κ κ 1, and d θ = c 2 1+θ/2+a1+θ) u θ 0, where c is the constant in 4.3). Then u n u ) 2an Proof. We prove the lemma by induction. First notice that 4.4) is true for n = 0. Then assume that 4.4) is true for fixed n N. In 4.3), we first estimate r 1/κ s 2 σ) κ 1)/2κ by R 1/κ s 2 s 1 ) κ 1)/2κ. Then we replace r with r n+1, R with r n, σ with σ n+1, s 1 with σ n, h with k n, and k with k n+1. With these replacements we arrive at u n+1 c R 1/κ s 2 s 1 ) κ 1)/2κ) k n+1 k n ) κ 1)/κ r n r n+1 ) u1+κ 1)/κ n.

16 816 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto Denote c := c R 1/κ s 2 s 1 ) κ 1)/2κ) so that we have u n+1 c u 1+θ n k n+1 k n ) θ r n r n+1 ). Since r n r n+1 = 2 n+2) R and k n+1 k n = 2 n+1) d, we obtain u n+1 c 2n+1)θ+n+2 d θ u 1+θ n R As 2s 2 s 1 ) = R 2, we have = 2c 2n+1)1+θ) d θ u 1+θ n 2c 2n+1)1+θ) R d θ R u0 2 an ) 1+θ. c := 2c R = 2 c R1/κ s 2 s 1 ) κ 1)/2κ) 1 R = 21+θ/2 c, the point being that the constant c is independent of R, s 1, and s 2. Finally, since 1 a)1 + θ) = a we arrive at 2n+1)1+θ) u n+1 c This completes the proof. d θ u0 2 an ) 1+θ = 2 an+1) u 0. Before proving the main result of this section, we need the following proposition. Proposition 4.2. For every number k 0 R we have where d is defined as in Lemma 4.1. uk 0 + d; R/2; s 1 + s 2 )/2, s 2 ) = 0, Proof. Since k n k 0 + d, R/2 r n R, and s 1 σ n s 1 + s 2 )/2, the doubling property implies that 0 uk 0 + d; R/2; s 1 + s 2 )/2, s 2 ) 2 N+1 uk n ; r n ; σ n, s 2 ) = u n. By Lemma 4.1, we have lim n u n = 0, and the claim follows. We close this section by proving local boundedness for functions in the De Giorgi class. Theorem 4.3. Suppose u DGΩ, T, γ). Then there is a constant c depending only on c d, γ, and the constants in the weak 1, 2) Poincaré inequality, such that for all B R s 1, s 2 ) Ω 0, T ), we have 1/2. ess sup u c u 2 dµ dt) B R/2 s 1 +s 2 )/2,s 2 ) B R s 1,s 2 )

17 Harnack s inequality for parabolic De Giorgi classes 817 Proof. Proposition 4.2 implies that ess sup u k 0 + d, B R/2 s 1 +s 2 )/2,s 2 ) where d is defined in Lemma 4.1. Then 1/2, ess sup u k 0 + c u k 0 ) 2 +dµ dt) B R/2 s 1 +s 2 )/2,s 2 ) B R s 1,s 2 ) with c = c 1/θ 2 1+θ/2+a1+θ), c the constant in 4.3). The previous inequality with k 0 = 0 can be written as follows: ess sup u c B R/2 s 1 +s 2 )/2,s 2 ) B R s 1,s 2 ) u 2 +dµ dt ) 1 2 c u 2 dµ dt B R s 1,s 2 ) Since also u DGΩ, T, γ) the analogous argument applied to u gives the claim. 5. Parabolic De Giorgi classes and Harnack inequality In this section we shall prove a scale-invariant parabolic Harnack inequality for functions in the De Giorgi class of order 2 and, in particular, for parabolic quasiminimizers. Proposition 5.1. Let ρ, θ > 0 be chosen such that the cylinder Q ρ,θ y, s) Ω 0, T ). Then for each choice of a, σ 0, 1) and θ 0, θ), there is ν +, depending only on N, γ, c, a, θ, and θ, such that for every u DG + Ω, T, γ) and m + and ω for which m + ess sup Q ρ,θ y,s) u and ω osc Q ρ,θ y,s) u, the following claim holds true: If ) ) µ L 1 {x, t) Q ρ,θ y, s) : ux, t) > m + σω} ν + µ L 1 Q ρ,θ y, s), then ux, t) m + aσω µ L 1 -a.e. in B ρ/2 y) s θρ 2, s]. Proof. Define the following sequences, h N: ρ h := ρ 2 + ρ 2 h+1 ρ 2, θ h = θ h θ θ) θ, B h := B ρh y), s h := s θ h ρ 2 s θρ 2, Q h := B h s h, s], σ h := aσ + 1 a 2 h σ aσ, and k h = m + σ h ω m + aσω. ) 1 2.

18 818 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto Consider a sequence of Lipschitz-continuous functions ζ h, h N, satisfying the following: ζ h 1 in Q h+1, ζ h 0 in Q ρ,θ y, s) \ Q h, g ζh 1 ρ h ρ h+1 = 2h+2 ρ, 0 ζ h) t 2h+1 θ θ 1 ρ 2. Denote A h := {x, t) Q h : ux, t) > k h}. We have u k h ) 2 +ζh 2 dµ dt u k h ) 2 + dµ dt u k h ) 2 + dµ dt A h+1 Q h Q h+1 k h+1 k h ) Ah+1 2 dµ dt = 1 a)σω)2 2 2h+2 µ L 1 A h+1 ), and consequently s u k h ) 2 +ζ 2 1 a)σω)2 µ L 1 A h+1 ) h dµ dt B h 2 2h ) µb h+1 ) s h On the other hand, if we use first Hölder s inequality and then Proposition 2.2, we obtain the following estimate: s µ L u k h ) 2 +ζh 2 1 A h ) ) κ 1 s ) 1 κ dµ dt u k h ) 2κ + ζh 2κ s h B h µb h ) dµ dt κ s h B h c 2/κ ρ 2/κ µ L 1 A h ) ) ) κ 1)/κ κ 1)/κ sup u k h ) 2 µb h ) +ζh 2 dµ t s h,s) B h s ) ) 1/κ 2 ζh 2 g2 u k h ) + + 2gζ 2 h u k h ) 2 + dµ dt s h B h c 2/κ ρ 2/κ µ L 1 A h ) µb h ) s 2 ) κ 1)/κ 1 µb h ) sup t s h,s) s ) κ 1)/κ u k h ) 2 + dµ B h ) 1/κ g s h Bh 2u kh)+ dµ dt + 22h+5 ρ 2 u k h ) 2 + dµ dt s h B h 2 1/κ c 2/κ ρ 2/κ µ L 1 A h ) ) κ 1)/κ 1 sup u k h ) 2 + dµ + µb h ) µb h ) t s h,s) B h s s ) + g Bh 2u kh)+ dµ dt + 22h+4 ρ 2 u k h ) 2 + dµ dt. B h s h s h

19 Harnack s inequality for parabolic De Giorgi classes 819 We continue by applying the energy estimate 3.2) with r = ρ h, R = ρ h 1, α = 0, s 2 = s, τ = s h, and s 1 = s h 1 and get s u k h ) 2 +ζh 2 dµ dt s h B h 2 1/κ c 2/κ ρ 2/κ µ L 1 A h ) ) κ 1)/κ 1 2 2h+4 s µb h ) µb h ) ρ 2 u k h ) 2 + dµ dt s h B h + γ 1 + 2h ) 2 2h+2 s ) θ θ ρ 2 u k h ) 2 + dµ dt s h 1 B h 1 C 1 ρ 2/κ µ L 1 A h ) ) κ 1)/κ 1 2 3h+4 µb h ) µb h ) ρ 2 u k h ) 2 + dµ dt, Q h 1 where C 1 = 2 1/κ c 2/κ 1+γ+γ/θ θ)). We also have that u k h m + k h = σ h ω, and then u k h ) 2 + dµ dt µ L 1 A h 1 )σ h ω) 2 µ L 1 A h 1 )σω) 2. Q h 1 This implies that s u k h ) 2 +ζh 2 dµ dt B h s h 2 2h+4 C 1 σω) 2 1 ρ 2 κ 1 κ µ L 1 A h ) ) κ 1 κ µ L 1 A h 1 ) µb h ) µb h ) 2 2h+4 C 1 σω) 2 θ κ 1 κ 2 N1+ κ 1 ) µ L 1 A h 1 ) κ µ L 1 B h 1 ) ) κ 1 κ µ L 1 A h 1 ) µb h 1 ) where we have estimated µb h 1 )/µb h ) 2 N. By the last inequality and 5.1), if we call C 2 the constant C 1 θ κ 1 κ 1 N1+ κ 2 κ )+6 1 a) 2, we obtain µ L 1 A h+1 ) µb h+1 ) C 2 2 4h µ L 1 A h 1 ) µ L 1 B h 1 ) ) κ 1 κ µ L 1 A h 1 ) µb h 1 ) finally, dividing by s s h+1 and since s s h 1 )/s s h+1 ) θ/ θ, we can summarize what we have obtain by writing y h+1 C 3 2 4h y 1+κ 1)/κ h 1, 5.2) where we have defined y h := µ L1 A h ) and C µ L 1 Q h ) 3 = C 2 ; i.e., θ θ C 3 = 2 1/κ c 2/κ 1 + γ + γ ) θ θ θ κ 1 κ 2 N1+ κ 1 κ )+6 1 θ 1 a) 2 θ. ;,

20 820 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto We observe that the hypotheses of Lemma 2.7 are satisfied with c = C 3, b = 2 4, and α = κ 1)/κ. Then if y 0 c 1/α b 1/α2 we would be able to conclude, since {y h } h is a decreasing sequence, that lim h y h = 0, since y 0 = µ L 1 A 0 )/µ L 1 Q 0 ), where Q 0 = B ρy) s θρ 2, s], and A 0 = {x, t) Q 0 : ux, t) > m + σω}. To do this it is sufficient to choose ν + to be ν + = C κ/κ 1) 3 16 κ2 /κ 1) 2. By definition of y h and A h we see that u m + aσω µ L 1 -a.e. in B ρ/2 y) s θρ 2, s ], which completes the proof. An analogous argument proves the following claim. Proposition 5.2. Let ρ, θ > 0 be chosen such that the cylinder Q ρ,θ y, s) Ω 0, T ). Then for each choice of a, σ 0, 1) and θ 0, θ), there is ν, depending only on N, γ, c, a, θ, and θ, such that for every u DG Ω, T, γ) and m + and ω for which m ess inf Q ρ,θ y,s) u and ω osc Q ρ,θ y,s) u, the following claim holds true: If ) ) µ L 1 {x, t) Q ρ,θ y, s) : ux, t) < m + σω} ν µ L 1 Q ρ,θ y, s), then ux, t) m + aσω µ L 1 -a.e. in B ρ/2 y) s θρ 2, s]. Proof. It is sufficient to argue as in the proof of Proposition 5.1 considering u ˆk h ) in place of u k h ) +, where ˆk h = m + σ h ω. The next result is the so-called expansion of positivity. Following the approach of DiBenedetto [9] we show that pointwise information in a ball B ρ implies pointwise information in the expanded ball B 2ρ at a further time level. Proposition 5.3. Let x, t ) Ω 0, T ) and ρ > 0 with B 5Λρ x ) [t ρ 2, t + ρ 2 ] Ω 0, T ). Then there exists θ 0, 1), depending only on γ, such that for every ˆθ 0, θ) there exists λ 0, 1), depending on θ and ˆθ, such that for every h > 0 and for every u DGΩ, T, γ) the following is valid: If ux, t ) h µ-a.e. in B ρ x ),

21 then Harnack s inequality for parabolic De Giorgi classes 821 ux, t) λh µ-a.e. in B 2ρ x ), for every t [t + ˆθρ 2, t + θρ 2 ]. From now on, let us denote A h,ρ x, t ) := {x B ρ x ) : ux, t ) < h}. Remark 5.4. Let x, t ) Ω 0, T ) and h > 0 be fixed. Then if ux, t ) h for µ-a.e. x B ρ x ) we have that A h,4ρ x, t ) B 4ρ x ) \ B ρ x ). The doubling property implies µa h,4ρ x, t )) 1 1 ) 4 N µb 4ρ x )). The proof of Proposition 5.3 requires some preliminary lemmas. Lemma 5.5. Given x, t ) for which B 4ρ x ) [t, t + θ ρ 2 ] Ω 0, T ), there exist η 0, 1) and θ 0, θ) such that, given h > 0 and u 0 in DGΩ, T, γ) for which holds then ux, t ) h µ-a.e. in B ρ x ), µa ηh,4ρ x, t)) < for every t [t, t + θρ 2 ] N+1 ) µb 4ρ x )) Proof. We may assume that h = 1; otherwise, we consider the scaled function u/h. We apply the energy estimate of Definition 3.1 with R = 4ρ, r = 4ρ1 σ), s 1 = t, s 2 = t + θρ 2 with θ to be chosen, τ = t, σ 0, 1), and α = 1. This gives us t + θρ 2 sup t <t<t + θρ 2 B 4ρ x ) B 4ρ1 σ) x ) u 1) 2 x, t) dµx) + u 1) 2 x, t ) dµx) + γ 16σ 2 ρ 2 t t + θρ 2 t B 4ρ1 σ) x ) B 4ρ x ) g 2 u 1) dµ dt u 1) 2 dµ dt. Since u 1 in B ρ x ), we deduce from Remark 5.4 that µ{x B 4ρ x ) : ux, t ) < 1}) < 1 1 ) 4 N µb 4ρ x )). Notice that u 1) 1; thus we have in particular u 1) 2 x, t) dµx) sup t <t<t + θρ 2 B 4ρ1 σ) x )

22 822 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto B 4ρ x ) u 1) 2 x, t ) dµx) + γ 16σ 2 ρ 2 t + θρ ) 4 N µb 4ρ x )) + γ t + θρ 2 16σ 2 ρ 2 t B 4ρ x ) 1 1 ) 4 N µb 4ρ x )) + γ θ 16σ 2 µb 4ρx )). Writing A h,ρ t) in place of A h,ρ x, t), decomposing t B 4ρ x ) u 1) 2 dµ dt u 1) 2 dµ dt A η,4ρ t) = A η,4ρ1 σ) t) {x B 4ρ x ) \ B 4ρ1 σ) x ) : ux, t) < η}, and using the doubling property we have µa η,4ρ t)) µa η,4ρ1 σ) t)) + µ B 4ρ x ) \ B 4ρ1 σ) x ) ). On the other hand, u 1) 2 x, t) dµx) B 4ρ1 σ) x ) A η,4ρ1 σ) t) Finally, we obtain u 1) 2 x, t) dµx) 1 η) 2 µa η,4ρ1 σ) t)). µa η,4ρ t)) µa η,4ρ1 σ) t)) + µ B 4ρ x ) \ B 4ρ1 σ) x ) ) 5.3) ) 1 η) 2 u 1) 2 x, t) dµ + µ B 4ρ x ) \ B 4ρ1 σ) x ) B 4ρ1 σ) x ) 1 η) N + γ θ 16σ 2 ) µb 4ρ x )) + µ B 4ρ x ) \ B 4ρ1 σ) x ) ). If the claim of the lemma were false, then for every θ, η 0, 1) there exists t [t, t + θρ 2 ] for which µa η,4ρ t)) 1 1 ) ) 4 N+1 µ B 4ρ x ). Applying this last estimate, then 5.3) for t = t and dividing by µ B 4ρ x ) ) we would have N+1 ) 1 η) N + γ θ 16σ 2 ) + µb 4ρx ) \ B 4ρ1 σ) x )) µb 4ρ x. )) Choosing, for instance, θ = σ 3 and letting η and σ go to zero we would have the contradiction 1 4 N N.

23 Harnack s inequality for parabolic De Giorgi classes 823 Lemma 5.6. Assume u DGΩ, T, γ), u 0. Let θ be as in Lemma 5.5 and h > 0. Consider x, t ) in such a way that B 5Λρ x ) [t θρ 2, t + θρ 2 ] Ω 0, T ) and assume that ux, t ) h, µ-a.e. x B ρ x ). Then for every ɛ > 0 there exists η 1 0, 1), depending on ɛ, c d, γ, θ, and the constant in the weak Poincaré inequality, such that µ L 1 {x, t) B 4ρ x ) [t, t + θρ ) 2 ] : ux, t) < η 1 h} < ɛµ L 1 B 4ρ x ) [t, t + θρ ) 2 ]. Proof. Apply the energy estimate 3.2) in B 5Λρ x ) t 2 θρ 2, t ) with R = 5Λρ, r = 4Λρ, s 2 = t + θρ 2, s 1 = t θρ 2, τ = t, and α = 0, at the level k = ηh2 m, where η > 0 and m N. We obtain t + θρ 2 t B 4ρ x ) g 2 B 4Λρ x u ηh ) γ θ γ θ 2 m ) dµ dt 5.4) t + θρ 2 ) 1 Λρ) 2 t θρ 2 B 5Λρ x ) ) 1 η 2 h 2 Λρ) 2 2 2m 2 θρ 2 µ B 5Λρ x )) u ηh ) 2 2 m dµ dt γ2 θ + 1) η2 h 2 2 2m µ B 5Λρx )). To simplify notation, let us write A h,ρ t) instead of A h,ρ x, t). Lemma 2.3 with parameters k = ηh/2 m, l = ηh/2 m 1, q = 1, and 2 ε < p < 2, implies u ηh ) ηh 2 m x, t) dµ 2 m µa ηh2 m,4ρt)) 5.5) 8C p ρµb 4ρ x )) 2 1/p µb 4ρ x ) \ A ηh2,4ρτ)) g p x, t) dµ m+1 u Ãt) ηh 2 m ) ) 1/p, for every t [t, t + θρ 2 ], where Ãt) := A ηh2 m+1,4λρt) \ A ηh2 m,4λρt). Clearly, B 4ρ x ) \ A ηh2 m+1,4ρt) B 4ρ x ) \ A ηh,4ρ t). If we choose η so that it satisfies the hypothesis of Lemma 5.5 and write µb 4ρ x ) \ A ηh,4ρ t)) + µa ηh,4ρ t)) = µb 4ρ x )),

24 824 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto then we deduce that µb 4ρ x ) \ A ηh,4ρ t)) > 4 N 1 µb 4ρ x )) for every t [t, t + θρ 2 ]. We finally arrive at u ηh ) B 4ρ x ) 2 m x, t) dµ ) 1/p. 8C p 4 N+1 µb 4ρ x )) 1 1/p ρ g p x, t) dµ u ηh 2 m ) Ãt) Integrating this with respect to t and defining the decreasing sequence {a m,ρ } m=0 as a m,ρ : = t + θρ 2 t µ A ηh2 m,ρt) ) dt = µ L 1{ x, t) B ρ x ) [t θρ 2, t ] : ux, t) < ηh }) 2 m, we get by the Hölder inequality t + θρ 2 u ηh ) t B 4ρ x ) 2 m x, t) dµ dt 5.6) 8C p 4 N+1 µb 4ρ x )) 1 1/p t + θρ 2 ) 1/p ρ g p x, t) dµ dt t u Ãt) ηh 2 m ) 8C p 4 N+1 µb 4ρ x )) 1 1/p t + θρ 2 ) 1/2 ρ g 2 dµ dt u ηh 2 m ) On the other hand, t + θρ 2 t a m 1,4Λρ a m,4λρ ) 2 p)/2. B 4ρ x ) t u ηh 2 m ) B 4Λρ x ) x, t) dµ dt ηh from which, using first 5.6) and then 5.4), we obtain a 2/2 p) m+1,4ρ ca m 1,4Λρ a m,4λρ ), 2 m+1 a m+1,4ρ where c = C 2 p16 N+3 γ2 θ + 1)µB 4ρ x )) 21 1/p) µb 5Λρ x ))ρ 2 ) 1/2 p). Hence, for every m N we have m m=1 a 2/2 p) m+1,4ρ ca 0,4Λρ a m,4λρ).

25 Harnack s inequality for parabolic De Giorgi classes 825 Since {a m,ρ } m=0 is decreasing the sum m=1 a2/2 p) m+1,4ρ converges, and consequently lim m a m,4ρ = 0. This completes the proof. Proof of Proposition 5.3. The proof is a direct consequence of Proposition 5.2 used with the right parameters. Fix θ = 1 and let θ be as in Lemma 5.5; choose also ˆθ 0, θ) and let ν be the constant in Proposition 5.2 determined by these parameters and a = 1/2. Apply Lemma 5.6 with ɛ = ν and obtain the constant η 1 for which the assumptions of Proposition 5.2 are satisfied with y = x, s = t + θρ 2, θ := θ ˆθ, m = 0, and σ = η 1 h ω. This concludes the proof with λ = 1 2 η 1. The following is the main result of this paper. Theorem 5.7 Parabolic Harnack). Assume u DGΩ, T, γ), u 0. For any constant c 2 0, 1], there exists c 1 > 0, depending on c d, γ, and the constants in the weak 1, 2) Poincaré inequality, such that for any Lebesgue point x 0, t 0 ) Ω 0, T ) with B 5Λρ x 0 ) t 0 ρ 2, t 0 + 5ρ 2 ) Ω 0, T ) we have ux 0, t 0 ) c 1 ess inf B ρx 0 ) ux, t 0 + c 2 ρ 2 ). 1 γ As a consequence, u is locally α-hölder continuous with α = log 2 γ and satisfies the strong maximum principle. Proof. Suppose t 0 = 0; up to rescaling, we may write ux 0, 0) = ρ ξ for some ξ > 0 to be fixed later. Define the functions Ms) = sup u, N s) = ρ s) ξ, s [0, ρ). Q s x 0,0) Let us denote by s 0 [0, ρ) the largest solution of Ms) = N s). Define M := N s 0 ) = ρ s 0 ) ξ, and fix y 0, τ 0 ) Q s 0 x 0, 0) in such a way that 3M 4 < sup u M, 5.7) Q ρ 0 /4 y 0,τ 0 ) where ρ 0 = ρ s 0 )/2; this implies that Q ρ 0 y 0, τ 0 ) Q ρ+s 0 )/2 x 0, 0), as well as that ρ + s0 ) sup u sup u < N = 2 ξ M. Q ρ 0 y 0,τ 0 ) Q ρ+s 0 )/2 x 2 0,0) Let us divide the proof into five steps.

26 826 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto Step 1. We assert that µ L 1{ x, t) Q ρ 0 /2 y 0, τ 0 ) : ux, t) > M }) ) > ν + µ L 1 Q 2 ρ 0 /2 y 0, τ 0 ), 5.8) where ν + is the constant in Proposition 5.1. To see this, assume to the contrary that equation 5.8) is not true. Then set k = 2 ξ M and m + = ω = k, θ = 1, ρ = ρ 0 2, σ = 1 2 ξ 1, and a = σ ). We obtain from 2 ξ+2 Proposition 5.1 that u 3M 4 in Q ρ 04 y 0, τ 0 ), which contradicts 5.7). and Step 2. We show that there exists t τ 0 ρ2 0 4, τ 0 ν + ρ ] such that { µ x B ρ0 /2y 0 ) : ux, t) M }) > ν µb ρ 0 /2y 0 )), 5.9) B ρ0 /2y 0 ) gux, 2 t) dµx) α µb ρ 0 y 0 )) ρ 2 k 2, 5.10) 0 for some sufficiently large α > 0. For this, we define the sets At), I, and J α as follows: { At) := x B ρ0 /2y 0 ) : ux, t) M }, 2 and J α := I := { t τ 0 ρ2 0 4, τ 0] : µat)) > ν } + 2 µb ρ 0 /2y 0 )), { t τ 0 ρ2 0 4, τ 0] : B ρ0 /2y 0 ) From 5.8) we have that gux, 2 t) dµx) α µb ρ 0 y 0 )) ρ 2 k 2}. 0 τ0 ν + µ L 1 Q ρ 0 /2 y 0, τ 0 )) < µat)) dt τ 0 ρ 2 0 /4 = µat))dt + µat)) dt I τ 0 ρ 2 0 /4]\I µb ρ0 /2y 0 )) I + ν + 2 µ L1 Q ρ 0 /2 y 0, τ 0 )) 4 ) = µ L 1 Q ρ 0 /2 y 0, τ 0 )) I ρ 2 + ν )

27 Harnack s inequality for parabolic De Giorgi classes 827 This implies the following lower bound I ν +ρ On the other hand, if we apply 3.2) with R = ρ 0, r = ρ 0 /2, α = 0, and θ = 1, we obtain gu 2 dµ dt = gu k) 2 dµ dt 5.11) Q ρ 0 /2 y 0,τ 0 ) 8γ ρ 2 0 Q ρ 0 y 0,τ 0 ) Q ρ 0 /2 y 0,τ 0 ) u k) 2 dµ dt 8γk2 µ L 1 Q ρ 0 y 0, τ 0 )) = 8γk 2 µb ρ0 y 0 )). This estimate implies 4γk 2 µb ρ0 y 0 )) ρ 2 0 τ 0 ρ 2 0 /4,τ 0] which in turn gives us J α ρ2 0 4 Choosing α = 64γ/ν +, we obtain dt gux, 2 t) dµ B ρ0 /2y 0 ) ) α µb ρ 0 y 0 )) ρ 2 k 2 ρ J α 1 16γ α ). I J α = I + J α I J α ν +ρ Then if we set T = τ 0 ρ2 0 4, τ 0 ν + ρ ], we get I J α T = I J α + T I J α ) T ρ2 0 ν + 4 8, and in particular I J α T. Step 3. We fix t T ; by Lemma 2.5 we have that for any δ 0, 1), there exist x B ρ0 /2y 0 ) and η 0, 1) such that { µ u, t) > M } ) B 4 ηρ0 /2x ) > 1 δ)µb ηρ0 /2x )). 5.12) Step 4. We show that for ε > 0 to be fixed, there exists x such that Q + εηρ 0 /4 x, t) Q ρ 0 y 0, τ 0 ) and µ L 1{ u M } ) Q + 8 εηρ 0 /4 x, t) ) 4 N+1 γε 2 + δ)µ L 1 Q + εηρ 0 /4 x, t). 5.13) Indeed, consider the cylinder Q = B ηρ0 /4x ) t, t+t ] with t = εηρ 0 /4) 2. Using the energy estimate 3.2) on Q with k = M/4, R = ηρ 0 /2, r = R/2,,

28 828 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto and α = 1, we obtain together with 5.12) that for any s t, t + t ] u M ) 2 x, s) dµx) B ηρ0 /4x ) 4 t+t dt 16γ η 2 ρ t B ηρ0 /2x ) B ηρ0 /2x ) ) 2 u M 4 u M ) 2 x, t) dµx) 4 x, t) dµx) M 2 16 γε2 + δ)µb ηρ0 /2x )). Define Bt) = { x B ηρ0 /4x ) : ux, t) M } 8, and we have that M ) 2 u 4 x, s) dµx) M ) 2 M 2 u x, s) dµx) 4 64 µbs)). B ηρ0 /4x ) Bs) Putting the preceding two estimates together we arrive at µbs)) 4γε 2 + δ)µb ηρ0 /2x )) for every s t, t + t ]. Integrating this inequality over s we obtain the estimate µ L 1{ u M } ) Q + 8 εηρ 0 /4 x, t) ) 2 N+2 γε 2 + δ)µ L 1 Q + εηρ 0 /4 x, t). We have to apply Proposition 5.1; we then have that there exists x so that Q + εηρ 0 /4 x, t) Q ρ 0 y 0, τ 0 ) satisfying equation 5.13). To see this, we take a disjoint family of balls {B εηρ0 /4x j )} m j=1 such that B εηρ 0 /4x j ) B ηρ0 /4x ) for every j = 1,..., m, and m B ηρ0 /4x ) B εηρ0 /2x j ). j=1 Given this disjoint family, there exists j 0 such that 5.13) is satisfied with x = x j0. Otherwise we would get a contradiction summing over j = 1,..., m. Step 5. Due to our construction, we are able to state osc u k = 2 ξ M. Q + εηρ 0 /4 x, t) We also have that if s = t + εηρ 0 /4) 2 then Q + εηρ 0 /4 x, t) = Q εηρ 0 /4 x, s); we apply Proposition 5.2 with ρ = εηρ 0 4, θ = 1, m = 0, ω = k, a = 1 2, and

29 Harnack s inequality for parabolic De Giorgi classes 829 σ = 2 ξ 3, so we can deduce that there exists ν > 0 such that if µ L 1{ u M } ) Q εηρ0/4 8 x, s) ν µ L 1 Q εηρ 0 /4 x, s)) 5.14) then ux, t) M 16, µ L1 -a.e. in Q r x, s), where r = εηρ 0 /8. Fix ε and δ in 5.13) small enough so that 5.14) is satisfied and t + εηρ/4) 2 < 0. With this choice of δ, we obtain the constants η and r that depend only on δ. Expansion of positivity, Proposition 5.3, implies ux, t) λ M 16, for all x B 2r x) and t [ˆt + ˆθr 2, ˆt + θr 2 ] for some ˆt t, t + εηρ 0 /4) 2 ], where θ depends only on γ, whereas λ depends on γ and ˆθ 0, θ) that we shall fix later. We can repeat the argument with r replaced by 2r and initial time varying in the interval [ˆt + ˆθr 2, ˆt + θr 2 ] to obtain the estimate ux, t) λ 2 M 16, for all x B 4r x) and t [ˆt + 5ˆθr 2, ˆt + 5 θr 2 ]. Thus iterating this procedure, we can show by induction that for any m N for all x B 2 m r x) and t [s m, t m ], where s m = ˆt + ˆθr 2 4m 1 3 ux, t) λ m M 16, 5.15) and t m = ˆt + θr 2 4m 1. 3 We fix m in such a way that 2ρ < 2 m r 4ρ; since x B ρ x 0 ), we then have the inclusion B ρ x 0 ) B 2 m r x). Recalling that r = εηρ s 0 )/16, we obtain 2 ρ s 0 ) ξ 4 ) ξ = εη r εη) ξ = 2 4ξ r ξ εη)ξ 2 m 6)ξ ρ ξ. Hence equation 5.15) can be rewritten as follows: ux, t) λ m M 16 = ρ s 0) ξ λm 2 ξ λ) m εη) ξ 2 6ξ 4 ρ ξ 16 = 2 ξ λ) m εη) ξ 2 6ξ 4 ux 0, 0), for any x B ρ x 0 ) and t [s m, t m ].

30 830 Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto We now fix c 2 > 0 and choose ˆθ in such a way that 16 3 ˆθ < c 2. With this choice, since 2 m r 4ρ, we have s m ˆθr 2 4m 3 ˆθ 16 3 ρ2 < c 2 ρ ) Once ˆθ has been fixed, we have λ; we now fix ξ = log 2 λ. With these choices also the radius r is fixed and so m is chosen in such a way that 1 log 2 r m 2 log 2 r. We draw the conclusion that ux, t) c 0 ux 0, 0) with c 0 := εη) ξ 2 6ξ 4 for all x B ρ x 0 ) and t [s m, t m ]. Notice that by 5.16) we have got two alternatives. Either c 2 ρ 2 [s m, t m ] or c 2 ρ 2 > t m. In the former case, the proof is completed by taking c 1 = c 1 0, whereas in the latter case we can select t [s m, t m ] such that ux, t) c 0 ux 0, 0) for all x B ρ x 0 ). We can assume that ˆθ is small enough such that t + ˆθρ 2 < c 2 ρ 2. By expansion of positivity, Proposition 5.3, we then obtain that ux, t) λc 0 ux 0, 0) for all x B 2ρ x 0 ) and t [ t + ˆθρ 2, t + θρ 2 ]. If c 2 ρ 2 < t + θρ 2, then the proof is completed by selecting c 1 = λc 0 ) 1. If this were not the case, we could restrict the previous inequality on B ρ x 0 ), and so iterating the procedure, adding the condition that ˆθ θ, using the fact that the estimate is already true on [ t + ˆθρ 2, t + θρ 2 ], ux, t) λ 2 c 0 ux 0, 0) for each x B ρ x 0 ) and t [ t+ ˆθρ 2, t+2 θρ 2 ]. By induction, if k is an integer such that t + k θρ 2 c 2 ρ 2, then ux, t) λ k c 0 ux 0, 0) for every x B ρ x 0 ) and t [ t + ˆθρ 2, t + k θρ 2 ]. It is crucial to select such an index k which depends only on the class and not on the function. We then take k in such a way that t + k θ c 2 ρ 2. As ρ 2 t c 2 ρ 2 the index k has to be chosen in such a way that both 1 + c 2 k θ and t + k θ remains in the domain of reference. Notice that 1 + c 2 2. Hence there exists k such that 2 k θ 3, and we are done with the proof. Acknowledgments. Miranda and Paronetto visited the Aalto University School of Science and Technology in February 2010, Marola visited the Università di Ferrara and Università degli studi di Padova in September 2010, and Kinnunen visited the Università of Padova in January It is a pleasure to thank these universities for their hospitality and the Academy

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