THE CAUCHY-DIRICHLET PROBLEM FOR A GENERAL CLASS OF PARABOLIC EQUATIONS

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1 THE CAUCHY-IRICHLET PROBLEM FOR A GENERAL CLASS OF PARABOLIC EQUATIONS PAOLO BARONI AN CASIMIR LINFORS ABSTRACT. We prove regularity results such as interior Lipschitz regularity and boundary continuity for the Cauchy-irichlet problem associated to a class of parabolic equations inspired by the evolutionary p-laplacian, but extending it at a wide scale. We employ a regularization technique of viscosity-type that we find interesting in itself. 1. INTROUCTION The aim of this paper is the study of the behaviour of solutions to a wide class of nonlinear parabolic equations modeled after g) ) u t div u = 0 in Ω T := Ω 0, T ) R n R, 1.1) n, where Ω is a bounded domain with C 1,β boundary and g : R + R + is a C 1 function satisfying g 0 1 O g s) := sg s) gs) g 1 1 for every s > 0 1.) with 1 < g 0 g 1 <. Notice that we can assume g 0 < g 1 without loss of generality. Indeed, if O g s) is constant, say O g s) = p 1 for some p > 1, a simple integration shows that gs) = s p 1 up to a constant factor, and therefore in this case 1.1) gives back the evolutionary p-laplacian widely studied in particular by ibenedetto, see the monograph [14]. This reveals that 1.1) is a natural generalization of the p-laplacian, and in effect this class of growth conditions was mathematically introduced exactly in these terms by Lieberman in [8], even if this kind of condition appears earlier in the applications, see the forthcoming lines. We stress that quite a comprehensive study of non-negative solutions to the equation where the function ϕ : [0, ) [0, ) satisfies u t div [ ϕ u)u ] = 0 1.3) 0 < a O ϕ s) := sϕ s) ϕs) 1 a, for s > 0; 1 + a O ϕs) for s > s 0 1.4) for some a 0, 1) and some s 0 > 0 has been provided by ahlberg and enig [10, 11]; see also the books [1, 35]. Clearly, while 1.3) is a generalization of the porous medium equation that happens when ϕu) = u m, m > 0, in the same spirit 1.1) can be seen as a generalization of the p-laplacian. As in 1.4), we shall also consider a more stringent growth assumption for g for large values of its argument. In addition to 1.), we shall assume that there exist constants c l, ɛ > 0 such that gs) c l s n n+ +ɛ for any s ) ate: September,

2 PAOLO BARONI AN CASIMIR LINFORS Note that in the p-laplacian case 1.5) reads precisely as p > n/n + ), a completely natural assumption in the theory of the evolutionary p-laplacian operator, see [14, 5, 1]. Note moreover that 1.5) is implied by assuming g 0 > n/n + ), see Paragraph.. The regularity for the elliptic and variational counterpart of 1.1) is quite well understood, see for instance [8, 3, 9, 16] for the first argument and [17, 0, 7, 8] for the second, just to cite some indicative references. In the parabolic setting, however, very few results are available, and some of them only in particular cases: to our knowledge, only [, 3, 9, 30], all by Lieberman and Hwang, and the recent [5]. The difficulty, in particular in finding zero-order results, stems from several facts, the main one perhaps being that the equation has very different behaviour, already in the p- Laplacian case, in the degenerate p ) and singular p < ) cases. In the degenerate case phenomena such as expansion of positivity occur, see [15, 6], and the diffusion dominates [13]. On the other hand, in the singular case the evolutionary character dominates [6] and extinction of positive solutions in finite time could happen, see [14]. In our general setting the degenerate case occurs when s gs)/s is increasing, and when it is decreasing we have the singular case. However, it might also happen that s gs)/s has no monotonicity whatsoever, making the handling of the equation all the more difficult. The comprehension of the interaction of these different phenomena is the key for a better understanding of the behaviour of local solutions to 1.1), and in this paper we hope to start to clarify this difficult point, which will be the object of future investigations. The class of differential operators we study, besides being quite a general extension of a well-known operator, finds important applications in the applied sciences, also in view of the following observation. Take the convex primitive G of g and consider the general minimization problem u u 0 + W 1,1 0 Ω) G) dx; 1.6) Ω it is often convenient to have energies with a precise dependence on of more general type than monomial that is, the case of the p-irichlet energy or appropriate extensions). For instance, in mechanics, fluid dynamics and magnetism, as first approximation it is customary to have dependencies of the energy on the modulus of the gradient of monomial type but with exponent depending on the size of, in order to have mathematical models fitting the experimental data. In this case g is given by the gluing of different monomials see the example in Paragraph.3). At this point, elliptic and parabolic equations having the growth described in 1.1) arise naturally as Euler equations or flows of the functional in 1.6). In [34], for instance, the two-dimensional stationary, irrotational subsonic flow of a compressible fluid is described using an energy defined in the following way: Gs) = 1 γ 1 s ) γ γ 1 for small s, Gs) = quadratic otherwise, 1.7) where γ 1, ) is the exponent in the law p ρ γ characteristic of polytropic gases. More in general, see [4, 18, 19], one is lead to consider quasilinear static equations in dimension two and three of the type div [ ρ )u ] = 0, with u representing the velocity field of the flow and q = being the speed of the flow. In this context one introduces the Mach number M [Mq)] := q ρq ) ρ q ) note that we must have ρ < 0). In our context, where gs) = ρs )s, we compute O g s) = 1 Ms). The general theory asserts that a point is elliptic if M < 1 and in this case the flow is subsonic, while if M > 1 the point is hyperbolic and the flow there is

3 THE CAUCHY-IRICHLET PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS 3 supersonic. If M = 1 the flow is called sonic. A solution of the boundary value problem is called a subsonic supersonic) flow according to whether all points are subsonic supersonic); note that mixed, or transonic flows can exists, with obvious meaning. However, if for some reason we know that the flow maintains a controlled, small speed q, then the problem falls in the class of operators we consider; the approximation in 1.7) is a way to study flows in the subsonic regime. The object of our study will be the Cauchy-irichlet problem u t div Au) = 0 in Ω T, u = ψ on p Ω T, 1.8) where A : R n R n is a C 1 vector field modeled after the one appearing in 1.1). In particular, we assume it satisfies the following ellipticity and growth conditions: Aξ)λ, λ ν g ξ ) λ ξ Aξ) L g ξ ), 1.9) ξ for any ξ R n \ {0}, λ R n and with structural constants 0 < ν 1 L; we assume without loss of generality that A0) = 0. The function g is a C 1 function as in 1.1), satisfying only 1.) and 1.5). For what concerns ψ, we assume it to be continuous in p Ω T with modulus of continuity ω ψ with respect to the natural distance dist par,g, that is, there exists a continuous, concave function ω ψ : R + R + with ω ψ 0) = 0 such that ψx, t) ψy, s) ω ψ max{ x y, [G 1 1/ t s )] 1 } ) for every x, t), y, s) p Ω T. As already mentioned, Ω is a bounded domain of R n, n, whose boundary is of class C 1,β for some β 0, 1); we shall provide some more details at the beginning of Section. In this setting, we state the main result of our paper, which concerns at the same time the existence and regularity of a unique) solution to 1.8). Theorem 1.1. There exists a unique solution u, in the sense of efinition, to the Cauchy- irichlet problem 1.8), where the vector field A satisfies the assumptions 1.9), with g C 1 R + ) satisfying 1.) and 1.5). In particular, u is continuous up to the boundary and moreover if the boundary datum ψ is Hölder continuous with respect to the natural metric dist par,g defined in.1), then so is u. The following theorem gives some properties together with quantitative estimates for the solution described in the previous statement. Theorem 1.. Let u be the solution to 1.8) given by Theorem 1.1. Then u is locally Lipschitz continuous and the following estimate holds: ) [ ] max{ 1, ɛn+)} u L Q R ) c G) + 1 dx dt 1.10) Q R for every parabolic cylinder Q R Ω T. The constant c depends on n, g 0, g 1, ν, L, ɛ and c l. Moreover, there exists a modulus of continuity ω u : R + R + depending on n, g 0, g 1, ν, L, ɛ, c l, ψ L, ω ψ, Ω such that { [ ux, t) uy, s) ω u max x y, G 1 1/ t s )] 1}) 1.11) for every x, t), y, s) Ω T p.

4 4 PAOLO BARONI AN CASIMIR LINFORS We refer the reader to Paragraph.1 for the definitions of the standard parabolic cylinders Q R x 0, t 0 ) and of the parabolic closure of Ω T. We also mention that in the standard case of the evolutionary p-laplacian our estimate 1.10) gives back exactly the gradient sup-estimate available for degenerate and singular equations, see [14, Chapter VIII, Theorems 5.1 & 5.]. Remark 1. Theorems 1.1 and 1. hold for a wider class of operators generalizing 1.1), which allow the presence of a function g that is not C 1 but merely Lipschitz. Indeed, we may consider Lipschitz functions g : R + R + satisfying 1.) almost everywhere and vector fields A : R n R n in W 1, R n ) satisfying the monotonicity and Lipschitz assumptions Aξ 1 ) Aξ ), ξ 1 ξ ν g ξ 1 + ξ ) ξ 1 ξ ξ 1 + ξ Aξ 1 ) Aξ ) L g ξ, 1.1) 1 + ξ ) ξ 1 ξ, ξ 1 + ξ for every ξ 1, ξ R n such that ξ 1 + ξ 0 and for some 0 < ν 1 L. For a proof of this fact see the end of Section Novelties and technical tools. We believe that the main interest of this paper, apart from the results of Theorems 1.1 and 1. themselves that will be used for instance in [31]), is the development of some tools for the treatment of the difficult equation 1.8) see Paragraph.3). We prove the Lipschitz estimate as an a priori estimate for problems enjoying further regularity. Instead of using a regularization of the type used in [8,, 9], the regularization we employ is of viscosity type, closer to that in []: we consider a vector field of the type A ε ξ) := φ ε A)ξ) + ε 1 + ξ ) p ξ, ξ R n, ε 0, 1), where p 1 is a large exponent and {φ ε } a family of mollifiers. This allows us to overcome the difficulties of deriving regularity estimates for the approximant problems, which we were not able to find in the literature. At this point continuity up to the boundary becomes an essential ingredient in the proof of the convergence, as well as the fact that we are solving a Cauchy-irichlet problem and therefore have a uniform bound on u ε L given by the maximum principle. We use the a priori Lipschitz continuity and the further regularity) of the approximating solutions in a way inspired by [7]. First, we employ the fact that the function v = is a subsolution to a similar problem, see Lemma 3.1. Then, we define an appropriate intrinsic geometry see 3.7)) depending on the growth of the approximating vector field A ε, which allows us to rebalance estimates, in the sense that the weight appearing in the Caccioppoli estimate for the equation satisfied by v turns out to be essentially constant, see 3.8). Here the fact that we can bound the supremum of u, and thus of v, from above is essential. Finally, we conclude the proof using an argument based on an alternative in order to get rid of the possible dependence on ε in terms of the aforementioned geometry, depending in turn on the growth of A ε.. PRELIMINARY MATERIAL: NOTATION, THE FUNCTION g, MISCELLANEA For what concerns Ω, we assume that there exists a radius R Ω > 0 such that for every point x 0 Ω there is a unit vector ê x0 such that the restriction of Ω is a graph of a C 1,β function in B RΩ along the ê x0 direction, in the following sense: with T being an orthogonal transformation that maps ê x0 into 0, 0,..., 0, 1), for every 0 < r R Ω it holds T 1 Ω x 0 ) B r r, r) ) = graph θ

5 THE CAUCHY-IRICHLET PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS 5 see below for the precise meaning of these symbols) with θ θ x0 C 1,β B r), θb r) r, r) and the C 1,β norm of θ uniformly bounded: [θ] C 1,β Θ. Note that without loss of generality, we can take ê x0 as the inner normal vector in x 0 : {v : v, ê x0 = 0} is the tangent hyperplane to Ω in x 0 ; therefore θ0) = 0. θ is the full gradient of θ with respect to its n 1 variables. Finally, by saying that a constant depends on Ω, we shall mean it depends on Θ..1. Notation. We denote by c a general constant always larger than or equal to one, possibly varying from line to line; relevant dependencies on parameters will be emphasized using parentheses, i.e., c 1 c 1 n, p, q) means that c 1 depends on n, p, q. For the ease of notation, we shall also use the following abbreviation: We denote by data := {n, g 0, g 1, ν, L}. B R x 0 ) := {x R n : x x 0 < R} the open ball with center x 0 and radius R > 0; when clear from the context or otherwise not important, we shall omit denoting the center as follows: B R B R x 0 ). The standard parabolic cylinder is defined as while we define the natural cylinder as Q R x 0, t 0 ) := B R x 0 ) t 0 R, t 0 ), Q G Rx 0, t 0 ) := B R x 0 ) t 0 [G1/R)] 1, t 0 ). The latter is strictly linked to the scaling of the equation, see Paragraph.6. Unless otherwise explicitly stated, different balls and cylinders in the same context will have the same center. We shall denote, for a factor α > 0, by αb R the ball B αr and by αq R x 0, t 0 ) the cylinder B αr x 0 ) t 0 αr), t 0 ); similarly for αq G R x 0, t 0 ). The parabolic boundary of a cylindrical domain = Γ, where is an open domain and Γ an open interval of the real line, is defined as p := inf Γ ) Γ ). Naturally, the parabolic closure of is then p := p. Accordingly with the customary use in the parabolic setting, when considering a sub-cylinder as above) compactly contained in Ω T, we shall mean that Ω and 0 < inf Γ < sup Γ T ; we will write in this case Ω T. By Ω x 0 we mean the set {x R n : x + x 0 Ω}. The standard parabolic distance is dist par x, t), y, s) ) := max { x y, t s } for any x, t), y, s) R n+1, while a distance strictly related to the scaling properties of the differential operator is ) [ dist par,g x, t), y, s) := max { x y, G 1 1 )] 1 }..1) t s Note that Q G R x 0, t 0 ) = {x, t) R n+1 : dist par,g x, t), x 0, t 0 )) < R, t < t 0 } and similarly for Q R x 0, t 0 ). Accordingly we define the parabolic distance between sets as ) dist par A, B) := inf dist par x, t), y, s) x,t) A y,s) B for A, B R n+1 ; similarly for dist par,g A, B). At a certain point it will be useful to split R n = R n 1 R. We agree here that we shall write a point x R n as x, x n ) R n 1 R; moreover, with B rx 0) we shall denote the ball of R n 1 with radius r and center x 0 R n 1.

6 6 PAOLO BARONI AN CASIMIR LINFORS With B R l being a measurable set, χ B denotes its characteristic function. If furthermore B has positive and finite measure and f : B R k is a measurable map, we shall denote by f) B fy) dy := 1 fy) dy B B B the integral average of f over B. If B is a cylinder, B := Γ R n+1, then we shall denote the slicewise average by f) τ) := fy, τ) dy. for almost every τ Γ. By sup we shall mean possibly the essential supremum, and similarly for inf. We shall also as usual denote osc B f := sup B f inf B f, [f] C 0,γ B) := sup x,y B x y fx) fy) x y γ. i f := f/ x i, for i {1,..., n}, will stand for the partial derivative of f in the ê i direction, and i,j f will denote f/ x i x j. Here ê i is the i-th element of the standard orthonormal basis of R n. By we shall denote the Sobolev conjugate exponent of, with the agreement that in the case n = we fix the value of as 4, i.e., n n >, := n.) 4 n =. With s being a real number, we shall denote s + := max{s, 0} and s := max{ s, 0}. For a vector ξ = ξ 1,..., ξ n ) R n, diag ξ denotes the diagonal matrix ξ i δ i,j ) n i,j=1. Finally, R + := [0, ), N is the set {1,,... } and N 0 = N {0}. By equation structurally similar to 1.8) 1 we mean an equation of the type t u div Ãu) = 0 with à satisfying assumptions 1.9) with ν, L and g replaced by ν, L and g. Both ν, L will depend on data, while g will satisfy 1.) and 1.5) with g 0, g 0, c l depending on data and c l... Properties of g. Without loss of generality we assume that 1 0 gρ) dρ = 1..3) Since 1.) implies that the map r gr)r g0 1) is increasing, while r gr)r g1 1) turns out to be decreasing, we have min { α g0 1, α g1 1} gr) gαr) max { α g0 1, α g1 1} gr) for every r, α > 0; clearly g0) = 0 and lim r gr) =. Since moreover g is strictly increasing, it has a strictly increasing inverse function g 1 C 1 R + ) with g 1 ) 1 r) = g g 1 for every r > 0. r)) Using 1.) we then see that also g 1 satisfies an Orlicz-type condition 1 g 1 1 rg 1 ) r) g 1 1 r) g 0 1 for every r > 0..4) Therefore, anything derived from 1.) for g holds for g 1 with g 0 1 and g 1 1 replaced by 1/g 1 1) and 1/g 0 1), respectively. efine the function G : R + R + as Gr) := r 0 gρ) dρ..5)

7 THE CAUCHY-IRICHLET PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS 7 Clearly G r) = gr) > 0 and G r) = g r) > 0 implying that G is both strictly increasing and strictly convex in 0, ). Moreover, G0) = 0 and G1) = 1 due to.3). We also define 1/G1/s) = 1/G 1 1/s) = 0 for s = 0. It is simple to check by integrating the function r rgr) by parts and using 1.) that also holds true for r > 0. efine the Young complement of G as Gr) = sup s>0 g 0 G r)r Gr) g 1.6) r rs Gs)) or Gr) := g 1 ρ) dρ;.7) in our setting these definitions are equivalent, see [33]. Note that the Young s inequality 0 sr Gs) + Gr).8) holds true for every r, s > 0 and by.4) and the second definition in.7) also G satisfies an Orlicz-type condition g 1 g 1 1 G r)r Gr) g 0 g ) Now starting from.6) and.9), we deduce precisely as for g the inequalities min{α g0, α g1 }Gr) Gαr) max{α g0, α g1 }Gr),.10) and } } min {α g 1 g 1 1, α g 0 g 0 1 Gr) Gαr) max {α g 1 g 1 1, α g 0 g 0 1 Gr) for every α, r 0. These, together with Young s inequality.8), imply for 0 < ε < 1 sr Gε 1 g 0 s) + Gε 1 g 0 r) εgs) + cg 0, ε) Gr). Another useful property is Gr) ) G Gr) for every r > 0, r see again [33] for the easy proof. From the second assumption of 1.9) we easily derive an upper bound for A. Indeed, when ξ R n \ {0} we have Aξ) ξ gs ξ ) Asξ) ds L ξ ds 0 s ξ cl, g 0 ) ξ 0 g r) dr cg 0, g 1, L) G ξ ) ;.11) ξ this holds also for ξ = 0 by our conventions, since A0) = 0. Similarly, the first assumption of 1.9) yields Aξ), ξ = 1 0 Asξ)ξ, ξ ds cg 1, ν) ξ We define the quantity V g : R n R n by ξ ) 1 g ξ ) V g ξ) = ξ ξ 0 g r) dr cg 0, g 1, ν)g ξ )..1)

8 8 PAOLO BARONI AN CASIMIR LINFORS when ξ 0 and set V g 0) = 0. Clearly V g is a continuous bijection of R n and, moreover, has a continuous inverse by the inverse function theorem. Furthermore, the following monotonicity formula holds true: Aξ 1 ) Aξ ), ξ 1 ξ c g ξ 1 + ξ ) ξ 1 ξ c V g ξ 1 ) V g ξ ).13) ξ 1 + ξ for a constant c cg 0, g 1, ν) and for every ξ 1, ξ R n, see [16, 17]..3. A concrete example. We give here a nontrivial example of a Lipschitz function g satisfying our assumptions - see Remark 1. This example is inspired by [8]. In particular we want to demonstrate the possibility that g oscillates between degenerate and singular behaviour. Suppose n/n + ) < g 0 < g 1 and set δ = g 1 g 0 )/3 > 0. efine the sequence s k = k for k N 0 and the function s g0 1+δ, 0 < s < gs) = s δ k+1 sg1 1, s k s < s k+1. s δ k+ sg0 1, s k+1 s < s k+ Clearly g is Lipschitz and it satisfies 1.). Moreover,.3) holds after scaling by a suitable normalization constant. We observe that gs), g 1 > + δ iff g 0 + g 1 > 6) lim sup = 1, g s s 1 = + δ iff g 0 + g 1 = 6), 0, g 1 < + δ iff g 0 + g 1 < 6) lim inf s gs) s, g 0 > δ iff g 0 + g 1 > 6) = 1, g 0 = δ iff g 0 + g 1 = 6). 0, g 0 < δ iff g 0 + g 1 < 6) By taking g 0 = 3 n, g 1 = + 3 n we obtain a particularly interesting case, that is, we have lim inf s gs)/s = 0 but lim sup s gs)/s =. Furthermore, if we consider the function gs) = 1 g1/s), we find similar behaviour as s 0. This is to say, we can build a structure function g and accordingly a vector field A as in 1.1)) that, for l N, along the sequence {l k } k N0 the function gs)/s is at the same time as large and as close to zero as we wish, and therefore it does not enjoy any monotonicity properties. This gives a clue about the difficulty of the application of e Giorgi-type methods, in particular when they have to be matched with intrinsic geometries: note that the expressions of the type Gs)/s gs)/s appear already in the energy estimate for 1.1), see Lemma.3. On the other hand, when the quantity g)/ is known to be under control, then the equation becomes treatable, see for instance Proposition 3.4 and in particular 3.8)..4. Orlicz spaces. For G as in.5), a measurable function u : A R, A R k, k N belongs to the Orlicz space L G A) if it satisfies G u ) dx <. A The space L G A) is a vector space, since G satisfies the -condition.10), and it can be shown to be a Banach space if endowed with the Luxemburg norm { u ) } u L G A) := inf λ > 0 : G dx 1. λ A

9 THE CAUCHY-IRICHLET PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS 9 A function u belongs to L G loc A), if u LG A ) for every A A. If also the weak gradient of u belongs to L G A), we say that u W 1,G A). The corresponding space with zero boundary values, denoted W 1,G 0 A), is the completion of Cc A) under the norm u W 1,G A) := u L G A) + u L G A). We denote by V G Ω T ) the space of functions u L G Ω T ) L 1 0, T ; W 1,1 Ω)) for which also the weak spatial gradient u belongs to L G Ω T ). The space V G Ω T ) is also a Banach space with the norm u V G Ω T ) := u L G Ω T ) + u L G Ω T ). Moreover, we denote by V0 G Ω T ) the space of functions u V G Ω T ) that belong to W 1,G 0 Ω) for almost every t 0, T ), while the localized version Vloc GΩ T ) is defined, as above, in the customary way. We also shorten V,G Ω T ) := L 0, T ; L Ω) ) V G Ω T ) and similarly for the localized and the zero trace versions. We shall moreover denote V,p Ω T ), for p > 1, the space V,G Ω T ) for the choice Gs) = s p..5. The concept of solution and consequences. We fix here the notions of solution employed in this paper. efinition 1. A function u is a weak solution to 1.8) 1 in a cylindrical domain R n+1, with the vector field A satisfying the assumptions 1.9), if u V,G loc ) and it satisfies the weak formulation [ u t η + Au), η ] dx dt = 0.14) for every test function η Cc ). If instead of equality we have the ) sign for every nonnegative η Cc ), we say that u is a weak subsolution supersolution) in. efinition. A function u is a solution to the Cauchy-irichlet problem 1.8) if u C 0 Ω T ) is a weak solution to 1.8) 1 in Ω T and moreover u = ψ pointwise on p Ω T. A very useful formulation, equivalent to.14), is the one involving Steklov averages. Indeed, the mild regularity of a solution does not allow us to use it as a test function. Furthermore, it is sometimes useful to have a weak formulation allowing for test functions independent of time, or test functions possibly vanishing only on the parabolic boundary of a cylinder. Apart from mollification, the possible way to have such properties involve the so-called Steklov averaging regularization of a function: for f : = t 1, t ) R measurable and 0 < h 1 appropriate, it is defined as f h x, t) := 1 h t t h fx, s) ds for x, t) t 1 + h, t ); note that we employ the backward regularization. If f L q ) for some q 1, then f h f in L q t 1 +ε, t )) for every ε > 0; the same holds in the L G spaces. Moreover, if f C 0 t 1, t ; L q )) then f h, τ) f, τ) in L q ) for a.e. τ t 1 + ε, t ) and for every ε > 0. At this point it is quite easy to infer the following slicewise formulation for weak solutions see [14]) using density arguments with respect to the spatial variable: [ t u h, τ)η + [Au)] h, τ), η ] dx = 0.15) for every η W 1,G 0 ), almost every τ t 1 + h, t ), and h > 0 such that the functions are well defined. Similar results hold also for weak super- and subsolutions.

10 10 PAOLO BARONI AN CASIMIR LINFORS Proposition.1. Comparison principle) Let := t 1, t ) Ω T and let u C 0 p ) be a weak subsolution to 1.8) 1 and v C 0 p ) a weak supersolution to 1.8) 1 in. If u v on p, then u v in p. Proof. For ε > 0 fixed define ϕ ε t) := t ε t) + and test.14) formally with η = u h v h ε) + ϕ ε. Note that η is compactly supported in due to the continuity of u and v and the fact that u v on p Q. Subtracting the Steklov version of the variational inequality of v from that of u and integrating over t 1, t ) yields t u h v h )η dx dt + [Au)] h [Av)] h, η dx dt 0. By the monotonicity of A, Lemma.13, we have [Au)] h [Av)] h, η dx dt Au) Av), u v) ϕ ε dx dt 0, {u>v+ε} and for the parabolic term we obtain using integration by parts t u h v h )u h v h ε) + ϕ ε dx dt = 1 u h v h ε) + t ϕ ε dx dt 1 t ε u v ε) + dx dt as h 0. Combining these gives t ε t 1 t 1 u v ε) + dx dt 0, which implies u v + ε almost everywhere in t 1, t ε). Since this holds for every ε > 0 and u, v CQ), the result follows. Observe that the uniqueness of a solution to the Cauchy-irichlet problem 1.8) follows immediately from the previous result. Moreover, we have the following corollary. Corollary.. Maximum principle) Let Ω T and let u C p ) be a weak solution to 1.8) 1 in. Then inf u u sup u p in p and, moreover, sup p u = sup u. p We recall the following standard energy inequality for local weak solutions. We give it in a more general form for future reference. Lemma.3 Caccioppoli s inequality). Let := t 1, t ) Ω T and let u be a weak solution to 1.8) in. Then there exists a constant c cg 0, g 1, ν, L) such that [ sup u k) ± ϕ g1], τ) dx + G u k) ± ) ϕ g1 dx dt τ t 1,t ) [ u k) ± ϕ g1], t 1 ) dx + c [ G ) ] ϕ u k) ± + u k) ± t ϕ dx dt for any k R and for every ϕ W 1, ) vanishing in a neighborhood of t 1, t ) and with 0 ϕ 1. The same inequality but only with the + sign holds for weak subsolutions.

11 THE CAUCHY-IRICHLET PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS 11 Proof. Fix ϕ W 1, ) as in the statement of the Lemma, call w := ±u k) ± and choose η = w h ϕ g1 as the test function in.15). Then we integrate over t 1, τ) for τ t 1, t ) to obtain t u h w h ϕ g1 χ t1,τ) dx dt + [Au)]h, w h ϕ g1 ) χ t1,τ) dx dt = 0..16) Integration by parts gives t u h w h ϕ g1 χ t1,τ) dx dt = 1 τ t w h)ϕ g1 dx dt.17) t 1 = 1 τ w hϕ g1 dx 1 τ w t=t 1 h t ϕ g1 ) dx dt t 1 1 τ w ϕ g1 dx 1 τ w t ϕ g1 ) dx dt t=t 1 t 1 as h 0. For the elliptic part we have by.1) [Au)]h, w h ϕ g1 ) χ t1,τ) dx dt τ t 1 τ c 1 t 1 τ Au), w ϕ g 1 dx dt + g 1 G w )ϕ g1 dx dt g 1 t 1 Au), ϕ w ϕ g 1 1 dx dt Au), ϕ w ϕ g 1 1 dx dt, where c 1 depends on g 0, g 1, ν. Furthermore, by.11), Young s inequality with ε 0, 1) to be chosen and the properties of g we obtain g 1 Au), ϕ w ϕ g 1 1 dx dt g 1 Aw) ϕ w ϕ g1 1 dx dt ) G w ) εc G ϕ g1 1 dx dt + cε) G ϕ w ) dx dt w εc G w )ϕ g1 dx dt + cε) G ϕ w ) dx dt,.18) where c depends on g 0, g 1, L and cε) depends on g 0, g 1, L as well as on ε. Now, combining.17)-.18) with.16) yields τ 1 w ϕ g1 dx 1 τ w t ϕ g1 ) dx dt + c 1 G w )ϕ g1 dx dt t=t 1 t 1 εc G w )ϕ g1 dx dt + cε) G ϕ w ) dx dt. We conclude by taking the essential supremum with respect to τ t 1, t ), choosing ε 0, 1) such that εc c 1 /, reabsorbing the term on the right-hand side and recalling the definition of w. The proof for subsolutions is very similar, taking into account that the test function η must be nonnegative..6. The geometry of the problem. In order to understand the equation, the first thing we want to stress is its scaling. Suppose u solves the model equation 1.1) in Q 1 = B 1 1, 0) and let κ > 0. Then the function x x0 ūx, t) := κu, 1 κ ) ) r κ G t t 0 ) r solves in Q κ r x 0, t 0 ) := B r x 0 ) t 0 κ [ κ )] 1, ) G t0 r

12 1 PAOLO BARONI AN CASIMIR LINFORS the equation where ḡ ū ) ) ū t div ū = 0,.19) ū ḡs) := κ r [ κ )] 1g κ ) G r r s..0) The function ḡ has the same structure as g, in the sense that it satisfies 1.) exactly with parameters g 0 and g 1 and moreover, we have G1) = 1, where s [ κ )] 1G κ ) Gs) := ḡσ) dσ = G r r s. 0 Conversely, if we have a solution w to 1.8) in Q κ r, then wx, t) := 1 x κ w 0 + rx, t 0 + κ [ κ )] 1t ) G r solves.19) in Q 1 with ḡ as in.0). In case we consider the general equation 1.8), the same scaling argument holds if we consider the vector field Aξ) := κ [ κ )] 1A κ ) G r r r ξ which satisfies the structural conditions 1.9) with g replaced by the function ḡ..7. Other auxiliary results. The following Lemma encodes the self-improving property of reverse Hölder inequalities. We take the form proposed in [7, Lemma 5.1] with slight changes in order to meet our purposes. Lemma.4. Let µ be a nonnegative Borel measure with finite total mass. Moreover, let γ > 1 and {σq} 0<σ 1 be a family of open sets with the property σ Q σq 1Q = Q whenever 0 < σ < σ 1. If w L Q) is a nonnegative function satisfying ) 1/γ) w γ dµ c 0 σ Q σ σ w dµ σq ) 1/ for all 1/ σ < σ 1, then for any 0 < q < there is a positive constant c cc 0, γ, q) such that 1/γ) 1/q w dµ) γ c σq 1 σ) ξ w dµ) q, Q for all 0 < σ < 1, where ξ := γ q qγ 1). The next one is a classic iteration Lemma. Lemma.5. Let φ : [R, R] [0, ) be a function such that φr) 1 φs) + where A, B 1 and β > 0. Then A + B for every R r < s R, s r) β φr) cβ) [ ] A R β + B.

13 THE CAUCHY-IRICHLET PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS A PRIORI LIPSCHITZ ESTIMATES In this section we impose on u an additional regularity assumption and prove intrinsic estimates for the gradient of u. To be precise, we shall suppose u, u ClocΩ 0 T ), u L loc0, T ; W, loc Ω)). 3.1) This is to say, we shall prove the estimates of this section as a priori estimates, leaving to Section 4 the approximation procedure which will explain how to deduce the desired estimates without the additional assumption 3.1). Notice that the continuity of u and u allows us to treat their pointwise values. ue to the assumed extra regularity it will be possible to differentiate the equation; this will be done by showing that the function is a subsolution to a similar equation. v := 3.) Lemma 3.1. Let u be a weak solution to 1.8) 1 in Ω T and, moreover, assume that the regularity assumptions 3.1) hold. Then v is a weak subsolution to t v divau)v) = 0 in Ω T. 3.3) Proof. Formally, the idea is to differentiate equation 1.8) 1 with respect to x j for j = 1,..., n, then multiply by j u, and finally sum over j. To this end, let 0 ϕ C c Ω T ), and test.14) with η = j j uϕ). This choice can be justified by using Steklov averages, as done previously in the paper; we shall proceed formally. Integration by parts yields 0 = u t j j uϕ)) dx dt + Au), j j uϕ)) dx dt Ω T Ω T = t j u) j uϕ dx dt + j Au), j uϕ) dx dt Ω T Ω T = 1 j u t ϕ dx dt + 1 Au) j u ), ϕ dx dt Ω T Ω T + Au) j u, j u ϕ dx dt. Ω T Now, since g) Au) j u, j u ϕ dx dt ν j u ϕ dx dt 0 Ω T Ω T by 1.9) 1, summing up over j = 1,..., n leads to t ϕ dx dt + Au), ϕ dx dt 0. Ω T Ω T This proves the claim. Next we prove a Caccioppoli inequality of porous medium type for the function v. Lemma 3.. Let u be a weak solution of 1.8) in Ω T and assume that 3.1) holds. Let := t 1, t ) Ω T and k R. Then there exists a constant c cν, L) such that sup [v k) +ϕ g) ], τ) dx + v k) + ϕ dx dt τ t 1,t ) ) g) c v k) + ϕ + t ϕ dx dt for every ϕ C ) vanishing in a neighborhood of p.

14 14 PAOLO BARONI AN CASIMIR LINFORS Proof. We can take η = v k) + ϕ χ t1,τ) for τ t 1, t ) as the test function in the weak formulation of 3.3), up to a regularization similar to the previous ones. For the parabolic part we have τ v t v k)+ ϕ ) dx dt = 1 τ t v k) t 1 +ϕ dx dt t 1 = 1 k) [v +ϕ ], τ) dx 1 τ v k) + t ϕ dx dt. t 1 The elliptic term can be estimated from below by using the assumptions 1.9) and Young s inequality with ε = ν/l). This gives τ Au)v, v k)+ ϕ ) dx dt t 1 = τ t 1 + τ ν t 1 Au)v k) +, v k) + ϕ dx dt τ L t 1 τ and thus, we obtain [v k) +ϕ ], τ) dx + ν Au)v k) +, ϕ v k) + ϕ dx dt g) v k) + ϕ dx dt t 1 g) v k) + ϕ v k) + ϕ dx dt ν τ g) v k) + ϕ dx dt t 1 τ cν, L) c t 1 τ t 1 g) ϕ v k) + dx dt, g) v k) + ϕ dx dt g) ϕ v k) + dx dt + Since τ t 1, t ) was arbitrary, the result follows. v k) + t ϕ dx dt. Combining the previous lemma with Sobolev s inequality leads to the following estimate. Lemma 3.3. Let the assumptions of Lemma 3. be in force. Then there exists a constant c cn, g 1, ν, L) such that g) v k) γ + ϕ γ dx dt c /n t t 1 ) γ 1 where - recall.) - v k) + g) γ := > 1. Proof. By Hölder s and Sobolev s inequalities we have g) v k) γ + ϕ γ dx dt ) γ ϕ + t ϕ dx dt), 3.4)

15 THE CAUCHY-IRICHLET PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS 15 1 t g) = v k) t t 1 t 1 +ϕ v k) +ϕ ) 1 / dx dt 1 t g) ) 1/v ) / k)+ ϕ) dx t t 1 t 1 ) 1 / v k) +ϕ dx dt ) 1 / cn) /n sup [v k) +ϕ ], τ) dx τ t 1,t ) A straightforward calculation yields g) ) 1/v k)+ ϕ) [ = v k)+ g ) 4v g) cg 1 ) g) g) ) 1/v k)+ ϕ) dx dt. 3.5) ) ] g) ) 1/v k)+ ϕ g) ) 1/v + k)+ ϕ v k) + ϕ + g) v k) + ϕ, and thus, integrating and estimating the first term using Lemma 3. yields g) ) 1 ) v k) + ϕ dx dt c v k) + g) ) ϕ + t ϕ dx dt, where the constant c depends only on g 1, ν, and L. From Lemma 3. it also follows that sup [v k) +ϕ ], τ) dx τ t 1,t ) ) g) c t t 1 ) v k) + ϕ + t ϕ dx dt; therefore, by inserting the previous two inequalities into 3.5) we obtain 3.4). Next the aim is to prove an intrinsic reverse Hölder s inequality. Q ρ x 0, t 0 ) Ω T, let λ 1 be such that To this end, let λ 1 4 sup, 3.6) Q ρx 0,t 0) and set We introduce the intrinsic cylinder θ λ := gλ) λ. Q λ ρ Q λ ρx 0, t 0 ) := min{1, θ λ } 1/ B ρ x 0 ) t 0 min{1, θ 1 λ }ρ, t 0 ). 3.7)

16 16 PAOLO BARONI AN CASIMIR LINFORS Note that we have the alternative expression B ρ x 0 ) t 0 θ 1 ) λ ρ, t 0, θλ 1 Q λ ρ = θ 1/ λ B ρx 0 ) ) t 0 ρ, t 0, 0 < θλ < 1, from which we easily see the analogy with the intrinsic geometry used to handle the parabolic p-laplacian, recalling that in this case gs)/s = s p and λ is dimensionally comparable to. Observe that we clearly have Q λ ρx 0, t 0 ) Q ρ x 0, t 0 ) in any case. Lemma 3.4. Let u be a weak solution to 1.8) 1 in Ω T, assume that 3.1) and 3.6) hold and let q > 0. Then there exists a constant c cn, g 1, ν, L, q) such that for every k λ. Q λ ρ/ 1/γ) ) 1/q v k) γ + dx dt) c v k) q + dx dt Q λ ρ Proof. Let 1/ σ < σ 1 and choose a cut-off function ϕ C σq λ ρ) vanishing in the neighborhood of p σq λ ρ) such that 0 ϕ 1, ϕ = 1 in σ Q λ ρ, and ϕ c ρσ σ ) min { 1, θ λ } 1/, t ϕ c ρ σ σ ) min { 1, θ 1 } 1. λ Observe that by the inclusion Q λ ρx 0, t 0 ) Q ρ x 0, t 0 ) and 3.6) we have 4λ in Q λ ρ. Moreover, we have λ in the support of v k) +, since k λ and v =. Thus, by using the properties of g we obtain in Q λ ρ {v k}. Now Lemma 3.3 yields v k) γ + dx dt cn) θ 1 λ σ Q λ ρ This is to say 1 4 θ λ g) cg 1 )θ λ 3.8) σq λ ρ g) v k) γ + ϕ γ dx dt c θ 1 λ min{1, θ λ } 1/ /n B σρ min{1, θ 1 λ }σρ)) γ 1 ) ) γ g) v k) + ϕ + t ϕ dx dt σq λ ρ c θ 1 λ min{1, θ λ} min{1, θ 1 λ }γ 1 ρ γ θλ min{1, θ λ } 1 + min{1, θ 1 λ } 1 ρ σ σ ) γ c = σ σ ) γ v k) + dx dt). σq λ ρ σ Q λ ρ σq λ ρ ) γ v k) + dx dt ) 1/γ) v k) γ + dx dt c 1/ σ σ v k) + dx dt), σq λ ρ where the constant c depends only on n, g 1, ν, L.

17 THE CAUCHY-IRICHLET PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS 17 Next we use Lemma.4 with w = v k) + and dµ = 1 dx dt. This gives for every Q λ ρ 0 < q < a constant c cn, g 1, ν, L, q) such that 1/γ) 1/q v k) γ + dx dt) c v k) q + dx dt) ; Q λ ρ/ the case q now follows from Hölder s inequality. Iterating the previous result yields the following pointwise estimate. Proposition 3.5. Let u be a weak solution to 1.8) in Ω T and assume that 3.1) holds. Then for every q > 0 there exists a constant c cn, g 1, ν, L, q) such that ux 0, t 0 ) λ + c λ ) ) 1/q) q dx dt Q λ ρ x0,t0) + holds for every λ satisfying 3.6). Proof. The idea is to apply e Giorgi s iteration method with the aid of Lemma 3.4. Let us first consider the case 0 < q <. To this end, choose for j N 0 ρ j = j ρ, Q λ ρ k j = λ + 1 j )d, where d > 0 is to be determined later. Observe that ρ 0 = ρ, k 0 = λ, and ρ j decreases to zero and k j increases to λ + d as j tends to infinity; clearly k j λ. enote Q j := Q λ ρ j x 0, t 0 ) and ) 1/q Y j := v k j ) q + dx dt for j N 0. Q j By Lemma 3.4 we have and since k j+1 > k j implies we obtain Y j+1 Q j+1 v k j ) γ + dx dt) 1/γ) c ) 1/q v k j ) q + dx dt, Q j v k j ) γ + k j+1 k j ) γ q v k j+1 ) q +χ {v kj+1}, ) γ/q c k j+1 k j ) β v k j ) γ + dx dt c d β βj Y 1+β j, Q j+1 for every j N 0, where β := γ/q 1 > 0 and c c n, g 1, ν, L, q). Then a standard hyper-geometric iteration lemma implies Y j 0 as j, provided that Y 0 c ) 1 β d and this can be guaranteed by choosing d = c ) 1 β v λ ) ) 1/q. q Q λ ρ x0,t0) + dx dt Now Lebesgue s differentiation theorem yields vx0, t 0 ) λ + d )) + = lim v λ + d )) ) 1/q q j + dx dt lim Y j = 0, Q j j which implies, recalling the choice of d, vx 0, t 0 ) λ + c v λ ) ) 1/q. q dx dt Q λ ρ x0,t0) + The case q follows again by Hölder s inequality.

18 18 PAOLO BARONI AN CASIMIR LINFORS 4. APPROXIMATION In this section we regularize the equation in order to apply the results of the previous section and show that the gradient of the solution to the regularized equation is uniformly bounded. Then all we have left to prove is that the approximating solutions converge to a function that solves the original equation. To this end, define for ε 0, 1) A ε ξ) := φ ε A)ξ) + ε 1 + ξ ) g 1 ξ, 4.1) where φ ε ξ) = φξ/ε)/ε n ; φ is a standard mollifier with R n φ dx = 1. That is, we mollify the vector field A and perturb it with the nondegenerate g 1 -Laplacian, where g 1 > max{g 1, }; we can take for example g 1 := g It is straightforward to see that A ε satisfies 1.9) with g replaced by g ε s) := gs + ε) s + ε s + ε1 + s) g1 s 4.) and L, ν replaced by L = cn, g 1 )L, ν = ν/cn, g 1 ), see also Paragraph 6.1. Now the key point is that O gε can be bounded independently of ε. Indeed, we have g 0 1 O gε s) g 1 1, where g 0 := min{g 0, }. Note that g ε also satisfies the lower bound in 1.5), since g ε s) gs)/ for s 1. Let u ε V, g1 Ω T ) C 0 Ω T ) be the solution to the Cauchy-irichlet problem { t u ε div A ε u ε ) = 0 in Ω T, u ε = ψ on p Ω T ; 4.3) for existence and uniqueness of such solutions see for instance [4]. Since ε1 + s) g1 g εs) s cg 1) 1 + s) g1, ε in addition to satisfying g ε -ellipticity and -growth conditions analogous to 1.9), the vector field A ε also enjoys nondegenerate p-laplacian growth conditions with p = g 1. Hence, by standard theory, u ε satisfies the assumption 3.1), see [14, 7]; therefore the results of the previous section are at our disposal for u u ε. Note that all the constants will turn out to be effectively independent of ε. Let us then show how to apply the result of the previous section in order to locally bound the gradient of the approximating solution uniformly in terms of ε. Here we also prove an estimate that, once convergence is established, leads to 1.10). Observe that the assumption 1.5) is crucial in this proof. We shall shorten ψ L ψ L pω T ). Proposition 4.1. Let u ε be a solution to 4.3) and let Ω T. Then u ε L ) is bounded by a constant depending on data, ɛ, c l, ψ L, and dist par p Ω T, ), but independent of ε. Proof. Let us consider a standard parabolic cylinder Q 4R Q 4R x, t ) Ω T and a subcylinder Q ρ x 0, t 0 ) Q R. Moreover, let λ 1 be such that λ 1 4 sup u ε. 4.4) Q ρx 0,t 0) We divide the proof into two cases depending on which term of g ε dominates at λ.

19 THE CAUCHY-IRICHLET PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS 19 Case I. Assume Setting we clearly have gλ + ε) λ + ε θ ε λ := g ελ) λ = ε1 + λ) g1. gλ + ε) λ + ε + ε1 + λ) g1 ε1 + λ) g1 θ ε λ ε1 + λ) g1. 4.5) By applying Proposition 3.5 to u ε with q = g 1 / we obtain ) 1/ g1 u ε x 0, t 0 ) λ + c uε λ ) g1/ Q λ ρ x0,t0) + dx dt max{1, θ ε 1/ g1 λ + c λ } min{1, θλ ε u g1 ε dx dt), }n/ Q ρx 0,t 0) since Q λ ρx 0, t 0 ) Q ρ x 0, t 0 ). We further distinguish two cases: in the case when θλ ε 1 we get max{1, θ ε λ } min{1, θ ε λ }n/ = θε λ ε1 + λ) g1, while when 0 < θλ ε < 1 we have max{1, θλ ε} min{1, θλ ε = θε λ) n/ ε1 + λ) g1 ) n/ = ε ε 1+/n 1 + λ) g1 ) n/ ; }n/ in both cases we have used 4.5). Since ε gλ + ε) 1 + λ) g1 λ + ε) c l λ + ε ) n n+ +ɛ λ ) g1 c l 1 + λ ) g1+min{ɛ 4/n+),0} =: cl 1 + λ ) η by 1.5) and the fact that λ 1, plugging this estimate into 4.6) yields ε 1+/n 1 + λ ) g 1 ) n/ cn, cl ) 1 + λ ) η1+/n)+ g 1 )n/ cn, c l ) 1 + λ ) g 1 min{ɛn+)/,} ; a direct computation shows indeed the relation between the exponents. Hence we have u ε x 0, t 0 ) λ + c 1 + λ ) 1 min{ɛn+)/ g 1),/ g 1} ) 1/ g1 ε u g1 ε dx dt Q ρx 0,t 0) ) max{ ɛn+), 1 } λ + c ε u g1 ε dx dt + 1 Q ρx 0,t 0) by Young s inequality; we also used g 1 >. 4.6)

20 0 PAOLO BARONI AN CASIMIR LINFORS Case II. Suppose then that gλ + ε) > ε1 + λ) g1. λ + ε Here we have gλ + ε) θ ε gλ + ε) λ λ + ε λ + ε, and again by Proposition 3.5 max{1, θ ε u ε x 0, t 0 ) λ + c λ } uε min{1, θλ ε λ ) ) 1 q }n/ + dx dt q. When θλ ε 1, choosing q = 1 leads to gλ + ε) u ε x 0, t 0 ) λ + c λ + ε λ + c λ + c Q ρx 0,t 0) The second inequality stems from the fact that Q ρx 0,t 0) Q ρx 0,t 0) uε λ ) + dx dt ) 1 g u ε ) uε λ ) ) 1 u ε + dx dt G u ε )χ { uε 1} dx dt Q ρx 0,t 0) ) 1. 1 λ + ε) u ε 4λ + ε) 4.7) in the set Q ρ x 0, t 0 ) { u ε λ} by 4.4), while for the last one we used.6) and the fact that λ 1. In the case 0 < θλ ε < 1 we choose q = ɛn + )/4 and use 1.5) and again 4.7) to obtain gλ + ε) u ε x 0, t 0 ) λ + c λ + ε λ + c Q ρx 0,t 0) λ + c = λ + c λ + c ) n uε λ ) ) 1 q dx dt q + Qρx0,t0) ) n uε uε λ ) ) 1 q g u ε ) dx dt q + n 1 u ε Q ρx 0,t 0) u ε Q ρx 0,t 0) 1+ n n+ ɛ) n +q χ { uε 1} dx dt n+ +ɛ χ { uε 1} dx dt G u ε )χ { uε 1} dx dt Q ρx 0,t 0) ) ɛn+) ) ɛn+) ; ) 1 q note that 1 n ) n 4 ) n n + ɛ + q = n + ɛ + ɛn + = 1 + n n + + ɛ. Therefore in both cases we have u ε x 0, t 0 ) λ + c G u ε )χ { uε 1} dx dt Q ρx 0,t 0) Combining Cases I and II and denoting η := max { 1, ɛn+)} yields u ε x 0, t 0 ) ) max{ 1, ɛn+)}.

21 since THE CAUCHY-IRICHLET PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS 1 ) λ + c G u ε )χ { uε 1} + ε u g1 ε dx dt) η + 1 Q ρx 0,t 0) R ) n+) η ) η λ + c G ε u ε ) dx dt + 1, 4.8) ρ Q R Gs) 1 g 0 gs + ε)s + ε) 4 g 0 gs + ε) s + ε s 4 g 0 sg ε s) 4 g 1 g 0 G ε s) for s 1 and trivially εs g1 ε1 + s) g1 s g 1 G ε s). The constant c in 4.8) depends only on data, ɛ, c l. Let us now choose two intermediate cylinders Q R Q r Q s Q R and fix λ := u ε L Q s) <, x 0, t 0 ) Q r, ρ := s r > 0. Clearly Q ρ x 0, t 0 ) Q s so that 4.4) holds. Then 4.8) implies u ε L Q r) 1 u ε L Q s) R ) n+) η ) η + c G ε u ε ) dx dt + 3. s r Q R Now, by choosing φr) = u ε L Q r), iteration Lemma.5 gives [ u ε L Q R ) c Gε u ε ) + 1 ] ) η dx dt. 4.9) Q R At this point, in order to get rid of the dependence on ε on the right-hand side, the idea is to use the Caccioppoli inequality of Lemma.3 to translate the dependence on u ε to one on u ε, and the latter in turn into a dependence on ψ. Indeed, take ϕ C Q 4R ) vanishing in a neighborhood of p Q 4R such that 0 ϕ 1, ϕ = 1 in Q R, and ϕ + t ϕ c/r. Since sup u ε sup u ε = sup ψ ψ L Q 4R pω T pω T by the maximum principle, Corollary., we can estimate by Lemma.3 [ G ε u ε ) dx dt c Gε ϕ u ε ) + u ε t ϕ ] dx dt Q R Q 4R c 1 + ψ ) g1 ) L ψ L + c R R = c data, ɛ, c l, ψ L, R ). 4.10) Note that the constant does not depend on ε. Therefore we conclude the proof of the Proposition, modulo a standard covering argument A uniform interior modulus of continuity via Lipschitz regularity. In this section we prove that the approximating solutions u ε are equicontinuous in the interior of the domain; in particular we shall show their equi-lipschitz regularity with respect to the parabolic metric. Proposition 4.. Let u ε be a solution to 4.3). Then u ε Lip1, 1/)Ω T ) locally, uniformly in ε; this is to say, for every subcylinder Ω T there exists a constant c depending on data, ɛ, c l, ψ L, and dist par p Ω T, ) such that u ε x, t) u ε y, s) c dist par x, t), y, s) ) 4.11)

22 PAOLO BARONI AN CASIMIR LINFORS for every x, t), y, s) and for every ε 0, 1). Proof. Fix an intermediate set such that Ω T and dist par ẑ, p Ω T ) = dist par, p Ω T )/ =: d/ for every ẑ p. Take also a cylinder Q r x 0, t 0 ) with x 0, t 0 ) ; this will happen for instance if r d/. Since u ε is continuous, by applying the divergence theorem and using the bound for A ε in.11) we infer B rx 0) u ε, τ) dx t τ=t 1 = n r c r t t 1 t t 1 B rx 0) B rx 0) x x 0 A ε u ε ), dh n 1 dt x x 0 g ε u ε ) dh n 1 dt for all t 0 r < t 1 t < t 0, where H n 1 stands for the n 1)-dimensional Hausdorff measure. We thus estimate t osc u ε) Brx τ t 0 r,t 0)τ) = sup 0) u ε, τ) dx t 0 r <t 1 t <t 0 B rx 0) τ=t 1 c t0 g ε u ε ) dh n 1 dt r t 0 r B rx 0) c r 1 + u ε L Q rx 0,t 0)) c r 1 + u ε L )) g1 1. Now by Proposition 4.1, in particular by 4.9)-4.10), we have ) g1 1 osc u ε) Brx τ t 0 r,t 0)τ) c data, c l, ɛ, ψ L, d ) r. 4.1) 0) At this point we simply split for x 1, t 1 ), x, t ) Q r x 0, t 0 ) u ε x 1, t 1 ) u ε x, t ) u ε x 1, t 1 ) u ε, t 1 ) dx B rx 0) + u ε, t 1 ) dx u ε, t ) dx + uε x, t ) B rx 0) B rx 0) B rx 0) u ε, t ) dx. While in order to bound the second term we shall use 4.1), the first and last terms can be estimated using the mean value theorem as follows: u ε x i, t i ) u ε, t i ) dx uε x i, t i ) u ε x, t i ) dx B rx 0) B rx 0) r u ε L ), for i {1, }. Therefore, using again Proposition 4.1, we have osc u ε c r 4.13) Q rx 0,t 0) with c as in 4.1), in particular not depending on ε. To conclude the proof, for x 1, t 1 ), x, t ), we simply check whether dist par x1, t 1 ), x, t ) ) d/4 holds true or not; if so, then there exists a cylinder Q r x 0, t 0 ) with r = dist par x1, t 1 ), x, t ) ) such that x 1, t 1 ), x, t ) Q r x 0, t 0 ) and we can apply 4.13) that directly yields 4.11). If on the other hand dist par x1, t 1 ), x, t ) ) > d/4, then, again simply using the maximum principle, we have u ε x, t) u ε y, s) u ε L Ω T ) 8 dist par x1, t 1 ), x, t ) ) ψ L ; 4.14) d the proof is concluded.

23 THE CAUCHY-IRICHLET PROBLEM FOR A CLASS OF PARABOLIC EQUATIONS 3 Remark. Notice that, tracking the dependence on d of the constant in Proposition 4. and in turn the dependence on R of estimate 4.10), and also slightly modifying the previous proof, we deduce that estimate 4.11) can be rewritten as u ε x, t) u ε y, s) c ) d γ dist par x, t), y, s), 4.15) z,w for an exponent γ γn, g 1, ɛ) 1 and a constant c depending only on data, ɛ, c l, ψ L, with z = x, t), w = y, s) and accordingly d z,w := min { dist par z, p Ω T ), dist par w, p Ω T ), 1 }. Indeed, if dist par z, w) d z,w /8, then we can apply the argument in the first part of the proof of Proposition 4. with r = dist par z, w) to get suppose s t) u ε z) u ε w) osc u ε c dist par z, w), Q rz) where γ = g 1 g 1 1) η, since we have Q r z) Q dz,w/8z), Q dz,w/z) Ω T and so u ε L Q rz)) u ε L Q dz,w /8 z)) c d γ z,w η d g1 z,w The case where d z,w < 8 dist par z, w) can be approached exactly as in 4.14). 5. CONTINUITY AT THE BOUNARY In this section we prove that the solution to the approximating problem 4.3) is continuous up to the boundary independently of ε by building an explicit barrier. We do not want to enter the details of the theory and the general relation between existence of barriers and regularity of the boundary points; the interested reader can see the nice paper [4] for the evolutionary p-laplacian, while [1, 3] summarize the results in the elliptic setting. We shall begin with the proof of the continuity at the lateral boundary; here we shall give all the details needed. For the continuity at the initial boundary we shall however only sketch the proof, which on the other hand is very similar and easier than the lateral case. Again, we will prove the existence of a uniform in the sense that it will be independent of ε) modulus of continuity for u ε ; in the last section we shall show that this modulus is easily inherited by the limit of u ε. Let us begin with the construction of an explicit barrier at the lateral boundary. ue to a scaling argument that will be clear soon it is enough to consider a very special case An explicit construction of a supersolution at the boundary. We define the function v + x, t) := x + M x n + t + 1), where M 1 is to be chosen depending on data. We aim to show that v + is a weak supersolution in Simple calculations show that Q := { x, t) R n+1 : x 1, x n [0, ], t [ 1, 0] }. v + = x, Mx 1/ n /), t v + = χ { 1<t< 1/}, v + = diag,...,, Mx 3/ n /4), and moreover, since i,j v+ = 0 whenever i j, we have n n div Av + ) = i A i v + ) = ξj A i v + )i,jv + i=1 i,j=1 n 1 = ξi A i v + ) M 4 ξ n A n v + )x 3/ n. i=1

24 4 PAOLO BARONI AN CASIMIR LINFORS The first term we estimate from above using 1.9) and for the second term we can apply 1.9) 1, since ξn A n v + ) = Av + )ê n, ê n. Furthermore, if we require M 3/ 16n 1)L/ν, we obtain div Av + ) n 1)L ν 4 Mx 3/ n ν 8 Mx 3/ n g v + ) v +. ) g v + ) v + Now, observe that since M 4 we also get v + = 4 x + Mx 1/ n / ) Mx 1/ n in Q. On the other hand, we have v + Mx 1/ n / 1. Using these estimates we obtain { g v + ) 1, g0 v + v + g0 1/) g0 Mx n g 0 <, and thus div Av + ) ν 8 M min{g0,} 1 x min{g0,}+1)/ n. 5.1) The exponent of x n is negative, so that by choosing M Mdata) large enough recall that g 0 > 1), we finally obtain t v + div Av + ) + ν 8 min{g0,}+1)/ M min{g0,} 1 0. It is easy to see that v + V,G loc Q) and thus v + is a weak) supersolution in Q. 5.. A reduction of the oscillation in a significant case. We set ourselves now in what seems to be a very particular, unitary case; it will be clear soon that, up to a simple rescaling procedure, this will be the significant case for the proof. Let Ω be a bounded C 1,β domain and Ω T := Ω 1, 0). Suppose that 0 Ω and the orthonormal system where the boundary is a graph is the standard cartesian one, with the direction where Ω is a graph given by ê n. We hence have Ω { x < 1, x n < 1} = graph θ, with θ : B 10) 1, 1) and θ0) = 0 and Ω { x < 1, x n < 1} is the epigraph of θ. Let ū be a weak solution to 1.8) 1 in Ω T Q 1 with Q 1 := B 1 1, 1) 1, 0) R n+1, such that ū = ψ in p ΩT Q 1. Moreover, we suppose ψ0) = ū0) = 0. Take δ 0, 1) to be fixed later. We assume that and moreover that the graph of θ over B 1 is contained in the cylinder B 1 δ, δ) 5.) osc ψ δ and osc ū 1 5.3) Ω B 1 1,1) Ω T Q 1 Let us take the barrier v + built in the previous paragraph and shift it in the ê n direction as follows: v + δ x, x n, t) := v + x, x n + δ, t) + δ. Now v + δ is defined and continuous, in particular, over the parabolic closure of Ω T Q δ, where Q δ = B 1 δ, 1) 1, 0), and there it is still a supersolution to an equation structurally similar to 1.8) 1. The aim is to prove that ū v + δ on p Ω T Q δ ) by considering the different pieces: