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Hindawi Publishing Corporation Boundary Value Problems Volume 2007, Article ID 35825, 28 pages doi:10.1155/2007/35825 Research Article Interior Gradient Estimates for Nonuniformly Parabolic Equations II Gary M. Lieberman Received 31 May 2006; Revised 6 November 2006; Accepted 9 November 2006 Recommended by Vincenzo Vespri We prove interior gradient estimates for a large class of parabolic equations in divergence form. Using some simple ideas, we prove these estimates for several types of equations that are not amenable to previous methods. In particular, we have no restrictions on the maximum eigenvalue of the coefficient matrix and we obtain interior gradient estimates for so-called false mean curvature equation. Copyright © 2007 Gary M. Lieberman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction A key step in the study of second-order quasilinear parabolic equations is establishing suitable aprioriestimates for any solution of the equation. This fact is the theme of many books on the subject [1–5] and our focus here is on one particular such estimate: a local pointwise gradient estimate for solutions of equations in divergence form: u t = divA(X,u,Du)+B(X,u,Du). (1.1) The role of this divergence structure has been noted many times under varying hypothe- ses on the functions A and B (see, in particular [6, Sections VIII.4 and VIII.5], [3,Section V.4 ] , [ 5, Section 11.5]). Our cur rent interest is deriving this estimate using a surprising variant (detailed below) of standard methods. Although this variant seems, at first, to be a purely technical modification, we mention here two quite different types of estimates which follow from this variant and which appear to be new. First, we derive a local gra- dient estimate for a class of equations which includes the parabolic false mean curvature 2 Boundary Value Problems equation, that is, the e quation with A(X,z, p) = exp  1 2  1+|p| 2   p (1.2) and some conditions on B. Such an operator does not fall under the hypotheses from, for example, [3], and the present author has, previously, given an incorrect proof of this estimate [7, page 569] (we will point out the error later), and then in [5, Section 11.5, page 281] a correct but weaker version of the estimate. Second, we estimate the gradient of a solution to a large class of equations only in terms of the structure of the equation and a bound for the gradient of the initial function. (Ordinarily, a gradient estimate is given in terms of a maximum estimate for the solution, which, in turn, depends on some estimate on the boundary and initial data.) Such an estimate was first proved by Ecker for the parabolic prescribed mean curvature equation [8, Theorem 3.1], but we also show that such an estimate is valid for the parabolic p-Laplacian if p<2, and this fact seems to be new . (In [9], a corresponding estimate was given for the L q norm of the solution in termsoftheL q norm of the initial data, and this estimate can be used to infer a gradient estimate, but our goal here is to give an estimate directly.) This gradient estimate provides an interesting counterpoint to known results on these equations (see [6, Chapter XII] for a detailed description of these results). In particular, it is known that for p>2n/(n +1), solutions of this equation are bounded (and have H ¨ older continuous spatial derivatives) at any positive time for quite general initial data, in particular for L 1 initial data. On the other hand, [6, Section XII.13-(i)] provides an initial datum in L 1 for which the solution is unbounded for all sufficiently small positive time. Although the counterexample is de- scribed in all of R n × (0,∞), it should be noted that it satisfies the boundary condition u = 0on{|x|=1,t>0}, so the regularity of the solution is affected only by that of the initial datum. An important point for our comparison is that the solution becomes infi- nite only at x = 0(fort>0 as well) and the initial function is smooth except at x = 0. Our result shows that this is the only configuration in which the solution can be unbounded since we obtain a gradient estimate at any x = 0. Of course, the additional surprise is that our gradient estimate also applies to some equations with p>2n/(n +1). The basic plan is to modify the Moser iteration technique [10] along the lines of Si- mon’s estimate for elliptic equations [11]. Of course, this is the plan followed by the au- thor before (especially [7]) but we add two important new twists. As in [12], we obtain an estimate that does not use an upper bound on the maximum eigenvalue of the matrix ∂A/∂p. Such an approach is also useful in studying anisotropic problems (see [13, 14]) and we present the calculations for this case in [15]. In addition, we use a modified version of the Sobolev inequality from [11]. This inequality will allow us to prove some unusual estimates (in particular the estimates for parabolic p-Laplace equations) and also to use some more standard notations, in particular, we will use a ij to denote the components of the matrix ∂A/∂p;in[7, 11, 16], a ij denoted the components of a slightly different matrix. Following [11], we break the estimate into several steps. After giving some notation in Section 2, we prove an energy-type inequality in Section 3. We then present the S obolev Gary M. Lieberman 3 inequality in Section 4, and we use the energy inequality along with the Sobolev inequal- ity in Section 5 to bound the maximum of the gradient in terms of an integral:  w  | Du|  q Du· AdX (1.3) for some function w and some exponent q, which we will detail in that section. This in- tegral is estimated in Section 6 in terms of the integral of Du · A, and this final integral is easily estimated; we will quote [5, Lemma 11.13]. Section 7 contains some examples, es- pecially the false mean curvature equation, to illustrate our structure conditions. We also discuss some interesting variants of our estimate. In Section 8, we examine the applica- tion of our Sobolev inequality to some equations satisfying structure conditions depend- ing on the maximum eigenvalue of ∂A/∂p; the most important of such equations are the parabolic prescribed mean curvature equation and parabolic p-Laplacian with p<2de- scribed above. Finally, we look at parabolic equations with faster than exponential growth in Section 9; our method is only partially successful in dealing with such problems. 2. Notation For the most part, we follow the notation in [5], so X = (x,t) denotes a point in R n+1 with |X|=  n  i=1  x i  2 + |t|  1/2 , (2.1) and, for R>0, we write Q(R) =  X ∈ R n+1 : |x| <R, −R 2 <t<0  , B(R) =  x ∈ R n : |x| <R  . (2.2) We also use ᏼQ(R) to denote the parabolic boundary of Q(R), that is, the set of X such that either |x|=R, −R 2 ≤ t ≤ 0, (2.3) or |x| <R, t =−R 2 . (2.4) Moreover, we use N to denote n if n>2 and an arbitrary constant greater than 2 if n = 2. We always assume that u ∈ C 2,1 (Q(R)) for some R>0andweset v =  1+|Du| 2  1/2 , ν = Du v , g ij = δ ij − ν i ν j . (2.5) We will also use this notation, without further comment, with p in place of Du to describe structural conditions on the functions A and B (and their derivatives). We also set a ij = ∂A i ∂p j , Ꮿ 2 = a ij g km D ik uD jm u, Ᏹ = a ij D i vD j v, (2.6) 4 Boundary Value Problems where we use the Einstein summation convention that repeated indices are summed from 1 to n.(Notethata ij , Ꮿ 2 ,andᏱ are not quite the same as in [7, 11, 16].) We also define the oscillation of u over a set S by osc S u = sup S u −inf S u. (2.7) In addition, for parameters τ>1andr ∈ (0,R], we write Q τ (r)andq τ (r,t)forthe subsets of Q(r)andB(r) ×{t}, respectively, on which v>τ. 3. The energy inequality In this section, we prove an energy inequality, that is, an inequality which estimates in- tegrals involving second spatial derivatives of u in terms of integrals involving only first derivatives. Before stating this inequality, we present some preliminary structure condi- tions. We suppose that there are matrices [C i k ]and[D i k ]suchthatD i k is differentiable with respectto(x,z, p)and C i k + D i k = ∂A i ∂z p k + ∂A i ∂x k + Bδ i k . (3.1) For simplicity, we set Ᏸ ij = ν k ∂D i k ∂p j , Ᏺ =  p i ∂D i k ∂z + ∂D i k ∂x i  ν k . (3.2) Our structure conditions are stated in terms of these expressions. We assume that there are nonnegative constants τ 0 ≥ 1, β 1 ,andβ 2 along with positive functions Λ 0 , Λ 1 ,andΛ 2 such that C i k g jk η ij ≤ β 1 Λ 1/2 0  a ij η ik η jk  1/2 , (3.3a) C i k ν k ξ i ≤ β 1 Λ 1/2 0  a ij ξ i ξ j  1/2 , (3.3b) vᏰ ij η ij ≤ β 1 Λ 1/2 0  a ij η ik η jk  1/2 , (3.3c) vᏲ ≤ β 2 1 Λ 0 , (3.3d) v   ν k D i k − ν i B   ≤ β 1 Λ 1 , (3.3e) |A|g ij η ij ≤ β 2 Λ 1/2 2  a ij η ik η jk  1/2 , (3.3f) |A|ν ·ξ ≤ β 2 Λ 1/2 2  a ij ξ i ξ j  1/2 , (3.3g) for all n × n matrices η,alln-vectors ξ,andall(X, z, p) ∈ Q(R) × R × R n such that z = u(X)andv>τ 0 . Note that conditions (3.3a)–(3.3d) are exactly the same as [5, (11.41a–d)] (except for a slight variation in notation). Our energy estimate is then a variant of [5, Lemma 11.10] (which in turn comes from [11, (2.11)]). Gary M. Lieberman 5 Lemma 3.1. Let χ be an increasing, nonnegative Lipschitz function defined on [τ, ∞) for some τ ≥ τ 0 and let ζ be a nonnegative C 2,1 (Q(R)) function which vanishes in a neighbor- hood of ᏼQ(R). Suppose conditions (3.3)hold,anddefine Ξ(σ) =  σ τ (ξ − τ)χ(ξ)dξ. (3.4) Then  q τ (R,s) Ξ(v)ζ 2 dx +  Q τ (R)  1 − τ v  Ꮿ 2 + Ᏹ  χζ 2 dX ≤ 20β 2 1  Q τ (R) Λ 0  (v − τ)χ  + χ  ζ 2 dX +4β 1  Q τ (R) Λ 1 χζ|Dζ|dX +4  Q τ (R) |A|χ    D 2 ζ   ζ + |Dζ| 2  vdX+32β 2 2  Q τ (R) Λ 2  (v − τ)χ  + χ  | Dζ| 2 dX +4  Q τ (R) Ξζζ t dX (3.5) for any s ∈ (−R 2 ,0). (Here, and in what follows, the argument v from χ and Ξ is suppressed.) Proof. Webeginjustasin[5, Lemma 11.10]. Let θ be a vector-valued C 2 function which vanishes in a neighborhood of ᏼQ(R), and set Q = B(R) × (−R 2 ,s). If we multiply the differential equation by divθ and then integrate by parts, we obtain  Q  − u t D k θ k + D k A i D i θ k + BD k θ k  dX = 0. (3.6) An easy approximation argument shows that this identity holds for any θ whichisonly Lipschitz (with respect to x only); in particular, we take θ = (v − τ) + χ(v)ζ 2 ν. (3.7) Just as in [5, pages 270-271], we see that  Q −u t D k θ k dX =  q τ (R,s) Ξζ 2 dx − 2  Q Ξζζ t dX. (3.8) Next, we have  Q D k A i D i θ k + BD k θ k dX =  Q  D k A i + Bδ i k  D i  (v − τ) + χν k  ζ 2 dX +  Q D k A i (v − τ) + χν k D i  ζ 2  dX +  Q B(v − τ) + χν · D  ζ 2  dX. (3.9) 6 Boundary Value Problems The first integr al is handled as usual. We set Ψ = ⎧ ⎨ ⎩ (v − τ)χ  + χ if v>τ, 0ifv ≤ τ, (3.10) and we note that D i  (v − τ)χν k  = ΨD i vν k +  1 − τ v  + χg kj D ij u. (3.11) It follows that  D k A i + Bδ i k  D i  (v − τ) + χν k  =  1 − τ v  + χ  Ꮿ 2 + C i k g kj D ij u  + Ψ  Ᏹ + ν k C i k D i v  + D i k D i  (v − τ) + χν k  . (3.12) An integration by parts then yields  Q D i k D i  (v − τ) + χν k  ζ 2 dX =−  Q  Ᏸ ij D ij u + Ᏺ  (v − τ) + χ  ζ 2 dX − 2  Q D i k (v − τ) + χν k ζD i ζdX. (3.13) For the second integral, we integrate by parts again (cf. the proof of [12, Lemma 2.3]):  Q D k A i (v − τ) + χν k D i  ζ 2  dX =−2  Q A i D k  (v − τ) + χν k  ζD i ζdX− 2  Q A i χ(v − τ) + ν k  ζD ik ζ + D i ζD k ζ  dX. (3.14) To simplify the notation, we now set I 1 =  q τ (R,s) Ξζ 2 dx, I 2 =  Q τ (R)  1 − τ v  Ꮿ 2 χ + Ᏹ  χ  (v − τ)+χ   ζ 2 dX. (3.15) Then I 1 + I 2 = 2  Q Ξζζ t dX + 10  j=3 I j , (3.16) Gary M. Lieberman 7 where I 3 =−  Q  1 − τ v  + C i k χg kj D ij udX, I 4 =−  Q ΨC i k ν k D i vζ 2 dX, I 5 =  Q Ᏸ ij D ij u(v − τ) + χζ 2 dX, I 6 =  Q Ᏺ(v − τ) + χζ 2 dX, I 7 = 2  Q  D i k − Bδ i k  (v − τ)χν k ζD i ζdX, I 8 = 2  Q  1 − τ v  + A i D i ζg kj D kj udX, I 9 = 2  Q A i (v − τ) + χν · DvζD i ζdX, I 10 = 2  Q A i χ(v − τ) + ν k  ζD ik ζ + D i ζD k ζ  dX. (3.17) These terms are estimated as in [5, Lemma 11.10] using (3.3) and Cauchy’s inequality. For the reader’s convenience, we give a brief estimate of each integral. First, from (3.3a), we have I 3 ≤ β 1  Q Λ 1/2 0  a ij D ik uD jk u  1/2  1 − τ v  + χζ 2 dX. (3.18) Since a ij D ik uD jk u = Ꮿ 2 + Ᏹ (3.19) and χ  ≥ 0, we have a ij D ik uD jk u  1 − τ v  + χ ≤  1 − τ v  + χᏯ 2 + ΨᏱ. (3.20) Therefore, by Cauchy’s inequality, I 3 ≤ 3β 2 1  Q Λ 0 Ψζ 2 dX + 1 12 I 2 . (3.21) Similarly, since Ꮿ 2 ≥ 0, we see from (3.3b) and Cauchy’s inequality that I 4 ≤ 3β 2 1  Q Λ 0 Ψζ 2 dX + 1 12 I 2 . (3.22) Next, we use (3.3c), (3.20), and Cauchy’s inequality to obtain I 5 ≤ 3β 2 1  Q Λ 0 Ψζ 2 dX + 1 12 I 2 . (3.23) 8 Boundary Value Problems Moreover, (3.3d)gives I 6 ≤ β 2 1  Q Λ 0 Ψζ 2 dX, (3.24) and (3.3e)gives I 7 ≤ 2β 1  Q Λ 1 χ(v − τ) + ζ|Dζ|dX. (3.25) From (3.3f) and Cauchy’s inequality, we infer that I 8 ≤ 8β 2 2  Q Λ 2 Ψ|Dζ| 2 dX + 1 8 I 2 , (3.26) and, finally, (3.3g), (3.20), and Cauchy’s inequality imply that I 9 ≤ 8β 2 2  Q Λ 2 Ψ|Dζ| 2 dX + 1 8 I 2 . (3.27) It follows that I 1 + I 2 ≤ 2  Q Ξζζ t dX +10β 2 1  Q Λ 0 Ψζ 2 dX +2β 1  Q Λ 1 χζ|Dζ|dX +16β 2  Q Λ 2 Ψ|Dζ| 2 dX +2  Q |A|χ    D 2 ζ   ζ + |Dζ| 2  dX + 1 2 I 2 . (3.28) Then (3.5) follows from this inequality by simple algebra.  In Section 6, we will need a sharper version of this lemma. To obtain this version, we note that (3.3d) is only needed to estimate the positive part of Ᏺ,so(3.5) also holds with an additional term of −  Q(R) Ᏺ − χ(v − τ) + ζ 2 dX (3.29) on the right-hand side. 4. The Sobolev inequality We now present our modified Sobolev inequality, which is an easy consequence of [17, Theorem 2.1]; however, for notational reasons (in particular the use of n and m), we quote a consequence of this theorem (see [5, Corollary 11.9]). Lemma 4.1. Let n ≥ 2,andletg ∈ L ∞ (Q(R)) be nonnegative. Set H i = D j (g ij ) and κ = (N +2)/N. Then  Q(R) |h| 2κ g 2/N dX ≤ C(N)  sup s∈(−R 2 ,0)  B(R)   h(x,s)   2 g(x,s)dx  2/N ×   Q(R)  g ij D i hD j h + h 2 |H| 2  dX  n/N   Q(R) h 2 dX  (N−n)/N (4.1) Gary M. Lieberman 9 for any h ∈ C(Q(R)) that vanishes on {|x|=R} and which is uniformly Lipschitz with re- spect to x. Proof. Let us set m = n +1andU = B(R). We define ν n+1 =−1/v and extend the defini- tion g ij = δ ij − ν i ν j for i and j in {1, ,m}.Withdμ = dx, it is easy to check that all the hypotheses of [5, Corollary 11.9] are satisfied, and this corollary gives  U |h| 2κ g 2/N dx ≤ C(N)   U |h| 2 gdx  2/N ×   U  g ij D i hD j h + h 2 |H| 2  dx  n/N   U h 2 dx  (N−n)/N (4.2) for each t ∈ (−R 2 ,0). (In this equation, all functions are evaluated at (x, t).) The proof is completed as in [5, Theorem 6.9]: note that  U h 2 gdx≤ sup s∈(−R 2 ,0)  U h(x,s) 2 g(x,s)dx, (4.3) integrate the resulting inequality with respect to t, and then apply H ¨ older’s inequality if n = 2.  Note that the vector H is not quite the usual mean curvature vector. For later reference, we observe that v 2 |H| 2 ≤ C(n)  g ij D ik ug km D jm u + g ij D i vD j vv  . (4.4) 5. Estimate of the maximum in terms of an integral From our energy inequality and the Sobolev inequality, we can now reduce our pointwise estimate of |Du| to an integral estimate of a suitable quantity. For this reduction, we introduce three positive C 1 [τ 0 ,∞) functions w, λ,andΛ. In addition to their smoothness, the functions w, λ,andΛ obey the following monotonicity properties: w is increasing, (5.1a) ξ −β w(ξ) is a decreasing function of ξ, (5.1b) w(ξ) −β   Λ(ξ)/λ(ξ)  N/2 ξ 2  is an increasing function of ξ, (5.1c) ξ −β  Λ(ξ) λ(ξ)  N/2 is a decreasing function of ξ (5.1d) for some nonnegative constant β. We also assume that Λ 0 ≤ vΛ, (5.2a) Λ 1 ≤ vΛ, Λ 2 ≤ vΛ, (5.2b) λ ≤ Λ, (5.2c) 1 ≤ Λ, (5.2d) 10 Boundary Value Problems and that |A|≤β 2 Λ. (5.3) Finally, we assume that λ  1+  vλ  λ  2  g ij ξ i ξ j ≤ va ij ξ i ξ j , (5.4) where (as before) we suppress the argument v from λ, Λ, and their derivatives. These hypotheses imply a pointwise estimate for the gradient in terms of an integral. Lemma 5.1. Suppose that conditions (3.3), (5.1), (5.2), (5.3), and (5.4)hold.Thenthereis a constant c 1 (n,β,β 1 R,β 2 ) such that sup Q τ (R/2)  1 − τ v  N+2 w ≤ c 1 R −n−2  Q τ (R) w  Λ λ  N/2 Λ v dX. (5.5) Proof. The proof is essentially the same as that of [5, Lemma 11.11], so we only give a sketch. First, for q ≥ 1+β a parameter at our disposal, we set χ =  Λ λ  N/2 w q   1 − τ v  +  (N+2)(q−1) v −2 . (5.6) Then conditions (5.1a), (5.1c)implythatχ is increasing while conditions (5.1b), (5.1d) imply that Ψ ≤ C(β)q 2 χ.Nowletζ be as in Lemma 3.1, and note that we can take ζ so that |Dζ|≤C/R, |D 2 ζ| + |ζ t |≤C/R 2 ,and0≤ ζ ≤ 1inQ(R). It then follows from Lemma 3.1 with ζ (N+2)q−N in place of ζ 2 that sup t∈(−R 2 ,0)  q τ (R,t) Ξ(v)ζ 2 dx +  Q τ (R)  1 − τ v  Ꮿ 2 + Ᏹ  χζ 2 dX ≤ C  β,β 1 R,β 2  q 2 R 2  Q τ (R) χΛζ (N+2)(q−1) vdX (5.7) by taking (5.2a), (5.2b), (5.2c), and (5.3) into account and observing that Ξ(v) ≤ 1 2 χ(v)(v − τ) 2 (5.8) (because χ is increasing). Now we define h by the equation h 2 = χλ  1 − τ v  2 vζ (N+2)q−N , (5.9) so g ij D i hD j h ≤ Cq 2 ζ (N+2)(q−1) χλ v  ζ 2  1+  vλ  λ  2  g ij D i vD j v + v 2 R 2  . (5.10) [...]... α = 0, we have the uniformly parabolic equations described in [16, Example 4] but without any assumptions on the maximum eigenvalue of the matrix [ai j ] In particular, we reproduce the usual gradient estimate for parabolic p-Laplacian equations once we observe that the condition Ψ(1) = 1 can be replaced by Ψ(τ ∗ ) = 1 for some τ ∗ ≥ 1 If we further assume that ε1 (v) = γ3 /v for some positive constant... 1/8 for v sufficiently large, while ai j ξi ξ j ≤ v−3 , so the structure function Λ2 needs to be at least (some multiple of) v3 and this choice of Λ2 clearly does not satisfy (6.3c) This example is important because it is the motivating case for the structure described in [11] Moreover, the hypotheses for gradient estimates in [11] and [5] are clearly satisfied for this choice of A and B 8 Gradient estimates. .. estimates for solutions of degenerate parabolic equations in divergence form,” Journal of Differential Equations, vol 113, no 2, pp 543–571, 1994 [8] K Ecker, Estimates for evolutionary surfaces of prescribed mean curvature,” Mathematische Zeitschrift, vol 180, no 2, pp 179–192, 1982 [9] G M Lieberman, “A new regularity estimate for solutions of singular parabolic equations, ” Discrete and Continuous... 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Order Equations of Elliptic and Parabolic Type, vol 171 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1998 [5] G M Lieberman, Second Order Parabolic Differential Equations, World Scientific, River Edge, NJ, USA, 1996 [6] E DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer, New York, NY, USA, 1993 [7] G M Lieberman, “Maximum estimates for. .. for their support References [1] A Friedman, Partial Differential Equations of Parabolic Type, Krieger, Malabar, Fla, USA, 1983 [2] N V Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, vol 7 of Mathematics and Its Applications, D Reidel, Dordrecht, The Netherlands, 1987 [3] O A Ladyzhenskaya, V S Solonnikov, and N N Ural’tseva, Linear and Quasilinear Differential Equations of Parabolic. .. ∞, we have a gradient estimate under these hypotheses In particular, the equation ut = div exp 1 2 v Du + B(X,u,Du) 2 (7.7) 22 Boundary Value Problems is included under these hypotheses if |B | = o(v2 exp((1/2)v2 )) as | p| → ∞: we take Ψ(v) = v exp((1/2)[v2 − 1]), and note that (7.2) is satisfied with ψ0 = 2 and α = 2 It would be of interest to know if a gradient estimate can be obtained for |B | = O(v2... [8], Ecker showed that the gradient of a solution to a prescribed mean curvature equation can be estimated, locally in space, just in terms of its initial data Here, we show how that result follows from a simple modification of our estimates In fact, we obtain a corresponding estimate for a larger class of equations To this end, we need to adjust our notation slightly First, for any R > 0 and T > 0, we . Corporation Boundary Value Problems Volume 2007, Article ID 35825, 28 pages doi:10.1155/2007/35825 Research Article Interior Gradient Estimates for Nonuniformly Parabolic Equations II Gary M. Lieberman Received. Vespri We prove interior gradient estimates for a large class of parabolic equations in divergence form. Using some simple ideas, we prove these estimates for several types of equations that are. coefficient matrix and we obtain interior gradient estimates for so-called false mean curvature equation. Copyright © 2007 Gary M. Lieberman. 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