PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES

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1 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES J. KINNUNEN, N. MAROLA, M. MIRANDA JR., AND F. PARONETTO Abstract. In this paper we study variational problems related to the heat equation in metric 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 the proof of a scale-invariant Harnack inequality for functions in parabolic De Giorgi classes. MSC: 30L99, 31E05, 35K05, 35K99, 49N60 Keywords: De Giorgi class; doubling measure; Harnack inequality; Hölder continuity; metric space; minimizer; Newtonian space; parabolic; Poincaré inequality; quasiminima, quasiminimizer. 1. Introduction The purpose of this paper is to study variational problems related to the heat equation u t u = 0 in metric spaces equipped with a doubling measure and supporting a Poincaré inequality. We give a notion of parabolic De Giorgi classes of order 2 and parabolic quasiminimizers and study local regularity properties of functions belonging to these classes. More precisely, we show that functions in parabolic De Giorgi classes, satisfy a scale 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 [19], 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 stronger 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 later time t 0 + t 1, where t 1 depends on the parabolic equation. Elliptic quasiminimizers were introduced by Giaquinta Giusti [13] and [14] 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 Ω) is a Q-quasiminimizer, Q 1, related to the power p in Ω if suppφ) loc u p dx Q u φ) p dx suppφ) for all φ W 1,p 0 Ω). Giaquinta and Giusti realized that De Giorgi s method [6] could be extended to quasiminimizers, obtaining, in particular, local Hölder continuity. DiBenedetto and Trudinger [10] 1

2 2 KINNUNEN, MAROLA, MIRANDA JR. AND PARONETTO proved the Harnack inequality for quasiminimizers. These results were extended to metric spaces by Kinnunen and Shanmugalingam [23]. Elliptic quasiminimizers enable the study of elliptic problems, such as the p-laplace equation and p-harmonic functions, in metric spaces. Compared with the theory of p-harmonic functions we have no differential equation, only the variational approach can be used. There is also no comparison principle nor uniqueness for the Dirichlet problem for quasiminimizers. See, e.g., J. Björn [4], Kinnunen Martio [22], Martio Sbordone [27] and the references in these papers for more on elliptic quasiminimizers. Following Giaquinta Giusti, Wieser [33] 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 [34, 35], Gianazza Vespri [12], Marchi [26], and Wang [32]. The present paper is using the ideas of DiBenedetto [8] and is based on the lecture notes [11] of the course held by V. Vespri in Lecce. We would like to point out that the definition for the parabolic De Giorgi classes of order 2 given by Gianazza and Vespri [12] is sligthly 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 [9], but as the proof uses the linear sructure of the ambient space a new proof in the metric setting was needed. Motivation for this work was to introduce a version of parabolic De Giorgi classes that include parabolic quasiminimizers, and provide the sufficiency of the Saloff-Coste Grigor yan theorem. Grigor yan [16] and Saloff-Coste [28] 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 [31] generalized this result to the setting of local Dirichlet spaces essentially following Saloff-Coste; such 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 show the sufficiency in general metric measure spaces without using Dirichlet spaces nor the Cheeger differentiable structure [5]. It would be very interesting to know whether also necessity holds in this setting. Such geometric characterization via the doubling property of the measure and a Poincaré inequality is not available for an elliptic Harnack inequality, see Delmotte [7]. 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. Acknowledgements Miranda and Paronetto visited the Aalto University School of Science and Technology in February 2010, and 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 the deparments of mathematics at these universities for the hospitality. This work and the visits of Kinnunen and Marola were also partially supported by the 2010 GNAMPA project Problemi geometrici, variazionali ed evolutivi in strutture metriche.

3 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES 3 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 1) 0 < µb 2r x)) cµb r x)) < 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 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, ) : 1) holds true}. The doubling condition implies that for any x X, we have ) N ) N µb R x)) R R 2) µb r x)) c d = 2 N, 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 [20] 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 3) uγ0)) uγl γ )) g ds. If 3) holds for p almost every curve in the sense of Definition 2.1 in Shanmugalingam [29] we say that g is a p weak upper gradient of u. 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 [24]. 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 gradients 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 a.e., see Corollary 3.7 in Shanmugalingam [30] and Haj lasz [18] for the case p = 1. Let Ω be an open subset of X and 1 p <. Following the definition of Shanmugalingam [29], we define for u L p Ω), u p N 1,p Ω) := u p L p Ω) + g u p L p Ω), 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. The space N 1,p Ω) is a Banach space and a lattice, see Shanmugalingam [29]. 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 a.e. in {x Ω : ux) = vx)}, in particular g min{u,c} = g u χ {u c} for c R. γ

4 4 KINNUNEN, MAROLA, MIRANDA JR. AND PARONETTO We shall also need a Newtonian space with zero boundary values; for the detailed definition and main properties we refer to Shanmugalingam [30, Definition 4.1]. For a measurable set E X, let N 1,p 0 E) = {f E : f N 1,p E) and f = 0 p a.e. on X \ 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, 4) B ρ u u Bρ dµ C 2 ρ B Λρ g 2 dµ) 1/2, where u B := u dµ := 1 u dµ. B µb) B It is noteworthy that by a result of Keith and Zhong [21] 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 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 [20] and Haj lasz Koskela [17]. 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 pn 5) p N p, p < N, = +, p N. In addition, we have that if u N 1,2 0 B ρ), B ρ Ω, then the following Sobolev type inequality is valid 6) B ρ u q dµ ) 1/q c ρ B ρ g 2 u dµ ) 1/2, 1 q 2 ; for a proof of this fact we refer to Kinnunen Shanmugalingam [23, 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 previuous 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 7) B ρ u q dµ ) 1/q C p ρ B ρ g p dµ ) 1/p, 1 q p, 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.

5 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES 5 Theorem 2.1. Assume u N 1,2 0 B ρ), 0 < ρ < diamx)/3; then there exist κ > 1 such that we have ) κ1 u 2κ dµ c 2 ρ 2 u 2 dµ gu B ρ B ρ B 2 dµ. Λρ Proof. Let κ = 2 2/2, where 2 is as in the Sobolev inequality 6). By Hölder s inequality and 6), we obtain the claim B ρ u 2κ dµ B ρ u 2 dµ) κ1 B ρ u 2 c 2 ρ 2 B ρ u 2 dµ ) κ1 dµ B Λρ g 2 u dµ. ) 2/2 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 ) κ1 s2 s2 s 1 B ρ u 2κ dµ dt c 2 ρ 2 sup t s 1,s 2) We shall also need the following De Giorgi-type lemma. B ρ ux, t) 2 dµx) 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/q1/p {k<u<l} B Λρ g p u dµ) 1/p. Remark The previous result holds in every open set Ω X, provided that 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 and consequently v v Bρ q dµ = B ρ B ρ\a 8) v Bρ q 1 µa) v v Bρ q dµ + v Bρ q dµ v Bρ q µa) A B ρ v v Bρ q dµ. On the other hand, we see that 9) v q dµ = B ρ l k) q dµ + {u>l} B ρ v q dµ {k<u l} B ρ l k) q µ{u > l} B ρ ),

6 6 KINNUNEN, MAROLA, MIRANDA JR. AND PARONETTO and using 8), we obtain ) 1/q ) 1/q v q dµ v v Bρ q dµ + v Bρ q µb ρ ) ) 1/q B ρ B ρ ) 1/q µb ρ ) 2 v v Bρ q dµ. µa) B ρ By 7) and the doubling property, we finally conclude that l k)µ{u > l} B ρ ) 1/q 2C p ρ µb ρ) 2/q1/p which is the required inequality. µa) 1/q B Λρ g p v dµ ) 1/p, The following measure-theoretic lemma is a generalization of a result obtained in [9] 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 N 1,2 loc X) satisfying gu 2 dµ β µb ρ 0 x 0 )) B ρ0 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 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 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 2ηρ x 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 { I + = i I : µ{u > 1} B 2ηρ x i )) > α } µb 2ηρ x i )) 2c d and I = { i I : µ{u > 1} B 2ηρ x i )) α } µb 2ηρ x i )). 2c d

7 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES 7 By assumption, we get 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 10) µbρ x 0 )) µb 1+η)ρ x 0 ) \ B ρ x 0 )) ) µ{u > 1} B 2ηρ x i )). 2 i I + Assume by contradiction that 11) µ{u > λ} B ηρ x i )) 1 δ)µb ηρ x i )), for every i I + ; this clearly implies that The doubling condition on µ also implies that µ{u λ} B ηρ x i )) µb ηρ x i )) µ{u λ} B 2ηρ x i )) µb 2ηρ x i )) By Lemma 2.3 with q = 2, k = λ and l = 1, we obtain that 12) δ. δ c d. 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 2η 2 ρ 2 1 λ) 2 gu 2 dµ. {λ<u<1} B 2Ληρx i) Summing up over I + and using the bounded overlapping property, from 10) we get α 2 1 δ λ)2 µbρ x 0 )) µb 1+η)ρ x 0 ) \ B ρ x 0 ) ) c d 16C2η 2 2 ρ 2 gu 2 dµ c η 2 ρ 2 i I + B ρ0 x 0) c βµb ρ0 x 0 ))η 2, {λ u<1} B 2Ληρx i) where the costant c is given by 16C 2 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 λ)2 δ c d µb ρ x 0 )). g 2 u dµ We conclude with a result which will be needed later; for the proof we refer, for instance, to [15, Lemma 7.1].

8 8 KINNUNEN, MAROLA, MIRANDA JR. AND PARONETTO 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 y h = 0. h 3. Parabolic De Giorgi classes and quasiminimizers We consider a variational approach related to the heat equation see Definition 3.3) u 13) u = 0 in Ω 0, T ) t and provide a Harnack inequality for a class of functions in a metric measure space generalizing the known result for positive solutions of 13) in the Euclidean case. The following definition is essentially based on the approach of DiBenedetto Gianazza Vespri [9] and also of Wieser [33]; we refer also to the book of Lieberman [25] for a more detailed description. Definition 3.1 Parabolic De Giorgi classes of order 2). Let Ω be a non-empty 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 locω)) L 2 loc0, T ; N 1,2 loc Ω)), and for all k R the following energy estimate holds 14) sup t τ,s 2) B rx 0) + γ u k) 2 +x, t) dµ α ) θ s2 τ 1 R r) 2 B rx 0) s2 s 1 g 2 uk) + dµ ds α B R x 0) u k) 2 + dµ ds, B R x 0) u k) 2 +x, s 1 ) dµx) 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 14) 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 14) given in Definition 3.1 is referred to as energy estimate or Caccioppolitype estimate. We also point out that our definition of parabolic De Giorgi classes of order 2 is sligthly different from that given in the Euclidean case by Gianazza Vespri [12]; 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 loc0, T ; N 1,2 loc Ω)) is said to be a parabolic Q-quasiminimizer, Q 1, related to the heat equation 13) if 15) suppφ) u φ dµ dt + Eu, suppφ)) QEu φ, suppφ)) t for every φ Lip c Ω 0, T )) = {f LipΩ 0, T )) : suppf) Ω 0, T )}. K

9 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES 9 In the Euclidean case with the Lebesgue measure it can be shown that u is a weak solution of 13) if and only if u is a 1-quasiminimizer for 13), see [33]. Hence 1-quasiminimizers can be seen as weak solutions of 13) in metric measure spaces. This motivates the following definition. 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 [13, 14] and as shown there, there exist many other examples as well. Remark It is possible to prove, by using the Cheeger differentiable structure and the same proof contained in Wieser [33, 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 ) and 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 [23] 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 uk)+ + 2u k) + /R r). We have B r σ,s 2) u k) 2 + dµ dt 2 N 2 N µ L 1 B r σ, s 2 )) B r σ,s 2) u k) 2 +η 2 dµ dt u k) q +η q dµ dt B r σ,s 2) ) 2/q µ L 1 Ak; r; σ, s 2 ) ) q2)/q µ L 2 N 1 ) q2)/q 2/q Ak; r; σ, s 2 )) µ L 1 u k) q B r σ, s 2 )) +η q 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 µ L r 2/κ 1 ) κ1)/κ Ak; r; σ, s 2 )) B r σ,s 2) µ L 1 B r σ, s 2 )) κ1)/κ ) 1/κ u k) 2 +η dµ) 2 guk) 2 +η B r dµ dt. sup t σ,s 2) B r σ,s 2)

10 10 KINNUNEN, MAROLA, MIRANDA JR. AND PARONETTO By applying 14) 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 B r σ,s 2) s 2 σ) 1/κ sup t σ,s 2) 2 N+2 c 2/κ r 2/κ s 2 σ) 1/κ B r u k) 2 +x, t) dµ) κ1)/κ 2 8 s2 + R r) 2 u k) 2 + dµ dt σ B r µ L 1 Ak; r; σ, s 2 )) µ L 1 B r σ, s 2 )) 6γ + s2 22N+3 R r) 2 u k) B 2 + dµ dt. R s 1 µ L 1 ) κ1)/κ Ak; r; σ, s 2 )) µ L 1 B r σ, s 2 )) ) 1/κ s2 σ ) κ1)/κ µbr ) µb r ) By the the choice of σ, we see that s 2 σ) 1 < 2s 2 s 1 ) 1, and consequently B r g 2 uk) + dµ dt 16) u k) 2 + dµ dt 2 N+4 3γ + 2 2N+2 )c 2/κ µ L r 2/κ 1 Ak; r; σ, s 2 )) B r σ,s 2) µ L 1 B r σ, s 2 )) µb R) µb r ) s 2 σ) κ1)/κ 1 R r) 2 u k) 2 + dµ dt B R s 1,s 2) ) κ1)/κ Consider h < k. Then k h) 2 µ L 1 Ak; r; σ, s 2 )) ) u h) 2 + dµ dt u h) 2 + dµ dt = Ak; r;σ,s 2) s2 Ah; r;σ,s 2) σ from which, using the doubling property 2), it follows that B r u h) 2 + dµ dt, 17) µ L 1 Ak; r; σ, s 2 )) 1 k h) 2 µ L 1 B r σ, s 2 )) ) uh; r; σ, s 2 ) 2 2N+1 k h) 2 µ L 1 B r σ, s 2 )) ) uh; R; s 1, s 2 ) 2,, where 1/2 ul; ρ; t 1, t 2 ) := u l) 2 +d µ dt). B ρ t 1,t 2) By plugging 17) into 16) and arranging terms we arrive at 18) uk; r; σ, s 2 ) c r1/κ s 2 σ) κ1)/2κ k h) κ1)/κ R r) uk; R; s 1, s 2 )uh; R; s 1, s 2 ) κ1)/κ, with c = 2 N+2+N+1)κ1)/2κ) 3γ + 2 2N+2 ) 1/2 c 1/κ.

11 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES 11 Let us consider the following sequences: for n N, k 0 R and fixed d we define k n := k 0 + d 1 1 ) k 0 + d, r n := R 2 + σ n := s 1 + s n R 2 n+1 R 2, and R2 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 ), and where c is the constant in 18). Then θ := κ 1 κ, a := 1 + θ = 1 + κ θ κ 1, d θ = c 2 1+θ/2+a1+θ) u θ 0, 19) u n u 0 2 an. Proof. We prove the lemma by induction. First notice that 19) is true for n = 0. Then assume that 19) is true for fixed n N. In 18), we first estimate r 1/κ s 2 σ) κ1)/2κ by R 1/κ s 2 s 1 ) κ1)/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 Denote c := c R 1/κ s 2 s 1 ) κ1)/2κ) so that we have c R 1/κ s 2 s 1 ) κ1)/2κ) k n+1 k n ) κ1)/κ r n r n+1 ) u1+κ1)/κ n. 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 ) κ1)/2κ) 1 R = 21+θ/2 c. 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) u0 ) 1+θ u n+1 c = 2 an+1) d θ 2 an u 0. This completes the proof. 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,

12 12 KINNUNEN, MAROLA, MIRANDA JR. AND PARONETTO Proof. Since k n k 0 + d, R/2 r n R, 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 Proof. The Proposition 4.2 implies that where d is defined in Lemma 4.1. Then 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) ess sup u k 0 + d, B R/2 s 1+s 2)/2,s 2) 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 18). The previous inequality with k 0 = 0 can be written as follows ess sup B R/2 s 1+s 2)/2,s 2) u c B R s 1,s 2) 1/2 1/2 u 2 +dµ dt) 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, 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.

13 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES 13 Consider a sequence of Lipschitz continuous functions ζ h, h N, satisfying the following: ζ h 1 in Q h+1, g ζh ζ h 0 in Q ρ,θ y, s) \ Q h 1 = 2h+2, 0 ζ h ) t 2h+1 ρ h ρ h+1 ρ θ θ 1 ρ 2. Denote A h := {x, t) Q h : ux, t) > k h}. We have Q h u k h ) 2 +ζh 2 dµ dt Q h+1 k h+1 k h ) Ah+1 2 dµ dt = u k h ) 2 + dµ dt u k h ) 2 + dµ dt A h+1 1 a)σω)2 2 2h+2 µ L 1 A h+1 ), and consequently 20) s s h u k h ) 2 +ζ 2 1 a)σω)2 µ L 1 A h+1 ) h dµ dt B h 2 2h+2. µb h+1 ) On the other hand, if we use first Hölder s inequality and then Proposition 2.2, we obtain the following estimate s s h B h uk h ) 2 +ζ 2 h dµ dt c 2/κ µ L ρ 2/κ 1 A h ) µb h ) s s h B h c 2/κ µ L ρ 2/κ 1 A h ) µb h ) 2 s µ L 1 ) κ1)/κ A h ) s µb h ) ) κ1)/κ sup t s h,s) ) 2 ζh 2 guk 2 h ) + + 2gζ 2 h u k h ) 2 + ) κ1)/κ 1 µb h ) 2 1/κ c 2/κ µ L ρ 2/κ 1 ) κ1)/κ A h ) 1 µb h ) + s + ζh 2κ u k h ) 2κ s h B h u k h ) 2 +ζh 2 dµ B h sup t s h,s) s dµ dt ) 1/κ ) 1/κ dµ dt ) κ1)/κ ) κ1)/κ u k h ) 2 + dµ B h ) 1/κ g s h Bh 2ukh)+ dµ dt + 22h+5 ρ 2 u k h ) 2 + dµ dt s h B h sup u k h ) 2 + dµ + µb h ) t s h,s) B h s ) g Bh 2ukh)+ dµ dt + 22h+4 ρ 2 u k h ) 2 + dµ dt. B h s h s h

14 14 KINNUNEN, MAROLA, MIRANDA JR. AND PARONETTO We continue by applying the energy estimate 14) with r = ρ h, R = ρ h1, α = 0, s 2 = s, τ = s h, s 1 = s h1 and get s uk h ) 2 +ζh 2 dµ dt s h B h 2 1/κ c 2/κ µ L ρ 2/κ 1 ) κ1)/κ A h ) 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 h1 B h1 µ L C 1 ρ 2/κ 1 ) κ1)/κ A h ) 1 2 3h+4 µb h ) µb h ) ρ 2 u k h ) 2 + dµ dt 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 + dµdt µ L 1 A h1 )σ h ω) 2 µ L 1 A h1 )σω) 2. Q h1 This implies that s uk h ) 2 +ζh 2 dµ dt 2 2h+4 C 1 σω) 2 1 B h s h 2 2h+4 C 1 σω) 2 θ κ1 κ 2 N1+ κ1 ρ 2 κ1 κ Q h1 µ L 1 A h ) µb h ) µ L κ ) 1 A h1 ) µ L 1 B h1 ) ) κ1 κ µ L 1 A h1 ) µb h ) ) κ1 κ µ L 1 A h1 ) µb h1 ) where we have estimated µb h1 )/µb h ) 2 N. By the last inequality and 20), if we call C 2 the constant C 1 θ κ1 κ1 κ N1+ 2 κ )+6 1 a) 2, we obtain µ L 1 A h+1 ) µ L C 2 2 4h 1 ) A h1 ) κ1 κ µ L 1 A h1 ) µb h+1 ) µ L 1 ; B h1 ) µb h1 ) finally, dividing by s s h+1 and since s s h1 )/s s h+1 ) θ/ θ, we can summarize what we have obtain by writing 21) y h+1 C 3 2 4h y 1+κ1)/κ h1 where we have defined i.e. y h := µ L1 A h ) θ µ L 1 Q h ) and C 3 = C 2 θ C 3 = 2 1/κ c 2/κ 1 + γ + γ ) θ θ θ κ1 κ 2 N1+ κ1 κ )+6 1 θ 1 a) 2 θ. 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 Since y 0 = µ L 1 A 0 )/µ L 1 Q 0 ), where lim y h = 0. h Q 0 = B ρy) s θρ 2, s], and A 0 = {x, t) Q 0 : ux, t) > m + σω}.

15 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES 15 To do this it is sufficient to choose ν + to be By definition of y h and A h we see that which completes the proof. ν + = C κ/κ1) 3 16 κ2 /κ1) 2. u m + aσω µ L 1 -a.e. in B ρ/2 y) s θρ 2, s ], 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, 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 uˆk h ) in place of uk h ) +, where ˆk h = m + σ h ω. The next result is the so called expansion of positivity. Following the approach of DiBenedetto [8] 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 ), then 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 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.

16 16 KINNUNEN, MAROLA, MIRANDA JR. AND PARONETTO 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 the following holds ux, t ) h µ a.e. in B ρ x ) then µa ηh,4ρ x, t)) < 1 1 ) 4 N+1 µb 4ρ x )) for every t [t, t + θρ 2 ]. 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 u 1) 2 x, t)dµx) + gu1) 2 dµ dt t <t<t + θρ 2 B 4ρ1σ) x ) t B 4ρ1σ) x ) u 1) 2 x, t γ t + θρ 2 )dµx) + 16σ 2 ρ 2 u 1) 2 dµ dt. B 4ρx ) Since u 1 in B ρ x ), we deduce from Remark 5.4 that µ{x B 4ρ x ) : ux, t ) < 1}) < Notice that u 1) 1; thus we have in particular u 1) 2 x, t) dµx) sup t <t<t + θρ 2 B 4ρ1σ) x ) B 4ρx ) N N u 1) 2 x, t γ ) dµx) + 16σ 2 ρ 2 ) µb 4ρ x )) + ) µb 4ρ x )) + Writing A h,ρ t) in place of A h,ρ x, t), decomposing γ 16σ 2 ρ 2 t B 4ρx ) 1 1 ) 4 N µb 4ρ x )). t + θρ 2 t t + θρ 2 t γ θ 16σ 2 µb 4ρx )) B 4ρx ) B 4ρx ) u 1) 2 dµ dt u 1) 2 dµ dt A η,4ρ t) = A η,4ρ1σ) t) {x B 4ρ x ) \ B 4ρ1σ) x ) : ux, t) < η}, and using the doubling property we have On the other hand, u 1) 2 x, t) dµx) B 4ρ1σ) x ) µa η,4ρ t)) µa η,4ρ1σ) t)) + µ B 4ρ x ) \ B 4ρ1σ) x ) ). A η,4ρ1σ) t) u 1) 2 x, t) dµx) 1 η) 2 µa η,4ρ1σ) t)).

17 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES 17 Finally, we obtain 22) µa η,4ρ t)) µa η,4ρ1σ) t)) + µ B 4ρ x ) \ B 4ρ1σ) x ) ) 1 η) 2 u 1) 2 x, t)dµ + µ B 4ρ x ) \ B 4ρ1σ) x ) ) B 4ρ1σ) x ) 1 η) N + γ θ ) 16σ 2 µb 4ρ x )) + µ B 4ρ x ) \ B 4ρ1σ) x ) ). If the claim of the lemma was 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 22) for t = t and dividing by µ B 4ρ x )) we would have 1 1 ) 4 N+1 1 η) N + γ θ ) 16σ 2 + µb 4ρx ) \ B 4ρ1σ) x )) µb 4ρ x. )) Choosing, for instance, θ = σ 3 and letting η and σ go to zero we would have the contradiction 14 N1 1 4 N. 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 14) 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 23) g 2 dµ dt u ηh 2 m ) t B 4Λρx ) γ θ γ θ ) 1 t + θρ 2 Λρ) 2 t θρ 2 B 5Λρx ) ) 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 m1, q = 1 and 2 ε < p < 2, implies u ηh ) 24) B 4ρx ) 2 m x, t) dµ ηh 2 m µa ηh2 m,4ρt)) 8C p ρµb 4ρ x )) 21/p ) 1/p µb 4ρ x g p x, t) dµ, ) \ A ηh2 m+1 u,4ρτ)) ηh Ãt) 2 m )

18 18 KINNUNEN, MAROLA, MIRANDA JR. AND PARONETTO 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 then we deduce that µb 4ρ x ) \ A ηh,4ρ t)) + µa ηh,4ρ t)) = µb 4ρ x )), µb 4ρ x ) \ A ηh,4ρ t)) > 4 N1 µb 4ρ x )) for every t [t, t + θρ 2 ]. We finally arrive at u ηh ) B 4ρx ) 2 m x, t) dµ 8C p 4 N+1 µb 4ρ x )) 11/p ρ ) 1/p g p x, t) dµ. u ηh Ãt) 2 m ) Integrating this with respect to t and defining the decreasing sequence {a m,ρ } m=0 as t + θρ 2 a m,ρ : = µ A ηh2 m,ρt) ) dt t { = 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 ) 25) t B 4ρx ) 2 m x, t) dµ dt t + θρ 2 8C p 4 N+1 µb 4ρ x )) 11/p ρ 8C p 4 N+1 µb 4ρ x )) 11/p ρ t Ãt) t + θρ 2 t B 4Λρx ) ) 1/p g p x, t) dµ dt u ηh 2 m ) ) 1/2 g 2 dµ dt a u ηh m1,4λρ a m,4λρ) 2p)/2. 2 m ) On the other hand, t + θρ 2 u ηh ) t B 4ρx ) 2 m x, t) dµ dt ηh 2 m+1 a m+1,4ρ from which, using first 25) and then 23), we obtain a 2/2p) m+1,4ρ ca m1,4λρ a m,4λρ ), where c = C 2 p16 N+3 γ2 θ + 1)µB 4ρ x )) 211/p) µb 5Λρ x ))ρ 2 ) 1/2p). Hence for every m N we have m m=1 a 2/2p) m+1,4ρ ca 0,4Λρ a m,4λρ). Since {a m,ρ } m=0 is decreasing the sum m=1 a2/2p) m+1,4ρ converges, and consequently This completes the proof. lim a m,4ρ = 0. m 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

19 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES 19 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 1h 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 ). 1γ As a consequence, u is locally α-hölder continuous with α = log 2 γ principle. and satisfies the strong maximum 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 and fix y 0, τ 0 ) Q s 0 x 0, 0) in such a way that 26) M := N s 0 ) = ρ s 0 ) ξ, 3M 4 < sup Q ρ 0 /4 y0,τ0) u M, where ρ 0 = ρ s 0 )/2; this implies that Q ρ 0 y 0, τ 0 ) Q ρ+s x 0)/2 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 x0,0) 2 Let us divide the proof into five steps. Step 1. We assert that { 27) µ L 1 x, t) Q ρ y 0/2 0, τ 0 ) : ux, t) > M }) ) > ν + µ L 1 Q 2 ρ y 0/2 0, τ 0 ), where ν + is the constant in Proposition 5.1. To see this, assume on the contrary that equation 27) is not true. Then set k = 2 ξ M and m + = ω = k, θ = 1, ρ = ρ 0 2, σ = 1 2ξ1, and a = σ ) 2 ξ+2. We obtain from Proposition 5.1 that u 3M 4 in Q ρ 04 y 0, τ 0 ), which contradicts 26). Step 2. We show that there exists t τ 0 ρ2 0 4, τ 0 ν + 8 ρ 2 ] 0 4

20 20 KINNUNEN, MAROLA, MIRANDA JR. AND PARONETTO such that 28) µ and 29) { x B ρ0/2y 0 ) : ux, t) M }) > ν µb ρ 0/2y 0 )), B ρ0 /2y 0) gux, 2 t) dµx) α µb ρ 0 y 0 )) ρ 2 k 2, 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 { I := t τ 0 ρ2 0 4, τ 0] : µat)) > ν } + 2 µb ρ 0/2y 0 )), and J α := From 27) we have that { t τ 0 ρ2 0 4, τ 0] : B ρ0 /2y 0) τ0 } gux, 2 t)dµx) α µb ρ 0 y 0 )) ρ 2 k 2. 0 ν + µ L 1 Q ρ y 0/2 0, τ 0 )) < µat)) dt τ 0ρ 2 0 /4 = µat))dt + µat)) dt I τ 0ρ 2 0 /4]\I This implies the following lower bound µb ρ0/2y 0 )) I + ν + 2 µ L1 Q ρ y 0/2 0, τ 0 )) ) 4 = µ L 1 Q ρ y 0/2 0, τ 0 )) I + ν ) +. 2 I ν +ρ On the other hand, if we apply 14) with R = ρ 0, r = ρ 0 /2, α = 0, and θ = 1, we obtain 30) g 2 Q ρ 0 /2 y0,τ0) u dµ dt = g 2 Q ρ 0 /2 y0,τ0) uk) dµ dt This estimate implies which in turn gives us 8γ ρ 2 0 Q ρ 0 y 0,τ 0) = 8γk 2 µb ρ0 y 0 )). 4γk 2 µb ρ0 y 0 )) u k) 2 dµ dt 8γk2 ρ 2 µ L 1 Q ρ 0 y 0, τ 0 )) 0 τ 0ρ 2 0 /4,τ0] dt B ρ0 /2y 0) α µb ) ρ 0 y 0 )) ρ ρ 2 k J α, J α ρ γ ). α ρ 2 0 g 2 ux, t) dµ

21 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES 21 Choosing α = 64γ/ν +, we obtain Then if we set we get I J α = I + J α I J α ν +ρ T = τ 0 ρ2 0 4, τ 0 ν + 8 ρ 2 ] 0, 4 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 { 31) µ u, t) > M } ) B ηρ0/2x ) > 1 δ)µb ηρ0/2x )). 4 Step 4. We show that for ε > 0 to be fixed, there exists x such that Q + εηρ x, 0/4 t) Q ρ 0 y 0, τ 0 ) and { 32) µ L 1 u M } ) Q + 8 εηρ x, 0/4 t) ) 4 N+1 γε 2 + δ)µ L 1 Q + εηρ x, 0/4 t). Indeed, consider the cylinder B ηρ0 /4x ) t+t Q = B ηρ0/4x ) t, t + t ] with t = εηρ 0 /4) 2. Using the energy estimate 14) on Q with k = M/4, R = ηρ 0 /2, r = R/2 and α = 1, we obtain together with 31) that for any s t, t + t ] u M ) 2 4 Define 16γ η 2 ρ 2 0 t x, s) dµx) dt B ηρ0 /2x ) M 2 16 γε2 + δ)µb ηρ0/2x )). and we have that B ηρ0 /4x ) Bt) = u M ) 2 x, t) dµx) + u M ) 2 x, t) dµx) 4 B ηρ0 /2x ) 4 { x B ηρ0/4x ) : ux, t) M }, 8 u M ) 2 x, s)dµx) u M ) 2 x, s)dµx) M 2 4 Bs) 4 64 µbs)). 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).

22 22 KINNUNEN, MAROLA, MIRANDA JR. AND PARONETTO We have to apply Proposition 5.1; we then have that there exists x so that Q + εηρ x, 0/4 t) Q ρ 0 y 0, τ 0 ) satisying equation 32). To see this, we take a disjoint family of balls {B εηρ0/4x j )} m j=1 such that B εηρ0/4x j ) B ηρ0/4x ) for every j = 1,..., m, and B ηρ0/4x ) m B εηρ0/2x j ). j=1 Given this disjoint family, there exists j 0 such that 32) 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 x, 0/4 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, σ = 2ξ3, so we can deduce that there exists ν > 0 such that if { 33) µ L 1 u M } ) Q εηρ0/4 8 x, s) ν µ L 1 Q εηρ 0/4 x, s)) then ux, t) M 16, µ L1 a.e. in Q r x, s), where r = εηρ 0 /8. Fix ε and δ in 32) small enough so that 33) 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 following 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 34) ux, t) λ m M 16, for all x B 2m r x) and t [s m, t m ], where s m = ˆt + ˆθr 2 4m 1 3 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 2m 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 m6)ξ ρ ξ.

23 PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES 23 Hence equation 34) can be rewritten as follows ux, t) λ m M 16 = ρ s 0) ξ λm 2 ξ λ) m εη) ξ 2 6ξ4 ρ ξ = 2 ξ λ) m εη) ξ 2 6ξ4 ux 0, 0). 16 for any x B ρ x 0 ) and t [s m, t m ]. We now fix c 2 > 0 and choose ˆθ in such a way that 16 ˆθ 3 < c 2. With this choice, since 2 m r 4ρ, we have 35) s m ˆθr 2 4m 3 ˆθ 16 3 ρ2 < c 2 ρ 2. 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 6ξ4 for all x B ρ x 0 ) and t [s m, t m ]. Notice that by 35) 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 was 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. References [1] M.T. Barlow and R.F. Bass and T. Kumagai, Stability of parabolic Harnack inequalities on metric measure spaces, J. Math. Soc. Japan, 58, 2006), n.2, pp [2] M.T. Barlow and A. Grigor yan and T. Kumagai, On the equivalence of parabolic Harnack inequalities and heat kernel estimates, J. Math. Soc. Japan, to appear. [3] A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics, European Mathematical Society EMS), Zürich, in press. [4] J. Björn Boundary continuity for quasiminimizers on metric spaces, Illinois J. Math., ), pp [5] J. Cheeger. Differentiability of Lipschitz functions on measure spaces, Geom. Funct. Anal ), no. 3, pp [6] E. De Giorgi, Sulla differenziabilità e l analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3) ), pp

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