Everywhere regularity of solutions to a class of strongly coupled degenerate parabolic systems

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Supplement Volume 2005 Website: http://AIMsciences.org pp 576–586 PARTIAL REGULARITY OF SOLUTIONS TO A CLASS OF STRONGLY COUPLED DEGENERATE PARABOLIC SYSTEMS Dung Le Department of Applied Mathematics University of Texas at San Antonio 6900 North Loop 1604 West San Antonio, TX 78249, USA Abstract Using the method of heat approximation, we will establish partial regularity results for bounded weak solutions to certain strongly coupled degenerate parabolic systems Introduction The aim of this paper is to study the partial regularity for weak solutions of nonlinear parabolic systems of the form ut = div(a(x, t, u)Du) + f (x, t, u, Du), (1.1) n+1 n in a domain Q = Ω × (0, T ) ⊂ R , with Ω being an open subset of R , n ≥ The vector valued functions u, f take values in Rm , m ≥ Du denotes the spatial nm derivative of u Here, a(x, t, u) = (Aαβ , Rnm ) ij ) is a matrix in Hom(R 1,0 A weak solution u to (1.1) is a function u ∈ W2 (Q, Rm ) such that f (x, t, u, Du)φ dz [−uφt + a(x, t, u)DuDφ] dz = Q Q for all φ ∈ Cc1 (Q, Rm ) Here, we write dz = dxdt It has been known that, in the case of systems of equations (i.e m > 1), one cannot expect that bounded weak solutions of (1.1) will be Hăolder continuous everywhere (see [7]) Partial regularity for (1.1), when a is regularly elliptic, was considered by Giaquinta and Struwe in [5] In this paper, we study the partial regularity for (1.1) when certain degeneracy is present In particular, we consider the case when a ceases to be regular elliptic at certain values of u Strongly coupled systems of porous medium type are included here For the sake of simplicity, we will only consider the homogeneous case f ≡ 0, and assume that a(x, t, u) depends only on u The nonhomogeneous case can be treated similarly modulo minor modifications In fact, we will assume the following structural conditions on (1.1) (A.1): There exists a C map g : Rm → Rm , with Φ(u) = Du g(u), such that for some positive constants λ, Λ > there hold a(u)Du · Du ≥ λ|Dg(u)|2 , |a(u)Du| ≤ Λ|Φ(u)||Dg(u)| 2000 Mathematics Subject Classification Primary: 35K65; Secondary: 35B65 Key words and phrases Parabolic systems, Degenerate systems, Partial Hă older regularity The author is partially supported by NSF Grant #DMS0305219, Applied Mathematics Program 576 PARTIAL REGULARITY OF SOLUTIONS 577 (A.2): (Degeneracy condition) Φ(0) = There exist positive constants C1 , C2 such that C1 (|Φ(u)| + |Φ(v)|)|u − v| ≤ |g(u) − g(v)| ≤ C2 (|Φ(u)| + |Φ(v)|)|u − v| (A.3): (Comparability condition) For any β ∈ (0, 1), there exist constants C1 (β), C2 (β) such that if u, v ∈ Rm and β|u| ≤ |v| ≤ |u|, then C1 (β)|Φ(u)| ≤ |Φ(v)| ≤ C2 (β)|Φ(u)| (A.4): (Continuity condition) Φ(u) is invertible for u = The map a(u)Φ(u)−1 is continuous on Rm \{0} Moreover, there exists a monotone nondecreasing concave function ω : [0, ∞) → [0, ∞) such that ω(0) = 0, ω is continuous at 0, and |a(v)Φ(v)−1 − a(u)Φ(u)−1 | ≤ (|Φ(u)| + |Φ(v)|)ω(|u − v|2 ), (1.2) |Φ(u) − Φ(v)| ≤ (|Φ(u)| + |Φ(v)|)ω(|u − v|2 ) (1.3) m for all u, v ∈ R In (A.4), we use ω to quantify our continuity hypothesis on a(u)Φ−1 (u) We would like to remark that the existence of the function ω also comes from the continuity of a(u)Φ−1 (u) and Φ(u) ( see [4, page 169]) To avoid certain technicalities in the presentation of our proof, we assume here (1.3) We will see at the end of this paper that it is not necessary Systems that satisfy the above structural conditions include the porous medium type systems: a behaves like certain powers of the norm |u| In this case, one may consider g(u) = |u|α/2 u for some α > As we mentioned before, partial regularity results for the regular case, when g is the identity map, were established in [5] There are three essential ingredients in their proof The first element is an inequality of Caccioppoli, or reverse-Poincar´e, type The second element of the proof is Campanato’s decay estimate for the averaged mean square deviation of solutions to systems of equations with constant coefficients Finally, the technique of freezing the coefficients was used One then had to control the deviation of the original solution of (1.1) from that of system of constant coefficients In this step, a variant version of the famous Gehring reverse Hăolder inequality played a crucial role Given the above assumptions, Cacciopoli’s inequality is not available More seriously, the crucial Gehring reverse inequality, and thus higher integrability of Du, does not seem to hold anymore However, we are able to prove the following partial regularity result Theorem Let u be a bounded weak solution to (1.1) Set Reg(u) = {(x, t) ∈ Ω ì (0, T ) : u is Hă older continuous in a neighborhood of (x, t)} and Sing(u) = Ω × (0, T )\Reg(u) Then Sing(u) ⊆ Σ1 Σ2 , where Σ1 = {(x, t) ∈ Ω × (0, T ) : lim inf |(u)QR (x,t) | = 0}, R→0 Σ2 = {(x, t) ∈ Ω × (0, T ) : lim inf R→0 QR |u − (u)QR (x,t) |2 dz > 0} 578 DUNG LE Here, for each R > 0, QR (x, t) = BR (x) × (t − R2 , t) and (u)QR (x,t) = u dz QR (x,t) Moreover, H n (Σ2 ) = 0, where H n is the n-dimensional Hausdorff measure We would like to describe briefly our approach here In order to deal with this degenerate situation, one needs to find another way to avoid the unavailable Gehring lemma Recently, the method of A-harmonic approximation has been successfully used by Duzaar et al [2, 3] to treat regular elliptic systems and p-Laplacian systems One of the advantages of this method is that it is more elementary and avoids the technical difficulties associated with the use of the Gehring lemma, the missing stone in our case Inspired by this method, we introduce in Section its parabolic variance: the heat approximation Basically, the point is to show that a function which is approximately a solution to a heat equation in a parabolic cylinder will be L2 -close to some heat solution in a smaller cylinder We also present a parabolic version of Giaquinta’s lemma [2, Lemma A.1] Finally, we prove Theorem in Section We apply a scaling argument to reflect the degeneracy of (1.1) and make use of the above heat approximation lemma in each scaled cylinders We start with the key assumption that the solution u is not averagely (in a scaled cylinder) too close to the singular point u = We then derive a decay estimate for the averaged mean square deviation of g(u) This estimate allows us to show that u is still averagely far away from the singular point in a smaller cylinder, and the argument then can be repeated Moreover, we then obtain the decay estimates for the averaged mean square deviation of u in a sequence of scaled and nested cylinders It is then standard to conclude that u is locally Hăolder continuous from these estimates A-heat approximation In this section, we present the parabolic version of the A-harmonic approximation lemma formulated in [2] The proof is a straightforward modification of that for the elliptic case modulo some careful choice of the pertinent norms We present some details here for the sake of completeness Fix (x0 , t0 ) ∈ Rn+1 For each R > 0, we consider the cylinder QR = BR (x0 ) × (t0 − R2 , t0 ) Let V (QR ) be the space of functions g ∈ W21,0 (QR , Rm ) with norm g V (QR ) g (x, t) dx + = sup R−n−2 t BR QR |Dg|2 dz (2.4) For A ∈ Bil(Hom(Rnm , Rnm )) and φ ∈ C (QR ) = C (QR , Rm ), we define LA (g, φ, QR ) = QR [A(Dg, Dφ) − gφt ] dz, and ∆(LA , g, QR ) = sup{|LA (g, φ, QR )| : φ ∈ Cc1 (QR ), sup |Dφ| ≤ QR 1 , |φt | ≤ } R R We shall consider the set of A-heat functions H(A, QR ) = {H ∈ V (QR ) : LA (H, φ, QR ) = 0, ∀φ ∈ Cc1 (QR )} The following A-heat approximation lemma is the parabolic version of [2, Lemma 2.1] PARTIAL REGULARITY OF SOLUTIONS 579 Lemma 2.1 Consider fixed λ, Λ > For any given ε > there exists δ ∈ (0, 1] that depends on λ, Λ, ε and has the following property: for any A ∈ Bil(Hom(Rnm , Rnm )) satisfying A(u, u) ≥ λ|u|2 , |A(u, v)| ≤ Λ|u||v| for all u, v ∈ Rnm , (2.5) ≤ 1, and |∆(LA , g, Qρ )| ≤ δ, (2.6) for any g ∈ V (Qρ ) satisfying g V (Qρ ) then there exists v ∈ H(A, Qρ ) such that ρ−2 Qρ |v − g|2 dz ≤ ε and Qρ |Dv|2 dz ≤ (2.7) Proof We assume first that ρ = If the conclusion is false, we can find ε > 0, {Ak } each satisfying (2.5) and {gk } ⊂ V (Q1 ) such that Q1 |vk − gk |2 dz ≥ ε for all vk ∈ H(Ak , Q1 ) with Q1 |Dvk |2 dz ≤ 1, (2.8) and gk |LAk (gk , φ, Q1 )| ≤ V (Q1 ) ≤ 1, (2.9) sup(|Dφ| + |φt |), any φ ∈ Cc1 (Q1 ) k Q1 (2.10) By (2.9), we can extract a weakly convergent sequence still denoted by {gk }, g ∈ V (Q1 ) and A such that: gk → g weakly in V (Q1 ), gk → g in L2 (Q1 ), Ak → A, and g V (Q2 ) ≤ For φ ∈ Cc1 (Q1 ), we have LA (g, φ, Q1 ) = LA (g − gk , φ, Q1 ) + Q1 (A − Ak )(Dgk , Dφ) dz + LAk (gk , φ, Q1 ) Letting k → ∞ and using (2.10), we see that g ∈ H(A, Q1 ) We then consider the solution vk in Q1 of the problem LAk (vk , φ, Q1 ) = for all φ ∈ Cc1 (Q1 ), with vk = g on the parabolic boundary of Q1 Set φk = vk − g We have λ Q1 On the other hand, |Dvk − Dg|2 dz ≤ Ak (Dφk , Dφk ) dz Q1 580 DUNG LE Ak (Dφk , Dφk ) dz = Q1 Q1 B1 = − B1 vk (φk )t dz − Q1 Q1 vk (φk )t dz − Q1 vk φk dx + + Q1 (A − Ak )(Dg, Dφk ) dz − Q1 (A − Ak )(Dg, Dφk ) dz − = A(Dg, Dφk ) dz (A − Ak )(Dg, Dφk ) dz Q1 = Ak (Dg, Dφk ) dz Q1 Ak (Dg, Dφk ) dz Q1 vk φk dx + = − Ak (Dvk , Dφk ) dz − ≤ |(A − Ak )| Q1 B1 φ2k dx + B1 φk (φk )t dz Q1 φ2k dx |Dg||Dφk | dz We conclude that Dvk −Dg L2 (Q1 ) → Since vk = g on the parabolic boundary of Q1 , a simple use of the Poincar´e inequality shows that vk − g L2 (Q1 ) → Let Vk = vk /mk , where mk = max{ Dvk L2 (Q1 ) , 1} Since lim Dvk L2 (Q1 ) = Dg L2 (Q1 ) ≤ 1, we have mk → Thus, Vk − g L2 (Q1 ) → and Q1 |DVk |2 dz ≤ This and the fact that gk − g L(Q1 ) → contradict (2.8) The proof for the case ρ = is complete Finally, for ρ = 1, we will use the following scalings: x = x0 + RX, t = t0 + R2 T ⇒ |dx| = Rn |DX|, dt = R2 dT 1 g(x0 + RX, t0 + R2 T ) = g(x, t), φ(X, T ) = φ(x, t) R R Therefore DX g¯ = Dx g, g¯T = Rgt , DX φ = RDx φ, φT = R2 φt , g¯(X, T ) = g¯ V (Q1 ) = g V (QR ) , Q1 ¯ − g¯|2 dz = R−2 |h QR |h − g|2 dz, and [−¯ g φt + A(DX g¯, DX φ] dz = RLA (g, φ, QR ) LA (¯ g , φ, Q1 ) = Q1 From these relations, it is easy to see that the case ρ = follows from the above proof We then have the following parabolic version of Giaquinta’s lemma [2, Lemma A.1] Lemma 2.2 Suppose that A satisfies the conditions of Lemma 2.1 For any ǫ > 0, there exists a constant C depending on λ, Λ, ε such that inf    Qρ |H − g|2 dz ≤ Cρ ∆(LA , g, Qρ ) + ǫρ g : H ∈ H(A, Qρ ) and V (Qρ ) Qρ |DH|2 dz ≤ g V (Qρ )    PARTIAL REGULARITY OF SOLUTIONS 581 Proof Assume first that ρ = Let δ be the constant found in Lemma 2.1 We consider first the case when ∆(LA , g, Q1 ) ≤ δ g We then have |H − g|2 dz inf Q1 = g V (Q1 ) Q1 ≤ǫ g | : H ∈ H(A, Q1 ) and H g V (Q1 ) V (Q1 ) − g g V (Q1 ) |2 dz Q1 |DH|2 dz ≤ g V (Q1 ) : H ∈ H(A, Q1 ) V (Q1 ) Otherwise, by the Poincar´e inequality, we have inf Q1 |H − g|2 dz 1/2 : H ∈ H(A, Q1 ) ≤ Q1 |(g)Q1 − g|2 dz 1/2 ≤ supτ ≤C g Q1 V (Q1 ) |(g)Q1 − (g(x, τ ))B1 |2 dz ≤ 1/2 + C δ ∆(LA , g, Q1 ) Q1 |(g(x, t))B1 − g|2 dz Combining these estimates, we prove the lemma when ρ = The case ρ = follows from the scaling argument used in Lemma 2.1 Partial Regularity Theorem We now consider the system (1.1) For R > 0, let µ = supQ |u| We can assume that µ > Let V0 be a constant vector in Rm with 34 µ ≤ |V0 | ≤ µ We will make a change of variables x ¯ = x − x0 , s = Φ2µ t − t0 , with Φµ = sup|u|≤µ |Φ(u)| (3.11) From now on we will work with the new variables (¯ x, s) in the cylinders QR = BR × JR , BR = {¯ x : |¯ x| ≤ R}, JR = (−R2 , 0) The system (1.1) becomes us = divx (a(u)Du) Φ2µ (3.12) Let φ be a cut-off function in Q2R That is, φ ∈ Cc1 (Q2R ) with φ ≡ in QR and |Dφ| ≤ 1/R, |φs | ≤ 1/R2 Testing (3.12) with (u − V0 )φ and using the fact that |a(u)Du(u − V0 )Dφ| ≤ ε|Dg(u)|2 + C(ε)|Dφ|2 Φ2µ |u − V0 |2 , which holds due to (A.1) and (A.3), we get the following degenerate Cacciopoli’s type estimate Φ2µ sup s∈JR BR |u − V0 |2 dx + λ QR |Dg(u)|2 d¯ z≤ CΦ2µ R2 Q2R |u − V0 |2 d¯ z, (3.13) where d¯ z = d¯ xds By (A.2), the above yields R2 g(u) − g(V0 ) V (QR ) ≤ CΦ2µ Q2R |u − V0 |2 d¯ z (3.14) 582 DUNG LE In order to apply Lemma 2.2, we define Φ(V0 )a(V0 )Φ−1 (V0 ) (3.15) Φ2µ Thanks to (A.3), there is a positive constant λ such that λΦµ ≤ |Φ(V0 )| ≤ Φµ This and (A.1) show that A satisfies the assumptions of Lemma 2.2 A= We also set IR = QR that |u − V0 |2 d¯ z for each R > Our first step is to show Lemma 3.3 For φ ∈ Cc1 (QR ), with |Dφ| ≤ 1/R and |φt | ≤ 1/R2 , we have R2 |LA (g(u), φ, QR )| ≤ CΦµ ω(I2R ) (I2R ) (3.16) Proof We note that [−(g(u) − g(V0 ))φs + A(Dg(u), Dφ)] d¯ z LA (g(u), φ, QR ) = QR Multiplying (3.12) by Φ(V0 ) and testing by φ, we get QR [−(Φ(V0 )(u − V0 )φs + Φ(V0 )a(x, u)DuDφ] d¯ z = Φ2µ Hence, ADg(u) − LA (g(u), φ, QR ) = QR − QR Φ(V0 )a(x, u)Du, Dφ d¯ z Φ2µ [g(u) − g(V0 ) − Φ(V0 )(u − V0 )]φs d¯ z Using (A.4), for u = 0, we estimate the first integrand on the right by Φ2µ Φ(V0 ){a(V0 )Φ(V0 )−1 − a(x, u)Φ(u)−1 }Dg(u), Dφ ≤ Φ1µ |Dφ||Dg(u)||a(V0 )Φ(V0 )−1 − a(u)Φ(u)−1 | ≤ 2|Dφ||Dg(u)|ω(|u − V0 |2 ) Similarly, the second integrand can be estimated by (Φ(tu + (1 − t)V0 ) − Φ(V0 ))(u − V0 )dt sup |φs | ≤ C Φµ ω(|u − V0 |2 )|u − V0 | R2 (3.17) Thus, |LA (g(u), φ, QR )| ≤ R + ≤ QR CΦµ R2 R CΦ + R2µ |Dg(u)ω(|u − V0 |2 )| d¯ z QR QR ω(|u − V0 |2 )|u − V0 | d¯ z z |Dg(u)|2 d¯ QR QR ω (|u − V0 | ) d¯ z This, (3.13) and the concavity of ω give the lemma z ω (|u − V0 |2 )| d¯ 2 QR |u − V0 | d¯ z PARTIAL REGULARITY OF SOLUTIONS Next, we have a decay estimate for g(u) 583 Hereafter, we will denote fR = f d¯ z QR Lemma 3.4 For ǫ > and σ ∈ (0, 1/4), we have QσR |g(u) − g(u)σR |2 d¯ z ≤ C[σ + σ −n−2 (ω (IR ) + ǫ2 )]Φ2µ IR (3.18) Proof From (A.1), we see that A satisfies the assumption of Lemma 2.2 Therefore, we can find H ∈ H(A, QR ) such that Q2R Q2R |H − g(u)|2 d¯ z ≤ ≤ |DH|2 d¯ z ≤ g(u) − g(V0 ) V (Q2R ) and CR2 |LA (g(u), φ, Q2R )|2 +Cε2 R2 g(u) − g(V0 ) 2V (Q2R ) CΦ2µ [ω (I4R )I4R + ǫ2 I4R ], using (3.14), (3.16) For σ ∈ (0, 1), using a decay result in [1] for the function H, we have QσR |g(u) − g(u)σR |2 d¯ z ≤ QσR |g(u) − HσR |2 d¯ z ≤ QσR |g(u) − H|2 d¯ z+ ≤ QσR |g(u) − H|2 d¯ z + σ2 QσR |H − HσR |2 d¯ z QR |H − HR |2 d¯ z By the general Poincar´e inequality [8, Lemma 3], we also have QR |H − HR |2 d¯ z ≤ CR2 Q2R |DH|2 d¯ z ≤ CR2 g(u) − g(V0 ) V2R ≤ CΦ2µ I4R Combining these estimates and the fact that QσR |g(u) − H|2 d¯ z ≤ Cσ −n−2 Q2R |g(u) − H|2 d¯ z, we obtain the lemma Finally, we will show that u is still averagely far away from the singular point in smaller cylinders so that the above argument can be repeated More importantly, this will allow us to derive the decay estimate for u from that of g(u) Lemma 3.5 Assume that µ ≤ |V0 | ≤ µ, IR ≤ ε2 µ2 (3.19) For any given θ ∈ (0, 1), there exist ε, σ > sufficiently small such that there is a sequence of vectors {Vi } satisfying i): Vi = (u)Qσi R and |Vi | ≥ 21 µ for all i > ii): Qσi+1 R |u − Vi+1 |2 d¯ z ≤ θ2 Qσ i R |u − Vi |2 d¯ z, (3.20) 584 DUNG LE |Vi+1 − Vi |2 ≤ (σ −n−2 + θ)2 Qσi R |u − Vi |2 d¯ z (3.21) Proof Given any θ ∈ (0, 1) By (3.18), (3.19) and the continuity of ω, we can find σ, ǫ, ε, in that order, sufficiently small such that QσR |g(u) − g(u)σR |2 d¯ z ≤ [θεΦµ µ]2 (3.22) We will fix such σ and make ε even smaller later on Let V¯ be a vector such that ¯ g(V ) = g(u)σR Thanks to (A.2) and by choosing ε sufficiently small, we have |g(V¯ ) − g(V0 )|2 ≤ QσR (|g(u) − g(V¯ )|2 + |g(u) − g(V0 )|2 ) d¯ z ≤ (C + θ2 )[εΦµ µ]2 We also have |g(V¯ ) − g(V0 )| ≥ CΦµ |V¯ − V0 | Therefore, |V¯ − V0 | ≤ (C + θ)εµ, so that |V¯ | ≥ |V0 | − (C + θ)εµ ≥ 12 µ, if ε is sufficiently small By (A.2) again, we have |g(u) − g(V¯ )| ≥ CΦµ |u − V¯ | for some universal constant C If ε ≤ C, we derive from (3.22) that Φ2µ QσR |u − uσR |2 d¯ z ≤ Φ2µ QσR |u − V¯ |2 d¯ z ≤ θ2 Φ2µ IR , which implies |u − uσR |2 d¯ z ≤ θ2 QσR QR |u − V0 |2 d¯ z (3.23) Hence, |V0 − uσR |2 ≤ QσR (|u − uσR |2 + |u − V0 |2 ) d¯ z ≤ (σ −n−2 + θ2 ) QR |u − V0 |2 d¯ z (3.24) We obtain ii) for i = If we can establish that |Vi | ≥ 21 µ, then it is clear that the above argument could be repeated with V0 being replaced by Vi = uσi R In particular, we take A = Φ12 a(Vi ) in (3.15) and replace V0 with Vi in the proof µ of Lemma 3.3 Using (A.3) we easily see that (3.18) continues to hold Thus, let assume that i) and ii) are true up to i − The above argument applies to give (3.20) We then have 1/2 Qσi R |u − Vi | d¯ z ≤ θi εµ, |Vi − Vi−1 | ≤ σ −(n+2) + θ2 θi−1 εµ Hence, ∞ |Vi | ≥ |V0 | − σ −(n+2) + i=0 θ2 θi εµ ≥ µ− √ σ −(n+2) + θ2 µ εµ ≥ , 1−θ if ε is sufficiently small Our proof is complete by induction Combining Lemma 3.5, Lemma 3.4 and (A.2), we obtain the following decay estimate for u in a sequence of nested cylinders PARTIAL REGULARITY OF SOLUTIONS 585 Proposition Suppose that there exist R > and V0 such that and QR 4µ ≤ |V0 | ≤ µ |u − V0 |2 d¯ z ≤ ε2 µ2 Let ǫ > be given If ε is sufficiently small, then there exist σ0 ∈ (0, 1) and a constant C such that for ρ = σ i R and Vi = uQρ , with < σ < σ0 and i ≥ 1, there holds Qσρ |u − Vi |2 d¯ z ≤ CK(σ, ǫ) where K(σ, ǫ) = σ + σ −n−2 [ω Qρ |u − Vi−1 |2 d¯ z, Qρ (3.25) |u − Vi−1 |2 d¯ z + ǫ2 ] This decay estimate allows us to give the proof of our main partial regularity result Σ2 For some β > we Proof of Theorem We consider a point (x0 , t0 ) ∈ Σ1 have βµ ≤ |(u)QR (x0 ,t0 ) | ≤ µ for all R > 0, and lim inf R→0 QR |u − uR |2 dz = Let V0 = (u)QR (x0 ,t0 ) It is clear that we can replace the condition 34 µ ≤ |V0 | ≤ µ in Proposition by βµ ≤ |V0 | ≤ µ, where β can be any real in (0, 1) Hence, in terms of the new variables (¯ x, s) defined in (3.11), we will verify that the conditions of Proposition are fulfilled and (3.25) holds Denote QR,à = BR (x0 ) ì [t0 − Φ12 R2 , t0 ] We see that the conditions of Proposiµ tion are satisfied if we can show that QR |u − V0 |2 d¯ z is small To see this, let ¯ = kR By the choice of k, we see that QR,µ ⊆ QR Hence, k = min{1, Φµ } and R ¯ QR ¯ |u − V0 |2 d¯ z= Therefore, lim inf R→0 QR QR,µ ¯ |u − V0 |2 dz ≤ Φ2µ k n+2 QR |u − V0 |2 dz |u − uR |2 d¯ z = and the conditions of Proposition are verified It is then standard to follow the argument of [5] and iterate (3.25) to assert that u is Hăolder continuous in a neighborhood of (x0 , t0 ) The Hăolder exponent may depend on (x0 , t0 ) and β, µ Going back to the original variables (x, t), we see that u is Hăolder continuous in a neighborhood of (x0 , t0 ) To see that the singular set Σ2 is small We simply use the following general Poincar´e inequality by Struwe [8, Lemma 3] QR |u − uR |2 dz ≤ CR2 Q2R |Du|2 dz We remark that this inequality was proven in [8] using only the fact that u satisfies (1.1) with |a(u)Du| ≤ C|Du| for some constant C This is true in our situation because of (A.1) and the fact that u is bounded Thus, Σ2 is a subset of Σ∗2 = {(x, t) ∈ Ω × (0, T ) : lim inf R→0 Rn QR |Du|2 dz > 0}, 586 DUNG LE which satisfies H n (Σ∗2 ) = by a result of [6, page 70] Our proof is then complete Finally, as we remarked in the introduction, the continuity condition (1.3) is not necessary A careful reading reveals that this condition is only used in the estimate (3.17) of the proof of Lemma 3.3 Without (1.3), the right hand side of (3.17) could ˆ µ = inf{|Φ(V )| : µ ≤ |V | ≤ µ} ˆ µ |Φ02 | ω(|u−V0 |2 )|u−V0 |, where Φ be replaced by C Φ R ˆ µ ω, and continue as before We then need only to replace the function ω with Φ REFERENCES [1] S Campanato, Equazioni paraboliche del second ordine e spazi L2,θ (Ω, δ), Ann mat Pura Appl., 73 (1966), 55–102 [2] F Duzaar and J Grotowski, Optimal interior partial regularity for nonlinear elliptic systems: The method of A-harmonic approximation, Manuscripta Math., 103 (2000), 267-298 [3] F Duzaar and G Mingione, The p-harmonic approximation and the regularity of p-harmonic maps, Calc Var to appear [4] M Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, NJ, 1983 [5] M Giaquinta and M Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems, Math Z., 179 (1982), 437–451 [6] E Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003 [7] O John and J Stara, Some (new) counterexamples of parabolic systems, Commentat math Univ Carol., 36 (1995), 503510 [8] M Struwe., On the Hă older continuity of bounded weak solutions of quasilinear parabolic systems, Manuscripta Math., 35 (1981), 125–145 Received September, 2004; in revised February, 2005 E-mail address: dle@math.utsa.edu ... p-harmonic approximation and the regularity of p-harmonic maps, Calc Var to appear [4] M Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton... is approximately a solution to a heat equation in a parabolic cylinder will be L2 -close to some heat solution in a smaller cylinder We also present a parabolic version of Giaquinta’s lemma [2,... Duzaar et al [2, 3] to treat regular elliptic systems and p-Laplacian systems One of the advantages of this method is that it is more elementary and avoids the technical difficulties associated