Current Issues on Singular and Degenerate Evolution Equations

Transcription

1 Current Issues on Singular and Degenerate Evolution Equations Emmanuele DiBenedetto Deartment of Mathematics, Vanderbilt University 136 Stevenson Center, Nashville TN , USA José Miguel Urbano Deartamento de Matemática, Universidade de Coimbra Coimbra, Portugal Vincenzo Vesri Diartimento di Matematica, Università di Firenze Viale Morgagni 67a, Firenze, Italy Contents 1 Introduction 1.1 Historical background on regularity and Harnack estimates A new aroach to regularity Limiting cases and miscellanea remarks Singular equations of the Stefan tye Outline of these notes Regularity of weak solutions 10.1 Weak solutions and local estimates Local energy and logarithmic estimates Some technical tools The classical aroach of De Giorgi The degenerate case > The geometric setting and the alternative Analysis of the first alternative Analysis of the second alternative The Hölder continuity The singular case 1 < < Rescaled iterations The first alternative The second alternative Proof of the main roosition The orous medium equation and other generalisations

2 3 Boundedness of weak solutions The degenerate case > Global estimates for solutions of the Dirichlet roblem Estimates in Σ T The singular case 1 < < Estimates near t = Global estimates: Dirichlet data A counterexamle Intrinsic Harnack Estimates Harnack estimates: the degenerate case Harnack estimates: the singular case Ellitic-tye Harnack estimates and extinction time Raleigh quotient and extinction rofile Stefan-like roblems The continuity of weak solutions A bridge between singular and degenerate equations Parabolic equations with one-oint singularity Parabolic equations with multile singularities The statement of the alternative A lemma of measure theory The geometric aroach The case N = The geometric aroach continued The case N The case N 3 continued References 97 1 Introduction Let Ω be an oen set in R N and consider the quasilinear, arabolic, artial differential equation of the second order 1.1) { u L loc 0, T ; W 1, loc Ω)), > 1; u t div Ax, t, u, u) = Bx, t, u, u) weakly in Ω T. Here T > 0 is given, Ω T = Ω 0, T ) and denotes the gradient with resect to the sace variables x = x 1,..., x N ). The functions A = A 1,..., A N ) and B are real valued, measurable with resect to their arguments, and satisfying the structure conditions { C0 u u C Ax, t, u, u) u ; 1.) Ax, t, u, u) + Bx, t, u, u) C 1 + u 1), where C 0 and C are given ositive constants. The quantity C 0 u is the modulus of elliticity of the equation. If > it vanishes whenever u = 0 and the equation

3 is said to be degenerate at those x, t) Ω T where this occurs. If 1 < < the modulus of elliticity becomes infinity whenever u = 0 and the equation is said to be singular at those x, t) Ω T where u = 0. Along with 1.1) 1.), consider also the quasilinear equation, 1.1) m { u L loc 0, T ; W 1, loc Ω)) ; u t div Ax, t, u, u) = Bx, t, u, u) weakly in Ω T, with structure conditions, { C0 u m 1 u C Ax, t, u, u) u ; 1.) m Ax, t, u, u) + Bx, t, u, u) C u m u ), where m is a given ositive number. The rototye examle of 1.1) m 1.) m is 1.3) m u t u m 1 u = 0 for some m > 0, and a the rototye examle of 1.1) 1.) is, 1.3) u t div u u = 0 for some > 1. The first is called the orous medium equation. If m > 1 the modulus of elliticity vanishes for u = 0 and the equation is degenerate at those oints of Ω T where the solution u vanishes. If 0 < m < 1 the modulus of elliticity is infinity whenever u = 0 and the equation is singular at those oints of Ω T where u = 0. If m > 1 the equation is referred to as the slow diffusion case of the orous medium equation. The case 0 < m < 1 is the fast diffusion. The equation in 1.3) is the Lalacian equation and its modulus of elliticity is u. If > such a modulus vanishes whenever u = 0 and at such oints the equation is degenerate. If 1 < < the modulus of elliticity becomes infinity whenever u = 0 and the equation is singular at these oints. If m = 1 in 1.3) m or = in 1.3) one recovers the classical heat equation for which information are essentially encoded in the fundamental solution } 1 x y 1.4) Γx, y; t, τ) = ex { x, y R N, t > τ. t τ) N/ 4t τ) The orous medium equation in 1.3) m admits an exlicit similarity solution that resembles the fundamental solution of the heat equation. Such a solution is called the Barenblatt similarity solution and it is given by [17]) 1.4) m Γ m x, y; t, τ) = where for a real number α, {α} + = max{α; 0}, and { [ ] } 1 m 1 1 x y 1 γ t τ) N/λ m t τ) 1/λ + t > τ 1.4) m λ = Nm 1)+, γ m = 1 λ m 1. 3

4 An examination of this solution reveals that it is well defined for all ositive values of m for which λ > 0. One also verifies that Γ m Γ as m 1. In this sense Γ m is the fundamental solution of the orous medium equation. Also the Lalacian equation 1.3) admits exlicit similarity solutions, 1.4) Γ x, y; t, τ) = where 1.4) κ = N )+, γ = { [ ] } 1 1 x y 1 1 γ t τ) N/κ t τ) 1/κ + ) κ t > τ This is well defined for all > 1 such that κ > 0 and Γ Γ as. In this sense Γ is the fundamental solution of the Lalacian equation. Issues of comact suort and regularity Assume first m > 1 and >. The first difference between these fundamental solutions and the fundamental solution of the heat equation is in their suort with resect to the sace variables. For fixed t > τ and y R N, the functions x Γ m x), Γ x) are comactly suorted in R N, whereas x Γx) is ositive in the whole R N. The moving boundaries searating the regions where Γ m and Γ are ositive from the regions where they vanish are, t τ) = γ m x y for Γ m τ R, y R N fixed. t τ) /κ = γ 1 x y for Γ The second difference is in their degree of regularity. For fixed t > τ and y R N the function x Γ m x) is Hölder continuous with Hölder exonent 1/m 1). The function x Γ x) is differentiable and its artial derivatives are Hölder continuous with Hölder exonent 1/ ). Such a modest degree of regularity is in contrast to the fundamental solution of the heat equation which is analytic in the sace variables. Let now 0 < m < 1 by keeing λ > 0. Then Γ m is ositive and locally analytic in the whole R N {t > τ}. Thus Γ m seems to share the same roerties as Γ. If 1 < < by keeing κ > 0 then Γ is ositive in the whole R N {t > τ} but still it maintains a limited degree of regularity. Issues of Harnack inequalities Non negative, local solutions of the heat equation in Ω T satisfy the Harnack inequality. This is a celebrated result of Hadamard [89] and Pini [144] and it takes the following form. Fix x 0, t 0 ) Ω T and for ρ > 0 consider the ball B ρ x 0 ) centered at x 0 and with radius ρ and the cylindrical domain 1.5) Q ρ x 0, t 0 ) = B ρ x 0 ) t 0 ρ, t 0 + ρ ). These are the arabolic cylinders associated with the heat equation and also to 1.3) m since these equations remain unchanged under a similarity transformation of the sace time variables that kees constant the ratio x /t. 4

5 There exists a constant C deending only uon N and indeendent of x 0, t 0 ) and ρ, such that 1.6) C ux 0, t 0 ) su ux 0, t 0 ρ ) rovided Q ρ x 0, t 0 ) Ω T. B ρx 0) The roof is based on local reresentations by means of the heat otentials 1.4). In articular, for fixed τ R and y R N, the fundamental solution x, t) Γx, t) satisfies such a Harnack estimate. It is then natural to ask whether the fundamental solution 1.4) m would satisfy a Harnack estimate. Take for examle the Γ m for τ = 0 and y the origin of R N. Assume first that m > 1 and fix x 0, t 0 ) on the moving boundary, so that Γ m x 0, t 0 ) = 0. If ρ and x 0 are sufficiently large, the ball B ρ x 0 ) intersects the suort of Γ m at t = t 0 ρ. Therefore for such choices the left hand hand side of 1.6) is zero and the right hand side would be ositive. If 0 < m < 1 and λ > 0, take x 0 = 0 and t 0 > 4ρ. Then a direct calculation shows that 1.6) cannot be verified for a constant C indeendent of ρ and t 0. Consider now 1.3) and the corresonding Γ. The similarity rescaling that kees 1.3) invariant is x /t = const. Therefore the natural arabolic cylinders associated with 1.3) are of the tye 1.5) Q ρ x 0, t 0 ) = B ρ x 0 ) t 0 ρ, t 0 +ρ ). Similar arguments show that Γ does not satisfy the Harnack inequality for all > 1 such that κ > 0, even in its own natural arabolic geometry 1.5). Local behavior of solutions These issues suggest a unifying theory of the local behavior of weak solutions of degenerate or singular arabolic equations. A cornerstone of such a unifying theory would be that weak solutions of 1.1) m, 1.) m, are Hölder continuous. Another key comonent would be an understanding of the Harnack estimate in the degenerate or singular setting of 1.1) m, 1.) m,. Whether for examle there is a form of such an estimate that relaces 1.6) and that would reduce to it when either m 1 or. The general structure in 1.1) m, 1.) m, is not an artificial requirement. To illustrate this oint we return briefly on the issue of regularity of solutions of 1.3). We have observed that the sace gradient of the fundamental solution Γ for > is locally Hölder continuous in R N τ, ). It is then natural to conjecture that the same would be true for local solutions u of 1.3). For such solutions, it turns out that v = u formally satisfies [55],[165]) v t a ij v vxi 0 where a ij = δ ij + ) )x u x i u xj j u. This is a quasilinear version of 1.3) m with m = /. Thus an investigation of the local regularity of solutions of 1.3) requires an understanding of degenerate or singular equations with the general quasilinear structure 1.1) m -1.) m. 1.1 Historical background on regularity and Harnack estimates Considerable rogress was made in the early 1950s and mid 1960s in the theory of ellitic equations, due to the discoveries of DeGiorgi [47] and Moser [13], [133]. Consider 5

6 local weak solutions of 1.7) a ij u xi ) xj = 0 weakly in Ω ; u W 1, loc Ω) where the coefficients x a ij x), i, j = 1,,..., N are only bounded and measurable and satisfying the elliticity condition 1.8) a ji ξ i ξ j C 0 ξ a.e. in Ω, ξ R N, for some C 0 > 0. DeGiorgi established that local solutions are Hölder continuous and Moser roved that non-negative solutions satisfy the Harnack inequality. Such inequality can be used, in turn, to rove the Hölder continuity of solutions. Both authors worked with linear.d.e. s. However the linearity has no bearing in the roofs. This ermits an extension of these results to ellitic quasilinear equations of the tye 1.9) div Ax, u, u) + Bx, u, u) = 0 weakly in Ω ; u W 1, loc Ω) ; > 1 with structure conditions 1.10) { C0 u C Ax, u, u) u ; Ax, u, u) + Bx, u, u) C 1 + u 1), for a given C 0 > 0 and a given non negative constant C. By using the methods of DeGiorgi, Ladyzhenskaja and Ural tzeva [10] established that weak solutions of 1.9) 1.10) are Hölder continuous, whereas Serrin [157] and Trudinger [163], following the methods of Moser, roved that non-negative solutions satisfy a Harnack rincile. The generalisation is twofold i.e., the rincial art Ax, u, u) is ermitted to have a non-linear deendence with resect to u, and a non-linear growth with resect to u. The latter is of articular interest since the equation in 1.9) might be either degenerate or singular. A striking result of Moser [134] is that the Harnack estimate 1.6) continues to hold for non-negative, local, weak solutions of 1.11) { u L 0, T ; L Ω) ) L 0, T ; W 1, Ω) ) ; u t a ij x, t)u xi ) xj = 0 in Ω T, where a ij L Ω T ) satisfy the analog of the elliticity condition 1.8). As before, it can be used to rove that weak solutions are locally Hölder continuous in Ω T. Since the linearity of 1.11) is immaterial to the roof, one might exect, as in the ellitic case, an extension of these results to quasilinear equations of the tye 1.1) u t div Ax, t, u, u) = Bx, t, u, u) in Ω T where the structure condition is as in 1.10). Surrisingly however, Moser s roof could be extended only for the case =, i.e., for equations whose rincial art has a linear growth with resect to u. This aears in the work of Aronson and Serrin [16] and Trudinger [164]. The methods of DeGiorgi also could not be extended. Ladyzenskaja et al. [11] roved that solutions of 1.1) are Hölder continuous, rovided the rincial art has exactly a linear growth with resect to u. Analogous results were established by Kruzkov [110], [111], [11] and by Nash [136] by entirely different 6

7 methods. Thus it aears that unlike the ellitic case, the degeneracy or singularity of the rincial art lays a eculiar role, and for examle, for the Lalacian equation in 1.3) one could not establish whether non-negative weak solutions satisfy the Harnack estimate or whether a solution is locally Hölder continuous. In the mid-1980, some rogress was made in the theory of degenerate.d.e. s of the tye of 1.1), for >. It was shown that the solutions are locally Hölder continuous see [51]). Surrisingly, the same techniques can be suitably modified to establish the local Hölder continuity of any local solution of quasilinear orous medium-tye equations. These modified methods in turn, are crucial in roving that weak solutions of the Lalacian equation 1.3) are of class C 1,α loc Ω T ). Therefore understanding the local structure of the solutions of 1.1) has imlications to the theory of equations with degeneracies quite different than 1.1). In the early 1990s the theory was comleted [37]) by establishing that solutions of 1.1) m, 1.) m, are Hölder continuous for all > 1 and all m > 0. For a comlete account see [55]. 1. A new aroach to regularity These results follow, one way or another, from a single unifying idea which we call intrinsic rescaling. The diffusion rocesses in 1.3) m, evolve in a time scale determined instant by instant by the solution itself, so that, loosely seaking, they can be regarded as the heat equation in their own intrinsic time-configuration. A recise descrition of this fact as well as its effectiveness is linked to its technical imlementations which we will resent in. The indicated regularity results assume the solutions to be locally or globally bounded. A theory of boundedness of weak solutions of 1.1) m, 1.) m, is quite different from the linear theory and it is resented in 3. For examle weak solutions of 1.1) 1.) are locally bounded only if κ = N ) + > 0 and weak solutions of 1.1) m 1.) m are locally bounded only if λ = Nm 1) + > 0. It is shown by counterexamles that these conditions are shar. The same notion of intrinsic rescaling is at the basis of a new notion of Harnack inequality for non negative solutions of 1.3) m, established in the late 1980s and early 1990s [54], [66]). Consider non negative weak solution of 1.3) m for m > 1. The Harnack inequality 1.6) continues to hold for such solutions rovided the time is rescaled by the quantity u m 1 x 0, t 0 ). Similar statements hold for 1.3) in their intrinsic arabolic geometry 1.5). In 4 we resent these intrinsic versions of the Harnack inequality and trace their connection to the Hölder continuity of solutions. A major oen roblem is to establish the Harnack estimate for non negative solutions of 1.1) m, with the full quasilinear structure 1.) m,. The roofs in [54], [66]) use in an essential way the structure of 1.3) m, as well as their corresonding fundamental solutions Γ m,. The lea forward of Moser s Harnack inequality was in byassing the classical aroaches based on heat otentials, by introducing new harmonic analysis methods and techniques. It is our belief that a roof of the intrinsic Harnack estimate for non negative solutions of 1.1) m, 1.) m, that would byass the otentials Γ m,, would have the same imact. The values of > 1 for which non negative solutions of 1.3) satisfy Harnack s inequality are those for which κ = N ) + > 0. Likewise the values of m > 0 7

8 for which non negative solutions of 1.3) m satisfy Harnack s inequality are those for which λ = Nm 1) + > 0. These limitations are shar for a Harnack estimate to hold 4). 1.3 Limiting cases and miscellanea remarks The cases κ, λ 0 are not well understood and form the object of current investigations. The case 1 < N/N + 1) seems to suggest questions similar to those of the limiting Sobolev exonent for ellitic equations see Brézis [30]) and questions in differential geometry. As 1, 1.3) tends formally to a.d.e. of the tye of motion by mean curvature. Investigations in this directions are due to Evans and Sruck [79]). As m 0 the orous medium equation 1.3) m for u 0 tends to the singular equation 1.13) u t ln u = 0 weakly in Ω T. When N = the Cauchy roblem for this equation is related to the Ricci flow associated to a comlete metric in R [90], [18], [59], [60]). A characterization of the initial data for which 1.13) is solvable has been identified and the theory seems fairly comlete [59], [46], [78]). The case N 3 however is still not understood and while solvability has been established for a rather large class of data [59]), a recise characterization of such a class still eludes the investigators. Degenerate and singular ellitic and arabolic equations are one of the branches of modern analysis both in view of the hysical significance of the equations at hand [8], [10], [11], [91], [9], [113], [114], [119], [16], [17], [161], [183]) and the novel analytical techniques that they generate [55]). The class of such equations is large, ranging from flows by mean curvature to Monge Amére equations to infinity-lalacian. These are imlicitly degenerate or singular equations in that the solution itself determines, imlicitly, the regions of degeneracy. Exlicitly degenerate equations would be those for which the degeneracy or singularity is a riori rescribed in the coefficients. For examle if the modulus of elliticity C 0 in 1.8) were a non negative function of x vanishing at some secified value x, such a oint would be a oint of exlicit degeneracy. There is a vast literature on all these asects of degenerate equations. We have chosen to resent a subsection of the theory that has a unifying set of techniques, issues, hysical relevance, and future directions. 1.4 Singular equations of the Stefan tye In this framework fall singular arabolic evolution equations where the singularity occurs on the time art of the oerator. These take the form { u Cloc 0, T ; W 1, loc 1.14) Ω T ) ) ; βu) t div Ax, t, u, u) Bx, t, u, u) in D Ω T ), where A and B have the same structure conditions as 1.) m for m = 1 and β ) is a coercive, maximal monotone grah in R R. The rototye examle is 1.15) βu) t u 0 in D Ω T ) 8

9 for a β ) given by s for s < 0 ; 1.16) βs) = [0,1] for s = 0 ; 1 + s for s > 0. Grahs β ) such as this one, i.e., exhibiting a single jum at the origin, arise from a weak formulation of the classical Stefan roblem modelling a solid/liquid hase transition such as water ice. In the latter case a natural question would be to ask whether the transition of hase occurs with a continuous temerature across water/ice interface. This issue, raised initially by Oleinik in the 1950s and reorted in the book [LSU] is at the origin of the modern and current theory of local regularity and local behaviour of solutions of degenerate and/or singular evolution equations. The coercivity of β ) for a solution to be continuous is essential, as ointed out by examles and counterexamles in [61]. It was established in [31], [48], [149], [150], [184] that for β ) exhibiting a single jum, the solutions of 1.14) are continuous with a given quantitative modulus of continuity not Hölder). This raises naturally the question of a grah β ) exhibiting multile jums and or singularities of other nature 5). For these rather general grahs, in the mid 1990s it was established in [7] that solutions of 1.14) are continuous rovided N =. For dimension N 3 the same conclusion holds rovided the rincial art of the differential equation is exactly the Lalacian, as in the first of 1.15). Several recent investigations have extended and imroved these results for secific grahs [86], [87]). It is still an oen question however, whether solutions of 1.14) with its full quasilinear structure and for a general coercive maximal monotone grah β ) and for N 3, are continuous in their domain of definition. 1.5 Outline of these notes The issues touched on here will be exanded in the next sections. We will rovide recise statements and self sufficient structure of roofs. In section we deal with the question of the regularity of the weak solutions of singular and degenerate quasilinear arabolic equations, roving their Hölder character. We start with the recise definition of weak solution and the derivation of the building blocks of the theory: the local energy and logarithmic estimates. In. we briefly resent the classical aroach of De Giorgi to uniformly ellitic equations. We introduce De Giorgi s class and show that functions in De Giorgi s class are Hölder continuous. The two main sections.3 and.4 deal, resectively, with the degenerate and the singular case. There we resent in full detail the idea of intrinsic scaling and, at least in the degenerate case, rove all the results leading to the Hölder continuity. We have decided to resent the theory for the model case of the -Lalace equation to bring to light what is really essential in the method, leaving aside technical refinements needed to deal with more general equations. We close the section with remarks on the ossible generalisations, namely to orous medium tye equations. Section 3 addresses the boundedness of weak solutions. The theory discriminates between the degenerate and the singular case. If >, a local bound for the solution is imlicit in the notion of weak solution. If 1 < <, local or global solutions need not be bounded in general. 9

10 In section 4, we first give a review about classical results concerning Harnack inequalities. Then we consider the degenerate case and we oint out the differences with resect to the nondegenerate one. We sketch a roof of the Harnack inequality both in the degenerate and singular case. We show that for ositive solutions of the singular -Lalace equation an ellitic Harnack inequality holds. We also analyze the henomenon of the extinction of the solution in finite time. Through a suitable use of the Raleigh quotient, we are also able to give shar estimates on the extinction time and to describe the asymtotic rofile of the extinction. In the whole section we oint out the major oen questions about Harnack inequalities for singular and degenerate arabolic equations. In section 5 we give hysical motivations concerning Stefan-like equations and show, through the Kruzkov- Sukorjanski transformation, the dee links between degenerate equations and Stefan-like equations. Then, we describe the aroaches made by Aronson, Caffarelli, DiBenedetto, Sachs and Ziemer in the 1980 s. Thanks to their contributions the case of only one singularity was comletely solved. Lastly we analyze the new ioneering aroach of [7] where, through a lemma of measure theory, the case of multile singularities was totally solved in the case N =. Moreover we show that this aroach also works in the case N 3 but only under strong assumtions. In this section we also oint out the major oen questions. We have chosen not to resent existence theorems for boundary value roblems associated with these equations. Theorems of this kind are mostly based on Galerkin aroximations and aear in the literature in a variety of forms. We refer, for examle, to [11] or [13]. Given the a riori estimates resented here these can be obtained alternatively by a limiting rocess in a family of aroximating roblems and an alication of Minty s Lemma. These notes can be ideally divided in three arts: 1. Hölder continuity and boundedness of solutions -3). Harnack tye estimates 4) 3. Stefan-like roblems 5) These arts are technically linked but they are concetually indeendent, in the sense that they deal with issues that have develoed in indeendent directions. We have attemted to resent them in such a way that they can be aroached indeendently. Acknowledgements. The research of J.M. Urbano was suorted by CMUC/FCT and Project POCTI/34471/MAT/000. Regularity of weak solutions We address the question of the regularity of weak solutions of singular and degenerate arabolic equations by roving that they are Hölder continuous. We will concentrate on quasilinear arabolic equations, with rincial art in divergence form, of the tye.1) u t div u u = 0, > 1. 10

11 If >, the equation is degenerate in the sace art, due to the vanishing of its modulus of elliticity u at oints where u = 0. The singular case corresonds to 1 < < : the modulus of elliticity becomes infinity at oints where u = 0. The results in this section extend to a variety of equations and, in articular, to equations with general rincial arts satisfying aroriate structure assumtions and with lower order terms. We have chosen to resent the results and the roofs for the articular model case.1) to bring to light what we feel are the essential features of the theory. Remarks on generalisations, which in some way or another corresond to more or less sohisticated technical imrovements, are made at the end of the section. Results on the continuity of solutions at a oint consist basically in constructing a sequence of nested and shrinking cylinders with vertex at that oint, such that the essential oscillation of the function in those cylinders converges to zero when the cylinders shrink to zero. At the basis of the roof is an iteration technique, that is a refinement of the technique by DeGiorgi and Moser cf. [47], [13] and [11]), based on energy also known as Caccioolli) and logarithmic estimates for the solution, that we briefly review in.. In the degenerate or singular cases these estimates are not homogeneous in the sense that they involve integral norms corresonding to different owers, namely the owers and. The key idea is then to look at the equation in its own geometry, i.e., in a geometry dictated by its intrinsic structure. This amounts to rescale the standard arabolic cylinders by a factor that deends on the oscillation of the solution. This rocedure, which can be called accommodation of the degeneracy, allows one to recover the homogeneity in the energy estimates written over these rescaled cylinders. We can say heuristically that the equation behaves in its own geometry like the heat equation. In the sequel, we first treat the degenerate case in.3 and then the more involved singular case in section.4. We conclude the section with some remarks on generalisations, namely to orous medium tye equations..1 Weak solutions and local estimates A local weak subsuer)-solution of.1) is a measurable function u C loc 0, T ; L loc Ω) ) ) L loc 0, T ; W 1, loc Ω) such that, for every comact K Ω and for every subinterval [t 1, t ] of 0, T ], t t {.) uϕ dx uϕt + u u ϕ } dx dt ) 0, K t 1 + t 1 K for all ϕ W 1, loc 0, T ; L K) ) ) L loc 0, T ; W 1, 0 K), ϕ 0. A function that is both a local subsolution and a local suersolution of.1) is a local solution of.1). It would be technically convenient to have at hand a formulation of weak solution involving the time derivative u t. Unfortunately, solutions of.1), whenever they exist, ossess a modest degree of time-regularity and in general u t has a meaning only in the sense of distributions. To overcome this limitation we introduce the Steklov average of a function v L 1 Ω T ), defined for 0 < h < T by 1 h v h = t+h t v, τ) dτ if t 0, T h] 0 if t T h, T ], 11

12 and observe that the notion.) of solution is equivalent to: for every comact K Ω and for all 0 < t < T h, {.3) uh ) t ϕ + u u ) h ϕ} dx ) 0, K {t} for all ϕ W 1, 0 K) L loc Ω), ϕ 0. We will show that locally bounded solutions of.1) are locally Hölder continuous within their domain of definition. No secific boundary or initial values need to be rescribed for u. Although the arguments below are of local nature, to simlify the resentation we assume that u is a.e. defined and bounded in Ω T and set M u L Ω T ). See section 3 for results on the boundedness of weak solutions..1.1 Local energy and logarithmic estimates Given a oint x 0 R N, denote by K ρ x 0 ) the N-dimensional cube with centre at x 0 and wedge ρ: { } K ρ x 0 ) := x R N : max x i x 0i < ρ ; 1 i N given a oint x 0, t 0 ) R N+1, the cylinder of radius ρ and height τ > 0 is x 0, t 0 ) + Qτ, ρ) := K ρ x 0 ) t 0 τ, t 0 ). Consider a cylinder x 0, t 0 )+Qτ, ρ) Ω T and let 0 ζ 1 be a iecewise smooth cutoff function in x 0, t 0 ) + Qτ, ρ) such that.4) ζ < and ζx, t) = 0, x K ρ x 0 ). We start with the energy estimates. Without loss of generality, we will state them for cylinders with vertex at the origin 0, 0), the changes being obvious for cylinders with vertex at a generic x 0, t 0 ). Proosition 1. Let u be a local weak solution of.1). C C) > 0 such that for every cylinder Qτ, ρ) Ω T, su τ<t<0 K ρ {t} 0 u k) ± ζ dx + τ u k) ± ζ dx dt K ρ K ρ { τ} 0 u k) ± ζ dx + C τ u k) ± ζ dx dt K ρ There exists a constant 0.5) + τ u k) ± ζ 1 t ζ dx dt. K ρ 1

13 Proof. Use ϕ = ±u h k) ± ζ as a testing function in.3) and erform standard energy estimates cf. [55, ages 4 7]). Given constants a, b, c, with 0 < c < a, define the nonnegative function { }) ψ ± {a,b,c} ln s) a a + c) s b) ± + { } a ln a+c)±b s) if b ± c < > s < > b ± a + c) = 0 if s b ± c whose first derivative is ) ψ ± {a,b,c} s) = 1 b s)±a+c) if b ± c < > s < > b ± a + c) 0 if s < > b ± c 0, and second derivative, off s = b ± c, is ) { } ψ ± {a,b,c} = ψ {a,b,c}) ± 0. Now, given a bounded function u in a cylinder x 0, t 0 ) + Qτ, ρ) and a number k, define the constant H ± u,k ess su x 0,t 0)+Qτ,ρ) u k) ±. The following function was introduced in [48] and since then has been used as a recurrent tool in the roof of results concerning the local behaviour of solutions of degenerate PDE s: ) Ψ ± H ± u,k, u k) ±, c ψ ± {H ±,k,c}u), 0 < c < H± u,k. u,k From now on, when referring to this function we will write it as ψ ± u), omitting the subscrits whose meaning will be clear from the context. Let x ζx) be a time-indeendent cutoff function in K ρ x 0 ) satisfying.4). The logarithmic estimates in cylinders Qτ, ρ) with vertex at 0, 0), are Proosition. Let u be a local weak solution of.1), k R and 0 < c < H ± u,k. There exists a constant C > 0 such that for every cylinder Qτ, ρ) Ω T, [ ψ ± u) ] ζ [ dx ψ ± u) ] ζ dx su τ<t<0 K ρ {t} K ρ { τ} 0.6) + C ψ ± u) ψ ± ) u) ζ dx dt. τ K ρ Proof. Take ϕ = ψ ± u h ) in time over τ, t) for t τ, 0). Since t ζ 0, t { [ ] t u h ψ ± u h ) ψ ± ) u h ) ζ } dx dt = τ K ρ [ ] ψ ± ) u h ) ζ as a testing function in.3) and integrate 13 t τ K ρ t { [ψ ± u h ) ] } ζ dx dt

14 [ = ψ ± u h ) ] ζ [ dx ψ ± u h ) ] ζ dx. K ρ {t} K ρ { τ} From this, letting h 0, t { [ ] t u h ψ ± u h ) ψ ± ) u h ) ζ } dx dt τ K ρ K ρ {t} K ρ { τ} [ ψ ± u) ] ζ dx [ ψ ± u) ] ζ dx. As for the remaining term, we first let h 0, to obtain t { [ ] u u ψ ± u) ψ ± ) u) ζ } dx dt τ K ρ t { = u 1 + ψ ± u) ) [ ] ψ ± ) u) ζ } dx dt τ K ρ t { [ ] + u u ζ ψ ± u) ψ ± ) u) ζ 1} dx dt τ K ρ t { u 1 + ψ ± u) ψ ± u) ) [ ] ψ ± ) u) ζ } dx dt τ K ρ t 1) 1 ψ ± u) ψ ± ) u) ζ dx dt τ K ρ t C ψ ± u) ψ ± ) u) ζ dx dt. τ K ρ Since t τ, 0) is arbitrary, we can combine both estimates to obtain.6)..1. Some technical tools We gather a few technical facts that, although marginal to the theory, are essential in establishing its main results. Given a continuous function v : Ω R and two real numbers k < l, we define.7) [v > l] {x Ω : vx) > l}, [v < k] {x Ω : vx) < k}, [k < v < l] {x Ω : k < vx) < l}. Lemma 1 DeGiorgi, [47]). Let v W 1,1 B ρ x 0 )) C B ρ x 0 )), with ρ > 0 and x 0 R N and k < l R. There exists a constant C, deending only on N and so indeendent of ρ, x 0, v, k and l), such that l k) [v > l] C ρn+1 v dx. [v < k] [k<v<l] 14

15 Remark 1. The conclusion of the lemma remains valid, rovided Ω is convex, for functions v W 1,1 Ω) CΩ). We will use it in the case Ω is a cube. In fact, the continuity is not essential for the result to hold. For a function merely in v W 1,1 Ω), we define the sets.7) through any reresentative in the equivalence class. It can be shown that the conclusion of the lemma is indeendent of that choice. The following lemma concerns the geometric convergence of sequences. Lemma. Let {X n }, n = 0, 1,,..., be a sequence of ositive real numbers satisfying the recurrence relation X n+1 C b n X 1+α n where C, b > 1 and α > 0 are given. If then X n 0 as n. X 0 C 1/α b 1/α Let V 0 Ω T ) denote the sace ) V 0 Ω T ) = L 0, T ; L Ω)) L 0, T ; W 1, 0 Ω) endowed with the norm u V Ω T ) = ess su 0 t T u, t),ω + u,ω T, for which the following embedding theorem holds cf. [55, age 9]): Theorem 1. Let > 1. There exists a constant γ, deending only uon N and, such that for every v V 0 Ω T ), v,ω T γ v > 0 N+ v V Ω T ). With C or C j we denote constants that deend only on N and and that might be different in different contexts.. The classical aroach of De Giorgi Results concerning the Hölder continuity of weak solutions u consist essentially in showing that for every oint x 0, t 0 ) Ω T we can find a sequence of nested and shrinking cylinders x 0, t 0 ) + Qτ n, ρ n ) such that the essential oscillation of u in these cylinders aroaches zero as n in a way that can be quantified. The aroach to regularity introduced by De Giorgi is based on the following embedding theorem see [47] for the ellitic case and [11] for the arabolic case): Proosition 3. Assume that u L 1, 1, loc 0, T ; Wloc Ω)) Wloc 0, T ; L loc Ω)) is locally bounded and satisfies the Cacciooli inequalities.5) with =. Then u is locally Hölder continuous, with the modulus of continuity deending only uon the data. 15

16 A solution of a non-degenerate arabolic equation with the full quasilinear structure satisfies these inequalities. One uses the structure of the equation to rove the Cacciooli estimates for a solution u and this is the only role of the equation. Once such inequalities are derived, the Hölder continuity of u is solely a consequence of.5) for =. Alternative aroaches are in Kruzkov [110], [111], [11]) and, through the use of the Harnack inequality, in Moser [13], [134]), Trudinger [164]), Aronson-Serrin [16]) and, for equations in non-divergence form, in Krylov-Safonov [117]). Set Q r = Qr, r), fix a oint x 0, t 0 ) Ω T, and let ρ 0 be the largest radius so that x 0, t 0 ) + Q ρ0 is contained in Ω T. For a constant δ 0, 1), consider the sequence of decreasing radii,.8) ρ n = δ n ρ 0, n = 0, 1,,... and the family of nested shrinking cylinders, with the same vertex.9) x 0, t 0 ) + Q ρn, n = 0, 1,,... Set µ n := ess inf x 0,t 0)+Q ρn u ; µ + n := ess su x 0,t 0)+Q ρn u ; ω n := ess osc x 0,t 0)+Q ρn u = µ + n µ n. Proosition 4. Let u satisfy the Cacciooli inequalities.5) for =. Then there exist constants C > 1 and δ, η 0, 1 ), that can be determined a riori only in terms of the data, such that for every x 0, t 0 ) Ω T and every n N, at least one of the following two inequalities holds.10) ess su x 0,t 0)+Q ρn+1 u µ + n ηω n,.11) ess inf x 0,t 0)+Q ρn+1 u µ n + ηω n. A nontrivial roof can be found in [11]. This roosition can be interreted as a weak maximum rincile. For examle.10) asserts that the suremum of u over the cylinder Q ρn+1 is strictly less than the suremum of u over the larger coaxial cylinder Q ρn. In other words, the suremum of u over Q ρn can only be achieved in the arabolic shell Q ρn \ Q ρn+1 that can be considered as a sort of arabolic boundary of Q ρn. A consequence of such a weak maximum rincile is: Proosition 5. Let u be as above. Then there exist constants C > 1 and δ, η 0, 1 ), that can be determined a riori only in terms of the data, such that for every x 0, t 0 ) Ω T and every n N,.1) ω n+1 1 η)ω n. This in turn imlies that u is locally Hölder continuous in Ω T. 16

17 Proof. Fix x 0, t 0 ) Ω T. Assume that.10) holds. By subtracting µ n+1 from the left hand side and µ n from the right hand side, ω n+1 = µ + n+1 µ n+1 µ+ n µ n ηω n = 1 η)ω n. If.11) holds one can argue in a similar way. By iteration,.13) ω n 1 η) n ω 0, n N. The numbers η and δ are related by 1 η) = δ α and α = ln1 η) ln δ ) α ρn.14) ω n ω 0, n N. ρ 0 0, 1). Therefore, Since x 0, t 0 ) Ω T is arbitrary, we conclude that u is locally Hölder continuous in Ω T with exonent α. Remark. The cylinder x 0, t 0 ) + Q ρ0 must be contained in Ω T. Thus from.14) it follows that the Hölder continuity can be claimed only within comact subsets of Ω T and that the Hölder constant ω 0 ρ α 0 deteriorates as x 0, t 0 ) aroaches the arabolic boundary of Ω T..3 The degenerate case > We go back to equation.1) and focus on the degenerate case >. The energy and logarithmic estimates of.1 are not homogeneous in the sace and time arts due to the resence of the ower. To go about this difficulty we will consider the equation in a geometry dictated by its own structure, which is designed, roughly seaking, to restore the homogeneity of the various arts of the Cacciooli inequalities.5). This means that, instead of the usual cylinders, we have to work in cylinders whose dimensions take the degeneracy of the equation into account, in a rocess that we call intrinsic rescaling. Let s make this idea recise..3.1 The geometric setting and the alternative Consider R > 0 such that QR, R) Ω T, define µ + :=ess su QR,R) u ; and construct the cylinder µ := ess inf QR,R) u ; ω :=ess osc QR,R) u = µ + µ.15) Qa 0 R, R) K R 0) a 0 R, 0) with a 0 = ω λ ), where λ > 1 is to be fixed later deending only on the data see.55)). Note that for =, i.e., in the non-degenerate case, these are the standard arabolic cylinders reflecting the natural homogeneity between the sace and time variables. 17

18 We will assume, without loss of generality, that.16) R < ω λ because if this doesn t hold there is nothing to rove since the oscillation is comarable to the radius. Now,.16) imlies the inclusion and the relation Qa 0 R, R) QR, R).17) ess osc Qa 0R,R) u ω which will be the starting oint of an iteration rocess that leads to our main results. Note that we had to consider the cylinder QR, R) and assume.16), so that.17) would hold for the rescaled cylinder Qa 0 R, R). This is in general not true for a given cylinder since its dimensions would have to be intrinsically defined in terms of the essential oscillation of the function within it - the stretching rocedure is commonly referred to as an accommodation of the degeneracy. We now consider subcylinders of Qa 0 R, R) of the form ω ).18) 0, t ) + QdR, R), with d = that are contained in Qa 0 R, R) if.19) λ )) R ω < t < 0. The roof now follows from the analysis of two comlementary cases. We briefly describe them in the following way: in the first case we assume that there is a cylinder of the tye 0, t ) + QdR, R) where u is essentially away from its infimum. We show that going down to a smaller cylinder the oscillation decreases by a small factor that we can exhibit. If that cylinder can not be found then u is essentially away from its suremum in all cylinders of that tye and we can comound this information to reach the same conclusion as in the revious case. We state this in a recise way. For a constant ν 0 0, 1), that will be determined deending only on the data, we will assume that either The first alternative: There is a cylinder of the tye 0, t ) + QdR, R) for which { } x, t) 0, t ) + QdR, R) : ux, t) < µ + ω.0) QdR ν 0, R) or The second alternative: For every cylinder of the tye 0, t ) + QdR, R) { } x, t) 0, t ) + QdR, R) : ux, t) > µ + ω.1) QdR < 1 ν 0, R) 18

19 .3. Analysis of the first alternative Lemma 3. Assume.0) holds for some t as in.19) and that.16) is in force. There exists a constant ν 0 0, 1), deending only on the data, such that ux, t) > µ + ω ) R a.e. x, t) 0, t ) + Q d, R ). 4 Proof. Take the cylinder for which.0) holds and assume, by translation, that t = 0. Let R n = R + R, n = 0, 1,..., n+1 and construct the family of nested and shrinking cylinders QdRn, R n ). Consider iecewise smooth cutoff functions 0 < ζ n 1, defined in these cylinders, and satisfying the following set of assumtions ζ n = 1 in Q dr n+1, R ) n+1 ζ n = 0 on Q drn, R n ) ζ n n+1 R 0 t ζ n n+1) dr. Write the energy inequality.5) for the functions u k n ), with k n = µ + ω 4 + ω, n = 0, 1,..., n+ in the cylinders Q drn, R n ) and with ζ = ζ n. They read su dr n<t<0 K Rn {t} 0 u k n ) ζn dx + drn u k n ) ζ n dx dt K Rn 0 0 C u k n ) ζ n dx dt + u k n ) ζn 1 t ζ n dx dt drn K Rn drn K Rn.) C n+1) R { 0 dr n K Rn u k n ) dx dt + 1 d 0 dr n K Rn u k n ) dx dt }. Next, observing that and we obtain from.) ω ) su drn<t<0 u k n ) = µ u) + ω 4 + ω n+ ω u k n ) = u k n ) K Rn {t} u k n ) ω ) u kn ), 0 u k n ) ζn dx + drn u k n ) ζ n dx dt K Rn 19

20 .3) C n+1) R { ω ) 1 ω ) } 0 + χ {u kn) d >0} dx dt. drn K Rn Recall that d = ) ω and divide.3) by d to get su dr n<t<0 K Rn {t}u k n ) ζ n dx + 1 d 0 dr n K Rn u k n ) ζ n dx dt.4) C n+1) ω ) 1 0 R χ {u kn) d >0} dx dt. drn K Rn Now we erform a change of the time variable in.4), utting t = t/d and defining u, t) = u, t) and ζ n, t) = ζ n, t), and obtain the simlified inequality.5) u kn ) ζ n C n ω ) 0 V 0 QR n,r n)) R χ {u kn) >0} dx dt. Rn K Rn Define, for each n, and observe that A n = 0 R n K Rn χ {u kn) >0} dx dt 1 ω ) An+1 k n+) n k n+1 A n+1 u k n ),QR n+1,rn+1) u k n ) ζ n,qr n,r n) C u k n ) ζ n V.6) C n ω ) 1+ A N+ R n. A N+ 0 QR n n,r n)) The first three of these inequalities follow from the definition of A n and k n+1 < k n ; the fourth inequality is a consequence of Theorem 1 and the last one follows from.5). Next, define the numbers A n X n = QRn, R n ), divide.6) by QR n+1, R n+1) and obtain the recursive relation X n+1 C 4 n X 1+ n N+, for a constant C deending only uon N and. convergence, if By Lemma on fast geometric.7) X 0 C N+ 4 N+) ν 0 0

21 then.8) X n 0. Therefore, { x, t) Q ) d R ), R : ux, t) µ + ω } = 0. 4 Our next aim is to show that the conclusion of lemma 3 holds in a full cylinder Qτ, ρ). The idea is to use the fact that at the time level.9) t := t d R the function ux, t ) is strictly above the level µ + ω 4 in the cube K R, and look at this time level as an initial condition to make the conclusion hold u to t = 0. As an intermediate ste we need the following lemma. Lemma 4. Assume.0) holds for some t as in.19) and that.16) is in force. Given ν 0, 1), there exists s N, deending only on the data, such that { x K R : ux, t) < µ + ω } K ν R, t t, 0). 4 s 4 ) Proof. We use the logarithmic estimate.6) alied to the function u k) in the cylinder Q t, R ), with the choices k = µ + ω 4 and c = ω n+ where n N will be chosen later. We have.30) k u H u,k = ess su u µ ω ) Q t, R ) 4 ω 4. If H u,k ω 8 the result is trivial for the choice s = 3. Assuming H u,k > ω 8 the logarithmic function is defined in the whole Q t, R ) and it is given by { } H u,k H u,k + u k + ln Ψ = ψ {H u,k,k, ω } u) = n+ From.30), we estimate.31) Ψ n ln since and.3) ω n+ if u < k ω n+ 0 if u k ω n+ H u,k H u,k + u k + ω n+ ψ ) u) = H u,k + u k + c ) ω ). ω 4 ω n+ = n. 1

22 Now observe that as a consequence of Lemma 3, we have ux, t) > k in the cube, which imlies that K R Ψ x, t) = 0, x K R Choosing a iecewise smooth cutoff function 0 < ζx) 1, defined in K R that ζ = 1 in K R and ζ 8 4 R, inequality.6) reads.33) su t<t<0 K R {t} [ ψ u) ] 0 ζ dx C t K R. and such ψ u) ψ ) u) ζ dx dt. The right hand side is estimated above, using.31) and.3), by ω ) ) 8 C nln ) t K R C n λ ) K R, R 4 since, by.9) t a 0 R = ω λ ) R. We estimate below the left hand side of.33) by integrating over the smaller set { St) = x K R : ux, t) < µ + ω } 4 n+ K R and observing that in S, ζ = 1 and since H u,k H u,k + u k + ω n+ ) H u,k ω 4 0. Therefore, ω 8 H u,k ω 4 ) + ω n+ ω 3 ω n+ = n 1, [ ψ u) ] [ ln n 1 )] = n 1) ln ) on St). Combining these estimates in.33) we get { x K R 4 : ux, t) < µ + ω n+ } C n n 1) λ ) K R 4 for all t ˆt, 0) and to rove the lemma we choose.34) s = n + with n > 1 + C ν λ ). We now state the main result of this section. Proosition 6. Assume.0) holds for some t as in.19) and that.16) is in force. There exist constants ν 0 0, 1), s 1 N, deending only on the data, such that, ux, t) > µ + ω, a.e. x, t) Q t, R ). s1+1 8,

23 Proof. Consider the cylinder for which.0) holds, let R n = R 8 + R, n = 0, 1,... n+3 and construct the family of nested and shrinking cylinders Q t, R n ), where t is given by.9). Take iecewise smooth cutoff functions 0 < ζ n x) 1, indeendent of t, defined in K Rn and satisfying ζ n = 1 in K Rn+1 ζ n n+4 R. Write the local energy inequalities.5) for the functions u k n ) in the cylinders Q t, ) R n, with k n = µ + ω s1+1 + ω, n = 0, 1,..., s1+1+n and ζ = ζ n. Observing that, due to Lemma 3, we have ux, t) > µ + ω 4 k n in the cube K R K Rn, which imlies that they read su t<t<0 u k n ) x, t) = 0, x K Rn, n = 0, 1,..., K Rn {t} 0 u k n ) ζn dx + u k n ) ζ n dx dt t K Rn 0.35) C u k n ) t KRn ζ 0 n dx dt C n+4) R u k n ) dx dt. t K Rn From.9) we estimate t a 0 R = ω λ ) R where a 0 is defined in.15). From this, u k n ) ω s1 ) u kn ) s 1 ) t λ R u k n) t R u k n), t rovided s 1 > λ. Dividing now.35), by R gives ) 0 u k n ) ζn dx + R u k n ) ζ n dx dt K Rn {t} t t K Rn su t<t<0 C n t 0 t K Rn u k n ) dx dt. The change of the time variable t = t R ), along with the new function t u, t) = u, t), 3

24 leads to the simlified inequality u k n ) ζ n C n ω ) 0 V 0 Q R ),R n)) R χ {u kn) ) >0} dx dt. s1 R ) K Rn Define, for each n, A n = 0 R ) K Rn χ {u kn) >0} dx dt. By a reasoning similar to the one leading to.6): 1 ω ) An+1 k n+) n k n+1 A n+1 s1 u k n ),Q R ),R n+1) u k n ) ζ n C u k,q R ),R n) n) ζ n A V 0 Q R ),R n)) n Next, define the numbers C n R ) ω s1 ) A 1+ A n n N+. X n = ), Q R ), R n divide the revious inequality by Q R ), R n+1 ) to obtain the recursive relations X n+1 C 4 n X 1+ By Lemma on fast geometric convergence, if n N+..36) X 0 C N+ 4 N+) ν 0, 1) then.37) X n 0. To verify.36), aly Lemma 4 with such a ν and conclude that there exists s s 1, deending only on the data, such that { x K R : ux, t) < µ + ω } K ν R, t t, 0). 4 s1 4 Since.37) imlies that A n 0, we conclude that { ) x, t) Q R ) R, 8 { = x, t) Q t, R ) 8 : ux, t) µ + ω s1+1 } : ux, t) µ + ω s1+1 } = 0. N+ 4

25 Corollary 1. Assume.0) holds for some t as in.19) and that.16) is in force. There exist constants ν 0, σ 0 0, 1), deending only on the data, such that.38) ess osc Qd R 8 ), R 8 ) u σ 0 ω. Proof. By Proosition 6 there exists s 1 N such that From this, Since d ) R 8 t = t + d R ess inf u µ + ω Q t, R 8 ) s1+1. ess osc Q t, R 8 ) u 1 1 ) s1+1 ω. ), t < 0, we have ) R Q d, R ) Q 8 8 and the corollary follows with σ 0 = 1 1 s 1 +1 ). t, R 8 ),.3.3 Analysis of the second alternative Assume now that the second alternative.1) holds true. We will show that a conclusion similar to.38) can be reached. Recall that the constant ν 0 has already been quantitatively determined by.7), and it is fixed. We continue to assume that.16) is in force. Lemma 5. Assume.1) holds and let.16) be in force. Fix a cylinder 0, t ) + QdR, R) Qa 0 R, R) for which.1) holds. There exists a time level [ t t dr, t ν 0 dr] such that { x K R : ux, t ) > µ + ω } ) 1 ν0 K R. 1 ν 0 / Proof. Suose not. Then, for all t [ t dr, t ν0 dr], { x, t) 0, t ) + QdR, R) : ux, t) > µ + ω } t ν 0 dr t dr which contradicts.1). { x K R : ux, τ) > µ + ω } dτ > 1 ν0 ) QdR, R), The next lemma asserts that the set where u, t) is close to its suremum is small, not only at the secific time level t, but for all time levels near the to of the cylinder 0, t ) + QdR, R). 5

26 Lemma 6. Assume.1) holds and let.16) be in force. There exists 1 < s N, deending only on the data, such that { x K R : ux, t) > µ + ω } ν0 ) ) 1 K R, s for all t [ t ν0 dr, t ]. Proof. The roof consists in using the logarithmic inequalities.6) alied to the function u k) + in the cylinder K R t, t ), with the choices k = µ + ω where n N will be chosen later. We have.39) u k H + u,k ln Ψ + = ψ + {H + u,k,k, ω } u) = n+1 = ess su K R t,t ) and c = ω n+1 u µ + + ω ) + ω. If H + u,k ω 8 the result is trivial for the choice of s = 3. Assuming H + u,k > ω 8 the logarithmic function Ψ + is defined in the whole K R t, t ), and it is given by { } H + u,k H + u,k u + k + if u > k + ω n+1 From.39), estimate ω n+1 0 if u k + ω n+1.40) Ψ + n ln since H + u,k H + u,k u + k + ω n+1 ω ω n+1 = n, and,.41) ψ +) u) = H + u,k u + k + c ) ω ). Choosing a iecewise smooth cutoff function 0 < ζx) 1, defined in K R and such that, for some σ 0, 1), inequality.6) reads su t <t<t ζ = 1 in K 1 σ)r and ζ σr) 1, K R {t} [ ψ + u) ] ζ dx K R {t } [ ψ + u) ] ζ dx t.4) + C ψ + u) ψ + ) u) ζ dx dt. t K R 6

27 The first integral on the right hand side can be bounded above using Lemma 5 and taking into account that Ψ + vanishes on the set {x K R : ux, ) < k}. This gives [ ψ + u) ] ) 1 dx n ln ) ν0 K R. 1 ν 0 / K R {t } To bound the second integral we use.40) and.41): t C ψ + u) ψ + ) u) ζ dx dt t K R ω ) C nln ) σr) t t ) K R ω ) ) 1 C n dr K R σr C n 1 σ K R, since t t dr. The left hand side is estimated below by integrating over the smaller set { S = x K 1 σ)r : ux, t) > µ + ω } n+ K R and observing that in S, ζ = 1 and since H + u,k H + u,k u + k + ω n+1 ) H + u,k ω 0. Therefore ω 8 H + u,k ω ) + ω n+ ω ω n = n 1, [ ψ + u) ] [ ln n 1 )] = n 1) ln ). From this su t <t<t K R {t} Combining these three estimates, we arrive at S [ ψ + u) ] ζ dx n 1) ln ) S. ) ) n 1 ν0 K R + C n 1 1 ν 0 / ) ) n 1 ν0 n 1 1 ν 0 / n 1 n 1) σ K R K R + C nσ K R. On the other hand { x K R : ux, t) > µ + ω } n+1 { x K 1 σ)r : ux, t) > µ + ω } + n+1 K R \ K 1 σ)r S + Nσ K R 7