Riesz potentials and nonlinear parabolic equations

Transcription

1 Dark Side manuscript No. (will be inserted by the editor) Riesz potentials and nonlinear parabolic equations TUOMO KUUSI & GIUSEPPE MINGIONE Abstract The spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric Riesz potential of the right hand side datum, exactly as in the case of the heat equation. Heat kernels type estimates persist in the nonlinear case. To Neil Trudinger for his 70th birthday Contents. Results Preparations Proof of Theorems. and Proof of Theorems.4 and Proof of Theorem Results In this paper we are going to consider nonlinear, possibly degenerate parabolic equations whose model is given by u t div ( Du p 2 Du) = µ (.) where µ denotes in the most general case a Borel measure with finite total mass. Although the kind of problems considered are nonlinear our goal is to provide a suitable linear potential theory aimed at describing, in a sharp way, the regularity

2 2 TUOMO KUUSI & GIUSEPPE MINGIONE properties of the gradient Du in terms of those of µ. More precisely, our description shows that sharp gradient pointwise estimates can be given in terms of classical Riesz caloric potentials of the right hand side µ. We will see that, surprisingly enough, bounds similar to those that hold for the heat equation u t u = µ (.2) actually hold for solutions to (.) and, more generally, for solutions to quasilinear equations of the type u t div a(x, t, Du) = µ in Ω T = Ω ( T, 0), (.3) provided suitable, actually optimal, regularity assumptions are made on the partial map (x, t) a(t, x, z). Here Ω R n is an open subset, n 2 and T > 0. Specifically, we shall consider a Carathéodory vector field a: Ω T R n R n which is C -regular in the third variable and satisfying the following parabolicity and continuity conditions: a(x, t, z) + z a(x, t, z) ( z 2 + s 2 ) /2 L( z 2 + s 2 ) (p )/2 ν( z 2 + s 2 ) (p 2)/2 ξ 2 z a(x, t, z)ξ, ξ (.4) a(x, t, z) a(x 0, t, z) Lω( x x 0 )( z 2 + s 2 ) (p )/2 whenever z, ξ R n, x, x 0 Ω, t ( T, 0), where 0 < ν L are positive numbers and p 2. The symbol ω( ) denotes a modulus of continuity meaning that ω: [0, ) [0, ] is a nondecreasing concave function such that ω(0) = 0. In the following s 0 is a parameter that will be used to distinguish the degenerate case (s = 0), which covers the model equation in (.), from the non-degenerate one (s > 0); the analysis made in the following will see no difference between these two cases. In the rest of the paper we shall assume that the partial map x is Dini-continuous in the sense that 0 a(x, t, z) ( z 2 + s 2 ) (p )/2 ω(ϱ) dϱ ϱ <. (.5) This assumption is optimal for the estimates we are going to derive in the following; see the comments at the beginning of Section.3. We anyway remark that, everywhere in this paper, we only assume measurability of the partial map t a(x, t, z), in other words we assume that time coefficients are merely measurable. Yet, in this paper we shall always consider the case p 2 as the case p < 2 has already been treated elsewhere [26] and involves an analysis which is different and somewhat simpler than the one which is necessary here. For more notation we refer the reader to Section 2 and to the rest of this introductory section.

3 Riesz potentials and nonlinear parabolic equations 3 Remark (On the notion of solution). Throughout the paper, when considering weak solutions to (.3) and unless otherwise stated, we shall mean energy weak solutions. An energy weak solution u belongs to the parabolic energy space, i.e. u C 0 ( T, 0; L 2 (Ω)) L p ( T, 0; W,p (Ω)), (.6) and it is a distributional solution to (.3) in the sense that uϕ t dx dt + Ω T a(x, t, Du), Dϕ dx dt = Ω T ϕ dµ Ω T (.7) holds whenever ϕ Cc (Ω T ). In view of the available approximation theory we shall assume that µ L without loss of generality, while upon letting in a standard way µ R n+ \Ω T = 0, we shall finally consider the case µ L (R n+ ). (.8) Assumptions (.6) and (.8) will be then finally removed in Section.2. There we shall deal with solutions to general measure data problems, where both (.6) and (.8) are no longer in force. In other words, we pursue the usual path that consists of first deriving a priori estimates for more regular problems and solutions, and then recovering the general case by approximation. Notice that under the assumptions (.6) and (.8) by standard density arguments the identity in (.7) remains valid whenever ϕ W,p (Ω T ) L (Ω T ) has a compact support... Intrinsic and explicit Riesz potential estimates Very recently, in [27, 28, 40], it has been shown that, surprisingly enough, the regularity theory of possibly degenerate quasilinear elliptic equations of the type div a(du) = µ completely reduces to that of standard Poisson equation u = µ (.9) up to the C -level, i.e. up to the gradient continuity. Moreover, in some sense the regularity theory can be actually linearized via Riesz potentials. In particular, the gradient of solutions can be pointwise bounded via classical Riesz potentials exactly as it happens for solutions to (.9), i.e., the inequality Du(x 0 ) p ci µ (x 0, r) + c holds for a.e. point x 0, where ( p ( Du + s) dx) (.0) B(x 0,r) I µ (x 0, r) := r 0 µ (B(x 0, ϱ)) ϱ n dϱ ϱ

4 4 TUOMO KUUSI & GIUSEPPE MINGIONE denotes the standard truncated Riesz potential of µ. We refer to [30] for a description of the current status of the theory. Our aim here is to build a related theory for general degenerate parabolic problems of the type in (.) and (.3). The main challenge here is to match the anticipated a priori Riesz potential estimate with the inhomogeneous nature of equations such as (.); it will be indeed part of the work to find the proper formulation, suited to the geometry of the equations considered, making this possible. We also remark that, even when applied to the stationary case, our results turn out to be more general than those contained in [27, 28] since the equations considered here are also allowed to have coefficients, that is the vector field a( ) is allowed to have an explicit dependence on (x, t). The ultimate outcome is that, once again, the (spatial) gradient regularity theory of solutions to (.3) can be unified in a natural way with the one of the usual heat equation (.2). The analysis here unavoidably involves the concept of the intrinsic geometry, introduced and widely employed by DiBenedetto [9,48,2]. According to this principle, the lack of scaling (for p 2) of equations as u t div ( Du p 2 Du) = 0 (.) can be locally rebalanced by performing the regularity analysis of the solution on certain special cylinders adapted to the solution itself, indeed called intrinsic parabolic cylinders. More precisely, instead of using standard parabolic cylinders Q r (x 0, t 0 ) := B(x 0, r) (t 0 r 2, t 0 ), (.2) one uses cylinders whose time-length is stretched accordingly to the size of the gradient on the cylinder itself. In other words, one is lead to consider cylinders of the type Q r (x 0, t 0 ) := B(x 0, r) (t 0 2 p r 2, t 0 ), > 0, (.3) on which it simultaneously happens that a condition of the type Q r (x0,t0) Du dx dt (.4) is satisfied. The use of the word intrinsic stems from the very basic fact that the parameter appears on both sides of (.4). Ultimately, this has the effect of rebalancing the local anisotropic character of the equation allowing for proving homogeneous regularity estimates: in some sense, the equation (.) looks like the heat equation when considered on Q r (x 0, t 0 ). For instance, when considering standard parabolic cylinders, for solutions to (.) it is only possible to prove bounds of the type sup Du c(n, p) ( Du + s + ) p dx dt, (.5) Q r/2 (x 0,t 0) Q r(x 0,t 0) whose lack of homogeneity precisely reflects that of the equation. In this sense the previous estimate is natural. When instead considering intrinsic cylinders with

5 Riesz potentials and nonlinear parabolic equations 5 (.3)-(.4) being in force, estimates become dimensionally homogeneous: ( ) /(p ) c(n, p) ( Du + s) p dx dt = Du(x 0, t 0 ). Q r (x0,t0) (.6) Both (.5) and (.6) are basic results of DiBenedetto [9] while we just remark that intrinsic geometries are nowadays a basic and common tool to treat degenerate parabolic equations [,9,20]. The previous considerations, together with (.5)-(.6), are actually the starting point for proving the desired potential estimates. Let us see how. Beside the usual caloric (truncated) Riesz potentials built upon standard parabolic cylinders as in (.2), that is I µ β (x 0, t 0 ; r) := r 0 µ (Q ϱ (x 0, t 0 )) ϱ N β we introduce the intrinsic Riesz potentials as I µ β, (x 0, t 0 ; r) := r 0 dϱ ϱ, 0 < β N := n + 2, µ (Q ϱ(x 0, t 0 )) ϱ N β (.7) dϱ ϱ, (.8) where N is called, as usual, the parabolic dimension. Note that in such a way we have I µ β (x 0, t 0 ; r) I µ β, (x 0, t 0 ; r). At this stage the word intrinsic merely refers to the fact that the additional parameter has been considered in the definition in (.8), while at the moment no local linkage with solutions of the type in (.3) has been considered yet. This will come in a few moments: indeed, the right way to give an intrinsic formulation of the linear potential bounds is inspired by (.6) and it is given in the following: Theorem. (Intrinsic Riesz potential bound). Let u be a solution to (.3) under the assumptions (.4)-(.5) and (.8). There exist a constant c > and a radius R 0 > 0, both depending only on n, p, ν, L, ω( ), such that the following implication holds: ( ) /(p ) ci µ, (x 0, t 0 ; r) + c ( Du + s) p dx dt Q r (x0,t0) = Du(x 0, t 0 ) (.9) whenever Q r (x 0, t 0 ) Ω T, (x 0, t 0 ) is Lebesgue point of Du, and r R 0. When the vector field a( ) is independent of x, no restriction occurs on r, i.e., R 0 =. Note that, as expected, (.9) allows to recover (.6) when µ = 0; this is a first sign of the fact that (.9) is the correct intrinsic extension of (.0). As a matter of fact Theorem. implies a gradient linear potential estimate involving standard Riesz potentials. Surprisingly enough, this is of the same type as the one which holds for the standard heat equation; moreover, when µ = 0, this reduces (.5). We indeed have the following:

6 6 TUOMO KUUSI & GIUSEPPE MINGIONE Theorem.2 (Riesz potential bound in classic form). Let u be a solution to (.3) in Ω T under the assumptions (.4)-(.5) and (.8). There exists a constant c, depending only on n, p, ν, L, ω( ), such that Du(x 0, t 0 ) ci µ (x 0, t 0 ; r) + c ( Du + s + ) p dx dt Q r(x 0,t 0) holds whenever (x 0, t 0 ) Ω T is a Lebesgue point of Du and whenever Q r (x 0, t 0 ) Ω T is a standard parabolic cylinder such that r R 0 ; here R 0 is the radius introduced in Theorem.. An immediate consequence of Theorem.2 is the following global bound via classical, non-truncated caloric Riesz potentials: Corollary.3. Let u be a weak solution to the equation u t div a(t, Du) = µ in R n+ (.20) under the assumptions (.4) and (.8); moreover, assume that the global integrability u L p (, t 0 ; W,p (R n )) holds for t 0 R. There exists a constant c, depending only on n, p, ν, L, such that the upper bound Du(x 0, t 0 ) c {t<t 0} holds whenever (x 0, t 0 ) is a Lebesgue point of Du. d µ (x, t) d par ((x, t), (x 0, t 0 )) N Remark 2. In Coirollary.3 d par ( ) denotes the standard parabolic distance in R n+, which is defined by { d par ((x, t), (x 0, t 0 )) := max x x 0, } t t 0. The previous result shows that Theorems. and.2 play the role of the usual representation formulae via heat kernels for solutions to the heat equation. In recent years there has been a large activity devoted to the understanding the extent to which heat kernel estimates are still valid when passing to more general settings, as for instance Lie groups and manifolds [,33,43]. In this paper we are interested, in a dual but yet related way, to see the extent to which estimates as those implied by well-behaving heat kernels can be recovered in the nonlinear degenerate setting. Our results also connects to those in [36], and concerning the p-superharmonicity of certain linear Riesz potentials. The proof of Theorem. opens the way to an optimal continuity criterion for the gradient still involving only classical Riesz potentials and that, as such, is again independent of p. Theorem.4 (Gradient continuity via linear potentials). Let u be a solution to (.3) in Ω T under the assumptions (.4)-(.5) and (.8). If lim r 0 Iµ (x, t; r) = 0 (.2) locally uniformly w.r.t. (x, t), then Du is continuous in Ω T.

7 Riesz potentials and nonlinear parabolic equations 7 An important corollary involves Lorentz spaces: Corollary.5 (Lorentz spaces criterion). Let u be a solution to (.3) in Ω T under the assumptions (.4)-(.5). If µ L(N, ), that is if then Du is continuous in Ω T. 0 {(x, t) Ω T : µ(x, t) > } /N d <, Corollary.5 substantially improves [23, Theorem.3] claiming only the boundedness of the gradient under the assumption µ L(N, ) and when no coefficients are present in the vector field; see also [6] for a related global boundedness result in the elliptic case. It might be interesting to note how the above result naturally extends to the parabolic case the classical gradient continuity results valid in the elliptic case, starting from those available for the Poisson equation u = µ in domain of R n. For this it is known that the condition µ L(n, ) is a sufficient one for the gradient continuity. This is in turn related to, and indeed implied by, a classical result of Stein [44] that claims the continuity of a function f whenever its distributional derivatives belong to L(n, ). Corollary.5 gives the precise nonlinear parabolic analog of such classical facts. As expected, the space dimension n is replaced by the parabolic one N = n + 2, which is naturally associated to the standard parabolic metric. Preliminary to the proof of the continuity criterion, there is another result which claims the VMO gradient regularity under weaker assumptions on the measure µ. Theorem.6 (VMO gradient regularity). Let u be a solution to (.3) under the assumptions (.4)-(.5) and (.8). If I µ (x, t; r) is locally bounded in Ω T and if µ (Q r (x, t)) lim r 0 r N = 0 (.22) locally uniformly in Ω T w.r.t. (x, t), then Du is locally V MO in Ω T, that is lim sup R 0 r R,Q r Q for every open subset Q Ω T. Q r Du (Du) Qr dx dt = 0 (.23).2. General measure data problems Solutions to measure data problems are usually found by approximation procedures via solutions to more regular problems. These are of the type (u h ) t div a(x, t, Du h ) = µ h L, h N, (.24) where (.6) holds for u h and {µ h } is a sequence of smooth functions obtained via convolution of µ with a sequence of suitable mollifiers (see also Remark 7 below). The point is that solutions to measure data problems do not belong, in general, to the energy space. This section is also aimed at justifying that we may actually work under the apparently additional assumption (.6). More precisely, the exact definition of SOLA, is given in the following:

8 8 TUOMO KUUSI & GIUSEPPE MINGIONE Definition ([4, 2, 22]). A SOLA (Solution Obtained as a Limit of Approximations) to (.3) is a distributional solution u L p ( T, 0; W,p (Ω)) to (.3) in Ω T, such that u is the limit of solutions u h C 0 ( T, 0; L 2 (Ω)) L p ( T, 0; W,p (Ω)) to equations as (.24) in the sense that u h u in L p ( T, 0; W,p (Ω)), L µ h µ weakly in the sense of measures and such that lim sup h µ h (Q) µ ( Q par ) (.25) for every cylinder Q = B (t, t 2 ) Ω T, where B Ω is a bounded open subset. In the right hand side of (.25) appears the symbol Q par, which denotes the parabolic closure of Q defined in (2.) below. For more on this kind of solutions see Remark 7 below; in particular, requiring (.25) is neither unnatural nor restrictive. Our estimates remain valid for SOLA and, in fact, the following holds: Theorem.7. The statements of Theorems.,.2,.4 and.6 continue to hold for SOLA u L p ( T, 0; W,p (Ω)) to (.3), under the only assumptions (.4)-(.5). As a consequence, the results in Corollaries.3 and.4 hold for SOLA as well..3. Comparison with nonlinear estimates Theorem. improves the previously potential estimates via nonlinear potentials [25], bringing them to the desired optimal level. Based on elementary dimension analysis we conjecture that the result of Theorem. cannot be improved by the use of any other nonlinear potential. Theorem. is optimal also with respect to the regularity assumed on the coefficients dependence x a(x, ), that is (.5). Indeed, already in the linear elliptic case div (ã(x)du) = 0, Dini-continuity of elliptic coefficients matrix ã(x) is essential in order to get gradient boundedness. Merely continuous coefficients are not sufficient to ensure that the gradient belongs even to BMO, see [6]. Now, let us see how Theorem. improves the previously known estimates via nonlinear potentials. In [25] a Wolff potential type gradient bound has been obtained for equations without coefficients, that is of the type u t div a(du) = µ. More precisely, in [25] we introduced the following intrinsic Wolff potentials: ( ) /(p ) r µ (Q W µ (x ϱ (x 0, t 0 )) dϱ 0, t 0 ; r) := 2 p ϱ N ϱ, 0 where N = n + 2 and > 0. See [4,5,8] for more on Wolff potentials. We then proved the existence of a universal constant c w c w (n, p, ν, L) for which c w W µ (x 0, t 0 ; r) + c w ( Q r (x0,t0) ( Du + s) p dx dt = Du(x 0, t 0 ) ) /(p ) (.26)

9 Riesz potentials and nonlinear parabolic equations 9 holds. Let us now show that (.9) implies (.26), for c w := [ 4 N+ (log 2) 2 p c ] /(p ) + 4 N c (.27) and radii r R 0, where c and R 0 are the constants appearing in the statement of Theorem. (no restriction on radii appears in the case of equations as in (.20)). With r i = r/2 i for integers i 0, Hölder s inequality for series (as it is p 2 here), gives I µ, (x 0, t 0 ; r/2) ri = µ (Q ϱ(x 0, t 0 )) dϱ i= r i+ ϱ N ϱ 2 N µ (Q r log 2 i (x 0, t 0 )) i= r N i ( ) /(p ) µ (Q 2 N log 2 ri (x 0, t 0 )) i= r N i p ( ) /(p ) µ (Q = 4 N (log 2) 2 p ri (x 0, t 0 )) ri i= 4 N (log 2) 2 p i= ri r i r N i ( µ (Q ϱ (x 0, t 0 )) ϱ N = 4 N (log 2) 2 p 2 p [W µ (x 0, t 0 ; r)] p 4 N (log 2) 2 p c p w 2c. r i dϱ ϱ ) /(p ) dϱ ϱ Notice that to derive the second-last estimate we have use the inequality in the first line of (.26), while in the last estimate we have used (.27). Using the inequality in the last display, again (.27) and finally the left hand hand side of (.26), a standard manipulation gives ( ) /(p ) ci µ, (x 0, t 0 ; r/2) + c ( Du + s) p dx dt Q r/2 (x0,t0) so that the right hand side inequality in (.26), that is Du(x 0, t 0 ), follows applying Theorem.. The improvement from (.26) to (.9) is rather strong both from the viewpoint of the theoretical significance - as now the regularity theory of quasilinear equations is unified with that of the heat equation up to spatial gradient continuity - and from the one of the consequences. For instance, when looking for sharp criteria for establishing Du L loc (and eventually in C0 ) in terms of Lorentz spaces, the result in (.26) gives that p p µ L(N, /(p )) = Du L loc, (.28)

10 0 TUOMO KUUSI & GIUSEPPE MINGIONE where we recall that µ L(N, /(p )) iff 0 2 p p {(x, t) ΩT : µ(x, t) > } N(p ) d <. The criterion of Corollary.5 is clearly stronger that the one in (.28), as L(N, ) L(N, /(p )), this inclusion being strict for p > 2; needless to say, here we also prove the gradient continuity. For further properties of Lorentz spaces we refer for instance to [46]. Moreover, it is easy too see that more refined criteria in terms of density/concentration are provided by (.9) with respect to (.26) when µ is genuinely a measure. We also remark that Wolff potentials play a major role in the analysis of the fine properties of quasilinear equations (see for instance [7, 8, 4, 42, 47]); since the estimates contained in this paper are stronger than those involving Wolff potentials, we expect they will have a similar, if not stronger, impact in the future..4. Techniques Finally, a few comments on the methods used in this paper. Several new ingredients are needed to deal with the parabolic case with respect to the previous elliptic one [24], and the proofs depart considerably from those proposed before. The proof of Theorem. involves a very delicate, double-step induction procedure based on a few ingredients that re-shuffle, in a pointwise manner, some classical methods used in linear Calderón-Zygmund theory and combine them with the use of intrinsic geometry. Extensive use of nonlinear potential theoretic methods and regularity theory is made throughout. Let us briefly describe the heuristic used here by specializing for simplicity to the model case (.) and considering µ L ; the essence relies in careful procedure that allows to linearize the equation and control the possible degeneracy in a precisely quantified way at every scale. The following argument will be purely formal. We consider a dyadic shirking sequence of intrinsic cylinders for i 0... Q r i+ (x 0, t 0 ) Q r i (x 0, t 0 ) Q r i (x 0, t 0 )... r i := σ i r where σ (0, ) is a constant depending only on n, p and is as in (.9). A suitable exit time argument, together with very careful regularity estimates for solutions to homogeneous equations, gives quantified error Du on Q r i (x 0, t 0 ) for i large enough. (.29) This is something that in a way we can always assume, starting from an exit time index, otherwise we are going to get an opposite inequality for the integral averages (Du) Q ri (x 0,t 0), that eventually leads to an immediate proof of Du(x 0, t 0 ), and therefore of (.9). Assuming (.29) leads to implement a delicate iteration procedure whose final outcome is the following inequality: Du p dx dt p (.30) Q r j (x 0,t 0)

11 Riesz potentials and nonlinear parabolic equations that again implies (.9). Note that proving (.30) not only allows to implement the iteration but also allows to use, at each scale, the intrinsic geometry methods (compare with (.4)). As emphasized, the key to the proof of Theorem. is the lower bound in (.29); let us now give a formal but yet convincing argument showing how a condition as (.29) allows to get (.9) and why intrinsic Riesz potentials and and conditions as (.9) naturally occur. Let us consider then (.29) to be satisfied on Q r (x 0, t 0 ) with a null error, i.e., Du, and let us assume the first inequality in (.9). The lower bound Du in turn allows to gain coercivity enough to treat the equation in display (.) as a heat equation with a coefficient, that is u t div ( p 2 Du) = µ that we can rewrite as 2 p u t u = 2 p µ in Q r (x 0, t 0 ). Now the effect of the use of intrinsic geometry and of the intrinsic Riesz potential shows up. Changing variables and introducing and v(x, t) := u(x 0 + rx, t p r 2 t) r µ(x, t) := 2 p rµ(x 0 + rx, t p r 2 t) for (x, t) Q = B (, 0), we have v t v = µ. (.3) Next, we apply the standard Riesz potential bound for solutions to (.3), that is ( /(p ) Dv(0, 0) I µ (0, 0; ) + Dv p dx dt). (.32) Q (0,0) Changing variables back to µ we notice I µ (0, 0; ) = 2 p r µ(x 0 + rx, t p r 2 t) dx dt dϱ = 2 p r 0 0 r = 2 p 0 Q ϱ(0,0) Q ϱr (x0,t0) µ(x, t) dx dt dϱ Q ϱ (x0,t0) µ(x, t) dx dt dϱ = 2 p I µ, (x 0, t 0 ; r), where in the last inequality we have used the first line in (.9). Finally, scaling back to u, using the previous inequality together with the first line of (.9), we conclude with Du(x 0, t 0 ) = Dv(0, 0), that is the proof of (.9). The one outlined in the last lines is only a heuristic argument used to show how intrinsic Riesz potentials play a decisive and natural role in this context, but its rigorous implementation is highly nontrivial and involves a refined double induction argument that exploits rather subtle aspects of regularity theory of degenerate parabolic

12 2 TUOMO KUUSI & GIUSEPPE MINGIONE equations. Several tools are used here. One of the main points is that the analysis of the relevant iterating quantities must be performed at two different levels, using different energy spaces. Indeed, since we are dealing essentially with measure data problems, the natural spaces involved are larger than L p ( T, 0; W,p ). This, together with the lack of reverse Hölder type inequalities and homogeneous estimates which is typical when dealing with degenerate parabolic equations, reflects in a simultaneous use of two different spaces, namely L ( T, 0; W, ) and L p ( T, 0; W,p ). Eventually, a very delicate interplay between local regularity of solutions to homogeneous equations and comparison estimates must be exploited in the framework of intrinsic geometries thanks to exit time arguments and the use of intrinsic Riesz potentials. The proof of Theorem. eventually opens the way to the continuity analysis and in particular to Theorem.4. For this we shall readapt the iteration procedure of Theorem. to estimate oscillations rather than the size of the gradient. This in turn imposes to consider a priori infinite many exit times arguments used to control the degeneracy of the equation via the oscillations of the gradient, and vice-versa. 2. Preparations 2.. General notation In what follows we denote by c a general positive constant, possibly varying from line to line; special occurrences will be denoted by c, c 2, c, c 2 or the like. All these constants will always be larger or equal than one; moreover relevant dependencies on parameters will be emphasized using parentheses, i.e., c c (n, p, ν, L) means that c depends only on n, p, ν, L. We denote by B(x 0, r) := {x R n : x x 0 < r} the open ball with center x 0 and radius r > 0; when not important, or clear from the context, we shall omit denoting the center as follows: B r B(x 0, r). Unless otherwise stated, different balls in the same context will have the same center. We shall also denote B B = B(0, ) if not differently specified. In a similar fashion standard and intrinsic parabolic cylinders with vertex (x 0, t 0 ) and width r > 0 have been defined in (.2) and (.3), respectively. When the vertex will not be important in the context or it will be clear that all the cylinders occurring in a proof will share the same vertex, we shall omit to indicate it, simply denoting Q r and Q r for the cylinders in (.2) and (.3), respectively. We recall that if Q = A (t, t 2 ) is a cylindrical domain, the usual parabolic boundary of Q is par Q := (A {t }) ( A [t, t 2 )), and this is nothing else but the standard topological boundary without the upper cap Ā {t 2}. Accordingly, we define the parabolic (topological) closure of Q as Q par := Q par Q. (2.)

13 Riesz potentials and nonlinear parabolic equations 3 With O R n+ being a measurable subset with positive measure, and with g: O R n being a measurable map, we shall denote by (g) O g(x, t) dx dt := g(x, t) dx dt O O its integral average; here O denotes the Lebesgue measure of O. A similar notation is adopted if the integral is only in space or time. In the rest of the paper we shall use several times the following elementary property of integral averages: ( /q g (g) O q dx dt) 2 O ( O /q g γ q dx dt), (2.2) O whenever γ R n and q. The oscillation of g on O is instead defined as osc O g := sup g(x, t) g( x, t). (x,t),( x, t) O Finally, we remark that we shall denote the partial derivative with respect to time of a function u both by u t and by t u; moreover, the letter will always denote a positive number. Further relevant notation is at the beginning of the next section Comparison maps The basic setup in this section is tailored to the needs of the proof of Theorem. and subsequent results. Therefore we shall consider u to be an energy solution to (.3) under the assumptions (.4)-(.5) and (.8) until the end Section 2.4; only in Section 2.5 we shall discuss the general case, thereby treating SOLA and discarding assumption (.8). With a point (x 0, t 0 ) Ω T being fixed, and given an intrinsic cylinder of the type Q r (x 0, t 0 ) B(x 0, r) (t 0 2 p r 2, t 0 ) such that Q 2r(x 0, t 0 ) Ω T, we consider a family of nested parabolic cylinders B j T j B(x 0, r j ) (t 0 2 p r 2 j, t 0 ) Ω T, r j := σ j r, (2.3) for a fixed decay parameter σ (0, /4). Accordingly, we consider their dyadic, parabolic dilations τ Q τr j (x 0, t 0 ) τb j τt j B(x 0, τr j ) (t 0 2 p (τr j ) 2, t 0 ) for τ > 0; notice that here, slightly abusing the notation, we are denoting τt j (t 0 2 p (τr j ) 2, t 0 ). (2.4) A similar notation will occur several times in rest of the paper. We also notice that, with respect to (2.3) we always have the inclusions

14 4 TUOMO KUUSI & GIUSEPPE MINGIONE Now, let w j C 0 (T j ; L 2 (B j )) L p (T j ; W,p (B j )) be the unique solution to the Cauchy-Dirichlet problem { t w j div a(x, t, Dw j ) = 0 in After having defined w j, we also define w j = u on par. v j C 0 ((/2)T j ; L 2 ((/2)B j )) L p ((/2)T j ; W,p ((/2)B j )) (2.5) as the unique solution to the frozen Cauchy-Dirichlet problem { t v j div a(x 0, t, Dv j ) = 0 in 2 ( v j = w j on par 2 Q ) (2.6) j A priori estimates for comparison maps We now derive various a priori estimates for w j and v j, starting from L - bounds. When turning our attention to w j we need to use the results recently established in [29], that allow to deal with equations with non-constant, Dini-continuous spatial coefficients. We start with a statement in terms of intrinsic geometry. Theorem 2. (Intrinsic gradient bound). Let w j be as in (2.5). There exists a positive radius R R (n, p, ν, L, ω( )) and a constant c c (n, p, ν, L) such that if ϱ (0, R ) and then the implication Q 0 ϱ (x, t ) := B(x, ϱ) (t 2 p 0 ϱ 2, t ), holds. ( /p c ( Dw j + s) p dx dt) 0 Q 0 ϱ (x,t ) = Dw j (x, t ) 0 (2.7) Proof. This result has been proved and used in [29, Theorem., Theorem.3, Theorem 4.]. The proof is exactly the one given in the proof of [29, Theorem.], once [29, Lemma 4.3] is used instead of [29, Lemma 4.2], see also [29, Remark 4.]. We also remark that no restriction on ϱ is necessary when the vector field a( ) is independent of x. The pointwise bound of Theorem 2. can be turned into an L -bound of exactly the same type proved by DiBenedetto [9] for equations with no coefficients, see also Theorem 2.4 below. Since we are going to cover the case of equations with measure data, where solutions with low degree of integrability naturally appear, we need to lower the p-integrability exponent to (p ) to get the correct form of a priori estimates. All this is done in the next

15 Riesz potentials and nonlinear parabolic equations 5 Corollary 2.2. Let w j be as in (2.5) and R as in Theorem 2.. There exists a constant c 2 c 2 (n, p, ν, L) such that if r (0, R ), then c 2 2 p sup Dw j + s c 2 ( + s) + τ m 2 n+2 ( τ m ) n+2 ( Dw j + s) p dx dt holds whenever τ m (0, ). In particular, we have sup Dw j + s c 2 ( + s) + c 2 2 p ( Dw j + s) p dx dt. 2 Qj Proof. Define 0 0 (τ, τ ) := 2 sup 2c p Dw j + + s + τ (τ τ 2 p ( Dw ) n+2 j + s) p dx dt whenever τ m τ < τ, where c c (n, p, ν, L) is as in Theorem 2.. As 0 and p 2, we clearly have that for δ := τ τ the inclusion Q 0 δr j (x, t ) := B(x, δr j ) (t 2 p 0 (δr j ) 2, t ) τ = τq r j (x 0, t 0 ) holds whenever (x, t ) τ. Furthermore, by using the very definition of 0 we may estimate ) /p ( c ( Dw j + s) p dx dt Q 0 δr (x j,t ) ( ) /p c 2 /p /p 0 ( Dw j + s) p dx dt Q 0 δr (x j,t ) ( ) /p ( c 2 /p /p 0 ( Dw j + s) p dx dt Q 0 δr j (x, t ) ( ) /p ( = c 2 /p /p 2 p 0 δ n+2 2 p ( Dw j + s) p dx dt 0 ( ) /p ( 0 δ n+2 c 2 /p /p 0 2 p δ n+2 2 p 0 2c p 2 p ) /p = 0. ) /p ) /p Therefore Theorem 2. implies that Dw j (x, t ) 0. But this holds for all (x, t ) τ and thus sup Dw j τ 2 sup Dw j ++s+ 2cp 2 p τ (τ τ ) n+2 ( Dw j +s) p dx dt (2.8) follows. Lemma 2.3 below applied with ϕ(τ) = sup τ Dw j then concludes the proof by properly choosing the constant c 2.

16 6 TUOMO KUUSI & GIUSEPPE MINGIONE The next result is a classical iteration lemma for the of which we refer to [32, Lemma 6.]. Lemma 2.3. Let ϕ : [τ m, ] [0, ), with τ m (0, ), be a function such that ϕ(τ ) 2 ϕ(τ) + K + B (τ τ ) n+2 holds for every τ m τ < τ, where B, K 0. Then ϕ(τ m ) c(n)k + ( τ m ) (n+2) B. Corollary 2.2 obviously holds for v j too, and in this case it is a by now classical estimate of DiBenedetto [9], as already mentioned above. See also for example [, 25] for similar bounds. We report the statement for completeness. Theorem 2.4. Let v j be as in (2.6). For a constant c 3 c 3 (n, p, ν, L) we have sup Dv j + s c 3 ( + s) + c 3 2 p ( Dv j + s) p dx dt. 4 Qj 2 Qj We now pass to give oscillation estimates for w j and v j. The next result provides a gradient oscillations estimate for solutions to homogeneous equations with Dinicontinuous coefficients. Theorem 2.5 (Oscillation reduction). Let w j be as in (2.5), then Dw j is continuous. Moreover, assume that sup 2 Qj Dw j + s A (2.9) holds for some A. Then, for every δ (0, ) there exists a positive constant σ σ (n, p, ν, L, A, δ, ω( )) (0, /4) such that osc σ Dw j δ. (2.0) Proof. The starting point of the proof is the work in [29], where the continuity of the gradient of solutions to parabolic equations as in (.3) has been proved under the assumption of Dini-continuity of the space coefficients; this by the way immediately implies the continuity of Dw j claimed in the statement. What we need here is a quantitative bound on the oscillations of Dw j. To this aim, let us briefly recall the main arguments in [29, proof of Theorem.3], where the continuity properties of Du are formulated and proved in terms of the auxiliary vector field V (z) := ( z 2 + s 2 ) (p 2)/4 z and the related field V (Dw j ). For the use of such maps in the present context we refer to [29] and related references; the only property we shall use here is the following inequality: V (z ) V (z 2 ) z z 2, (2.) c v (s 2 + z 2 + z 2 2 )(p 2)/4

17 Riesz potentials and nonlinear parabolic equations 7 that holds for c v c v (n, p) and for all vectors z, z 2 R n which are not simultaneously null; see for instance [39]. By following the arguments developed for [29, (5.5)] it can be proved that for every ε (0, ) there exists a positive radius R ε R ε (n, p, ν, L, ω( ), ε) (0, /6) such that and ( Q A ϱ ( x, t) (V (Dw j )) Q A τ ( x, t) (V (Dw j)) Q A ϱ ( x, t) (A)p/2 ε (2.2) V (Dw j ) (V (Dw j )) Q A ϱ ( x, t) 2 dx dt ) /2 (A) p/2 ε (2.3) hold whenever ( x, t) 4 and 0 < τ ϱ R ε r j ; notice that R ε is in particular independent of, A and the considered cylinder. Letting τ 0 in (2.2) and recalling that V (Dw j ) is continuous yields V (Dw j ( x, t)) (V (Dw j )) Q A ϱ ( x, t) (A)p/2 ε ϱ (0, R ε r j ]. (2.4) We are now ready to finish the proof with the choices ε := δ p/2 c v 2 p/2 48 N A p/2, σ := A(2 p)/2 R ε 32. The constant c v is the one appearing in (2.) Notice that the dependence of σ upon n, p, ν, L, A, δ, ω( ), as described in the statement, appears through the one implicitly contained in R ε. Take now (ỹ, s), ( x, t) σ ; we can assume that t s otherwise we can interchange the roles between the two points in the next lines. It obviously follows that Q A R εr j/8 (ỹ, s) QA R εr j ( x, t) 4. Using this last fact, thanks to Jensen s inequality, the one in display (2.3) and using also (2.2), we have (V (Dw j )) Q A Rεr j /8 (ỹ, s) (V (Dw j )) Q A Rεr j /8 ( x, t) V (Dw j ) (V (Dw j )) Q A Q A Rεr j /8 (ỹ, s) ( x, t) dx dt Rεr j /8 2 V (Dw j ) (V (Dw j )) Q A Q A Rεr j /8 (ỹ, s) Rεr ( x, t) dx dt j 6 N Q A Rεr j ( x, t) V (Dw j ) (V (Dw j )) Q A Rεr ( x, t) dx dt j ( 6 N V (Dw j ) (V (Dw j )) Q A Q A Rεr ( x, t) 2 dx dt Rεr ( x, t) j j 6 N (A) p/2 ε. ) /2

18 8 TUOMO KUUSI & GIUSEPPE MINGIONE By using the previous estimate and (2.4) (actually used also for (ỹ, s) instead of ( x, t)) together with the triangle inequality we easily gain V (Dw j ( x, t)) V (Dw j (ỹ, s)) 48 N (A) p/2 ε. (2.5) We are now ready to show (2.0) proving that Dw j ( x, t) Dw j (ỹ, s) δ (2.6) whenever (ỹ, s), ( x, t) σ. First of all, observe that we can assume that either Dw j ( x, t) δ/2 or Dw j (ỹ, s) δ/2 holds otherwise we are done. The inequalities in (2.) and (2.5) then imply (2.6) as follows: Dw j ( x, t) Dw j (ỹ, s) c v 48 N (A) p/2 (δ/2) p/2 ε = δ and the proof is complete. Remark 3. The proof of the previous result allows in fact, together with the argument given in [29], to get an explicit modulus of continuity for the gradient of solutions of equations with Dini-continuous coefficients. Indeed, the choice of the radius R ε r j making (2.2) is made to meet a condition of the form Rεr j ω(r ε r j ) + ω(ϱ) dϱ 0 ϱ εb c for some positive constants b and c depending only on n, p, ν, L, A. This gives a modulus of continuity involving a power of the function r ω(r) + r 0 ω(ϱ) dϱ ϱ which is in accordance to the known results in the classical elliptic regularity theory. We now collect a few results from [25,26], that is Theorems below, which are aimed to provide oscillation estimates for the functions v j. Theorems have actually been presented in [25,26] for solutions to equations of the type u t div a(du) = 0. (2.7) On the other hand, by following the arguments in [25,26] it is not difficult to see that all the proofs carry out for solutions to equations of the type in display (.20). Therefore Theorems apply to the functions v j as well, that indeed solve equations as in (.20). The next statement is a slight variant of [25, Theorem 3.].

19 Riesz potentials and nonlinear parabolic equations 9 Theorem 2.6. Let v j be as in (2.6). Consider numbers A, B and ε (0, ). Then there exists a constant σ 2 (0, /4) depending only on n, p, ν, L, A, B, ε such that if holds, then B sup Dv j s + sup Dv j A (2.8) σ 2Q j 4 Qj Dv j (Dv j ) τqj dx dt ε Dv j (Dv j ) (/4)Qj dx dt (2.9) τ (/4) holds too, whenever τ (0, σ 2 ]. Remark 4. The essence of the previous result lies in the fact that once the bounds (2.8) are satisfied, then solutions to evolutionary p-laplacean type equations satisfy elliptic type decay estimates as in (2.2) when framed in the proper intrinsic geometry dictated by (2.8). Indeed, let us denote by E(f, Q) the usual excess functional E(f, Q) := f (f) Q dx dt (2.20) Q which is defined whenever f is an integrable function and Q a measurable set with positive measure; this functional gives an integral measure of the oscillations of f in a subset Q. Estimate (2.9) now reads as E(Dv j, τ ) εe(dv j, (/4) ). (2.2) Theorem 2.6 gives the natural analog, when passing to the framework of degenerate parabolic equations of p-laplacean type, of the classical results known for solutions to the heat equations. Indeed, Theorem 2.6 holds without assuming (2.8) for solutions to (.2). This is a classical result of Campanato [5]. Using Theorem 2.6 it is possible to give a proof of the Hölder continuity of the gradient of solutions to frozen equations, as for instance shown in [26, Theorem 3.2]; see also [25, Theorem 3.2]. Theorem 2.7. Let v j be as in (2.6). For every A there exist constants c 4 c 4 (n, p, ν, L, A) and α α(n, p, ν, L, A) such that sup 4 Qj Dv j + s A = osc τqj Dv j c 4 τ α τ (0, /4).

20 20 TUOMO KUUSI & GIUSEPPE MINGIONE 2.4. Comparison estimates We start this section by a reformulation of a result established in [25, Lemma 4.] and [25, (4.5), (4.6)]. We remark that the result there was presented only for equations without coefficients as in (2.7). Nevertheless, the proof works directly for general equations with merely measurable coefficients; the crucial point is the strict monotonicity in the gradient variable. Lemma 2.8. Let u be as in Theorem. and w j as in (2.5) with j 0. Let ε (0, /(n + )]. There exist constants c c (n, p, ν, ε) and c 2 c 2 (n, p, ν) such that ( Du Dw j q dx dt ) /q [ ] (n+2)/[(p )n+p] µ ( ) c r N (2.22) j holds for any 0 < q p + /(n + ) ε. Moreover, the inequalities and sup τ T j B j u w j (x, τ) dx µ ( ) (2.23) ( Du + Dw j ) Qj p 2 Du Dw j 2 α ξ (α + u w j ) ξ dx dt c 2 ξ hold for any α > 0 and ξ >. [ µ ( ) 2 p r N j ] (2.24) We here recall a parabolic Sobolev-Poincaré inequality that will be useful in the sequel; we refer to [9, Chapter, Proposition 3.] for the proof. Proposition 2.9. Let v L (T j ; L m (B j )) L q2 (T j ; W,q2 0 (B j )) for q 2, m. There exists a constant c depending only on n, q 2, m such that the following inequality holds for q = q 2 (n + m)/n: ( ) ( ) q2/n v q dx dt c Dv q2 dx dt sup τ T j B j v(x, τ) m dx Using the previous result and Lemma 2.8 we get another comparison estimate. Lemma 2.0. Let u be as in Theorem. and w j, w j as in (2.5), with j. Then, for any ε (0, /(n + )], there exists a constant c 3 c 3 (n, p, ν, ε, σ) such that the inequality ( u w j q dx dt ) /q [ ] (n+p)/[(p )n+p] µ ( ) c 3 r j r N (2.25) j holds whenever 0 < q p + p/n ε and ( ) /q [ ] (n+2)/[(p )n+p] Dw j Dw j q µ ( ) dx dt c 3 r N j (2.26).

21 Riesz potentials and nonlinear parabolic equations 2 holds whenever 0 < q p + n + ε. Proof. To prove (2.25), we use Proposition 2.9 with the choice m =, q := p + p n ε, q 2 := n n + q = p + n + n ε n + and v u w j L (T j ; L 2 (B j )) L q2 (T j ; W,q2 0 (B j )); recall (2.23). This yields ( ) /q u w j q dx dt ([ ] /q2 [ c Du Dw j q2 dx dt sup τ T j ] /n ) n/(n+) u w j dx. B j Let c be as in Lemma 2.8 be the constant corresponding to the choice εn/(n + ) instead of ε. Substituting (2.22) and (2.23) into the previous estimate and computing the exponents leads to ( u w j q dx dt c c n/(n+) c 3 r j [ ) /q [ ] (n+2)/[(p )n+p] [ µ ( )] /n µ ( ) r N j ] (n+p)/[(p )n+p] µ ( ) r N, j n/(n+) which, together with Hölder s inequality, proves (2.25). Here we have also used that N = n +, that r j = σr j and the identity [ n n + 2 n + (p )n + p + ] n + p = n (p )n + p. As for (2.26) we instead argue as follows: ( Dw j Dw j q2 dx dt ) /q2 ( ) ( /q2 Qj Du Dw j q2 dx dt ( ) /q2 + Du Dw j q2 dx dt ) /q2

22 22 TUOMO KUUSI & GIUSEPPE MINGIONE [ ( Qj ) /q2 c + rn j c 3 [ µ ( ) r N j r N j ] [ ] (n+2)/[(p )n+p], ] (n+2)/[(p )n+p] µ ( ) r N j where we have repeatedly applied (2.22). Now (2.26) follows, again by Hölder s inequality, as ε is arbitrary. The following lemma provides one of the key estimates to obtain Theorem.. Lemma 2.. Let u be as in Theorem. and w j, w j as in (2.5), with j. Suppose further that µ ( ) r N j (2.27) and that the bounds A Dw j A in (2.28) hold for some A. Then there exists a constant c 4 depending only on n, p, ν, σ and A such that [ ] µ ( ) Du Dw j dx dt c 4 r N. (2.29) j Proof. Let us begin by fixing several parameters appearing in the proof: γ := 4(p )(n + ), ξ := + 2γ, ε := 2(n + ), (2.30) and, throughout the proof, we will apply Lemmas 2.8 and 2.0 with exponents 0 < q q := ξ(p ) = p + 2(n + ). Let c, c 2, c 3 be as in Lemma 2.8 and Lemma 2.0, respectively, corresponding to these choices of σ and ε; therefore they ultimately depend only on n, p, ν, σ. We also set w j := w j, w j := w j. (2.3) In what follows constants denoted by c will only depend on n, p, ν, σ, A and will in general vary from line to line. We start to estimate the term on the left in (2.29) with the aid of (2.28) as follows: Du Dw j dx dt A (p 2)(+γ) D w j (p 2)(+γ) Du Dw j dx dt c D w j D w j (p 2)(+γ) Du Dw j dx dt +c D w j (p 2)(+γ) Du Dw j dx dt. (2.32)

23 Riesz potentials and nonlinear parabolic equations 23 Appealing to Hölder s inequality, together with (2.22) and (2.26), gives us D w j D w j (p 2)(+γ) Du Dw j dx dt But now, as (p 2)(+γ) ( Dw j Dw j (p )(+γ) dx dt c(σ) c (+γ)(p 2) 3 c [ ( ) (p 2)/(p ) ) /(p ) Du Dw j p dx dt ] [(+γ)(p 2)+](n+2)/[(p )n+p]. µ ( ) r N j [( + γ)(p 2) + ] (n + 2) (p )n + p precisely for p 2, (2.27) implies > (p )(n + 2) (p )n + p (2.33) D w j D w j (p 2)(+γ) Du Dw j dx dt c µ ( ) r N j with c c(n, p, ν, σ). Therefore (2.32) gives us Du Dw j dx dt c D w j (p 2)(+γ) Du Dw j dx dt + c µ ( ) r N j. (2.34) We then continue to estimate the first term on the right in the above display. Applying Hölder s inequality, together with (2.24), and recalling that ξ = + 2γ, we obtain for any α > 0 that D w j (p 2)(+γ) Du Dw j dx dt [ 2 p ( Du + Dw j ) p 2 Du Dw j 2 ] /2 (α + u w j ) ξ [ D w j (+2γ)(p 2) (α + u w j ) ξ] /2 dx dt ( (2 p)/2 ( Du + Dw j ) p 2 Du Dw j 2 (α + u w j ) ξ dx dt ( ) /2 D w j ξ(p 2) (α + u w j ) ξ dx dt ) /2

24 24 TUOMO KUUSI & GIUSEPPE MINGIONE ( ) [ ] /2 /2 c2 α ( ξ)/2 µ ( ) ξ rj N ( D w j ξ(p 2) (α + u w j ) ξ dx dt As the choice of α is still in our disposal, we set α := ( D w j ξ(p 2) u w j ξ dx dt ) /2. (2.35) ) /ξ + δ (2.36) for some small positive δ (0, ) to get, enlarging the constants involved ( D w j ξ(p 2) (α + u w j ) ξ dx dt ) /2 2α ξ/2 ( D w j ξ(p 2) dx dt) /2 + 2α ξ/2. Notice that since w j belongs to the parabolic Sobolev space L p (T j ; W,p (B j )) and ξ(p ) < p by (2.30), we have that α is finite by Hölder s inequality. The presence in (2.36) of the parameter δ, which shall be sent to zero at the end of the proof, guarantees that α is positive. In the above display, the integral on the right can be estimated by means of Lemma 2.0 as D w j ξ(p 2) dx dt c D w j D w j ξ(p 2) dx dt + c D w j ξ(p 2) dx dt [ ] ξ(p 2)(n+2)/[(p )n+p] µ ( ) r N + ca ξ(p 2) c, j c c ξ(p 2) 3 owing to (2.27) and (2.28), while the last constant c depends only on n, p, ν, σ, A. Thus (2.35), together with the last two displays, yields [ ] /2 α D w j (p 2)(+γ) µ ( ) Du Dw j dx dt c r j r N j so that applying Young s inequality together with an obvious estimation in turn gives D w j (p 2)(+γ) Du Dw j dx dt c β µ ( ) r N j + βα r j (2.37)

25 Riesz potentials and nonlinear parabolic equations 25 for all β (0, ), where c c(n, p, ν, σ, A). Inserting this into (2.34) leads to Du Dw j dx dt c µ ( ) β r N + βα (2.38) r j j again for all β (0, ), where c c(n, p, ν, σ, A) is in particular independent of β. We then focus on α, which has been defined in (2.36), and split as follows: ( α c D w j D w j ξ(p 2) u w j ξ dx dt ( ) /ξ +c D w j ξ(p 2) u w j ξ dx dt + δ ) /ξ =: I + I 2 + δ. (2.39) By (2.30), as ξ(p ) = q, we get by (2.25)-(2.26), together with Hölder s inequality, that Since I c ( D w j D w q j dx dt [ c c p 3 r j ) (p 2)/ q ( u w j q dx dt ] [(p 2)(n+2)+n+p]/[(p )n+p] µ ( ) r N. j (p 2)(n + 2) + n + p (p )n + p = + precisely for p 2 we have, in view of (2.27), that [ ] µ ( ) I cr j r N j 2(p 2) (p )n + p, for c c(n, p, ν, σ). On the other hand, using condition (2.28) we obtain ) / q I 2 ca p 2 ( u w j ξ dx dt) /ξ. Using Proposition 2.9, estimate (2.23), and Young s inequality we get ( ) /ξ ( ) n/(n+) u w j ξ dx dt u w j (n+)/n dx dt [ ] /n c Du Dw j dx dt sup u w j dx t T j B j n/(n+)

26 26 TUOMO KUUSI & GIUSEPPE MINGIONE ( ) n/(n+) [ µ ( ) c Du Dw j dx dt r j r N j [ ] µ ( ) cr j Du Dw j dx dt + cr j r N. j ] /(n+) Combining the estimates contained in the last three displays with (2.39) leads to α c r j Du Dw j dx dt + c r j [ ] µ ( ) r N + δ j with c c (n, p, ν, σ, A). Inserting this finally into (2.38) with β = /(2c ) and then reabsorbing terms and sending δ to zero finishes the proof. Next, a comparison estimate between w j and v j. Lemma 2.2. Let w j and v j be as in (2.5) and (2.6), respectively, with j 0. For every A there exists a constant c 5 c 5 (n, p, ν, L, A) such that the following holds: sup Dw j + s A 2 Qj = ( Dw j + Dv j ) p 2 Dw j Dv j 2 dx dt 2 Qj + Proof. The result is based on the following estimate: Dw j Dv j p dx dt c 5 [ω(r j )] 2 p. (2.40) 2 Qj ( Dw j + Dv j ) p 2 Dw j Dv j 2 dx dt 2 Qj c [ω(r j )] 2 ( Dw j + s) p dx dt 2 Qj that has been proved in [29, Lemma 4.3] and [], with a constant c c(n, p, ν, L). The inequality in display (2.40) follows by trivially estimating [ω(r j )] 2 ( Dw j + s) p dx dt [ω(r j )] 2 sup( Dw j + s) p c 5 [ω(r j )] 2 p 2 Qj 2 Qj and taking into account that p 2. Finally, along the lines of Lemma 2., we yet prove another comparison estimate, this time between u and v j.

27 Riesz potentials and nonlinear parabolic equations 27 Lemma 2.3. Let u be as in Theorem. and let w j and v j be as in (2.5) and (2.6), respectively, with j. Suppose further that (2.27) holds together with sup Dw j + s A 2 Qj (2.4) A Dw j A in, for some A. There exist positive constants c 6 c 6 (n, p, ν, L, A) and c 7 c 7 (n, p, ν, σ, A) such that the following inequality holds: [ ] µ ( ) Du Dv j dx dt c 6 ω(r j ) + c 7 r N. (2.42) 2 Qj j Proof. We shall keep the notation introduced for Lemma 2.; in particular, we recall (2.3). Inequality (2.4) allows to use (2.40) so that ( Dw j + Dv j ) p 2 Dw j Dv j 2 dx dt 2 Qj + Dw j Dv j p dx dt c 5 [ω(r j )] 2 p. (2.43) 2 Qj With p = p/(p ), by (2.4) 2 we continue estimating as Dw j Dv j dx dt 2 Qj A (p 2)/p c D w j (p 2)/p 2 Qj Dw j Dv j dx dt (2.44) D w j (p 2)/p Dw j Dv j dx dt 2 Qj +c D w j D w j (p 2)/p Dw j Dv j dx dt (2.45) 2 Qj with c depending only on p and A and we are using the notation in (2.3). As for the first term in the right hand side of (2.45), we notice that since p 2 we have (p 2)/2 (p 2)/p so that (2.4) allows to estimate D w j (p 2)/p 2 Qj Dw j Dv j dx dt A (p 2)2/(2p) (2 p)/2 Dw j (p 2)/2 Dw j Dv j dx dt 2 Qj so that, using Hölder s inequality and (2.43) yields D w j (p 2)/p Dw j Dv j dx dt 2 Qj ) /2 ( c (2 p)/2 ( Dw j + Dv j ) p 2 Dw j Dv j 2 dx dt 2 Qj /2 c (2 p)/2 c 5 ω(r j ) p/2 = cω(r j ) (2.46)

28 28 TUOMO KUUSI & GIUSEPPE MINGIONE for c c(n, p, ν, L, A). The second term in the right hand side of (2.45) is estimated with the aid of Hölder s inequality, (2.26), (2.43) and finally Young s inequality, with conjugate exponents (p/2, p/(p 2)) when p > 2; this means D w j D w j (p 2)/p Dw j Dv j dx dt 2 Qj ( ) /p c D w j D w j p 2 dx dt ( c (p 2)/p 3 c /p 5 c [ [ µ ( ) r N j Dw j Dv j p dx dt 2 Qj µ ( ) r N j ) /p ] [(p 2)/p](p )(n+2) [n(p )+p] ] (p )(n+2)/[n(p )+p] + ω(r j ) [ω(r j )] 2/p for c c(n, p, ν, L, σ, A). Thanks to (2.27) and (2.33), we get [ ] D w j D w j (p 2)/p µ ( ) Dw j Dv j dx dt c r N + ω(r j ) 2 Qj j with c c(n, p, ν, L, σ, A). By using the last inequality together with (2.45) and (2.46) we conclude with [ ] µ ( ) Dw j Dv j dx dt cω(r j ) + c r N 2 Qj j for c c(n, p, ν, L, A) and c c(n, p, ν, L, σ, A). Note next that the assumptions of this lemma fulfill also the assumptions of Lemma 2.. Appealing then to (2.29) and and to the triangle inequality finishes the proof Comparison lemmas for SOLA In this section we show that the basic comparison Lemmas 2.8 and 2.0 hold for SOLA in a suitably modified way. This fact ultimately allows to prove Theorem.7 by mean of the same proofs already given for the other theorems when considered for energy solutions. Before starting, since the setting here is the one defined in Theorem.7, we assume the existence of functions u h being local weak solutions to the equations considered in (.24) and such that { Duh L p, Du h Du in L p (2.47) u h u and Du h Du a.e.