On the existence of W^{1,2}_{p} solutions for fully nonlinear parabolic equations under either relaxed or no convexity assumptions
aa r X i v : . [ m a t h . A P ] O c t ON THE EXISTENCE OF W , p SOLUTIONS FOR FULLYNONLINEAR PARABOLIC EQUATIONS UNDER EITHERRELAXED OR NO CONVEXITY ASSUMPTIONS
N.V. KRYLOV
Summary.
We establish the existence of solutions of fully nonlinearparabolic second-order equations like ∂ t u + H ( v, Dv, D v, t, x ) = 0 insmooth cylinders without requiring H to be convex or concave withrespect to the second-order derivatives. Apart from ellipticity noth-ing is required of H at points at which | D v | ≤ K , where K is anyfixed constant. For large | D v | some kind of relaxed convexity as-sumption with respect to D v mixed with a VMO condition with re-spect to t, x are still imposed. The solutions are sought in Sobolevclasses. We also establish the solvability without almost any conditionson H , apart from ellipticity, but of a “cut-off” version of the equation ∂ t u + H ( v, Dv, D v, t, x ) = 0. Introduction and main results
In this paper we consider parabolic equations ∂ t v ( t, x ) + H [ v ]( t, x ) = 0 , (1.1)where H [ v ]( t, x ) = H ( v ( t, x ) , Dv ( t, x ) , D v ( t, x ) , t, x ) , in subdomains of R d +1 = { ( t, x ) : t ∈ R , x ∈ R d } . Let Ω ∈ C , be an open bounded subset of R d . Fix T ∈ (0 , ∞ ) and setΠ = [0 , T ) × Ω(if the t -axis is directed vertically, [0 , T ) × Ω is indeed looking like a pie).Fix p > d and a measurable function ¯ G ≥ R d +1 . One of our main results implies that, for d = 3, equation (1.1) with( a ∧ b = min( a, b )) H ( D u, x ) := ¯ G ( t, x ) ∧ | D u | + ¯ G ( t, x ) ∧ | D u | + ¯ G ( t, x ) ∧ | D u | +∆ u − f ( t, x ) (1.2)in Π with zero boundary condition on its parabolic boundary has a uniquesolution u ∈ W , p (Π), provided that ¯ G, f ∈ L p (Π) with p > d + 2. Recall Mathematics Subject Classification.
Key words and phrases.
Fully nonlinear parabolic equations, cut-off equations. that W , p (Π) denotes the set of functions v defined in Π such that ∂ t v , v , Dv , and D v are in L p (Π). Observe that H in (1.2) is neither convexnor concave with respect to D u . So far, there are only two approachesto such equations: the theory of ( L p ) viscosity solutions and the theory ofstochastic differential games, provided H has a somewhat special form. Thepast experience shows that it is hard to expect getting sharp quantitativeresults using probability theory. On the other hand, the theory of viscositysolutions indeed produced some remarkable quantitative results (see, forinstance, [5], [8] and the references therein). However, to the best of theauthor’s knowledge the result stated above about (1.2) is either very hardto obtain by using the theory of ( L p ) viscosity solutions or is just beyond it,at least at the current stage. It seems that the best information, that theoryprovides at the moment, is the existence of the maximal and minimal ( L p )viscosity solution (see [8]), no uniqueness of ( L p ) viscosity solutions can beinferred for (1.2) and no regularity apart from the classical C α -regularity(see [4]).The current paper is a natural continuation of [12] where similar resultsare obtained for elliptic equations.Fix some constants K , K F ∈ [0 , ∞ ), δ ∈ (0 , S the set ofsymmetric d × d matrices and let S δ be the subset of S consisting of matrices a such that δ | λ | ≤ a ij λ i λ j ≤ δ − | λ | ∀ λ ∈ R d . Here are our assumptions about H . Assumption 1.1.
The function H ( u , t, x ), u = ( u ′ , u ′′ ) , u ′ = ( u ′ , u ′ , ..., u ′ d ) ∈ R d +1 , u ′′ ∈ S , ( t, x ) ∈ R d +1 , is measurable with respect to ( u ′ , t, x ).The following assumptions contain (small) parameters ˆ θ, θ ∈ (0 ,
1] whichare specified later in our results.
Assumption 1.2.
There are two measurable functions F ( u , t, x ) = F ( u ′ , u ′′ , t, x ) , G ( u , t, x )such that H = F + G. For u ′′ ∈ S , u ′ ∈ R d +1 , and ( t, x ) ∈ R d +1 we have | G ( u , t, x ) | ≤ ˆ θ | u ′′ | + K | u ′ | + ¯ G ( t, x ) , F (0 , t, x ) ≡ . Define B R ( x ) = { x ∈ R : | x − x | < R } , B R = B R (0) ,C r ( t , x ) = [ t , t + r ) × B r ( x ) , C r = C r (0) , and for Borel Γ ⊂ R d +1 denote by | Γ | the volume of Γ. ULLY NONLINEAR PARABOLIC EQUATIONS 3
Assumption 1.3. (i) The function F is Lipschitz continuous with respectto u ′′ with Lipschitz constant K F and is measurable with respect to ( t, x ).Moreover there exist R ∈ (0 ,
1] and τ ∈ [0 , ∞ ) such that, for any u ′ ∈ R , z = ( t , x ) ∈ Π and r ∈ (0 , R ], one can find a convex function ¯ F ( u ′′ ) =¯ F z ,r, u ′ ( u ′′ ) (depending only on u ′′ ) for which(ii ) We have ¯ F (0) = 0 and at all points of differentiability of ¯ F we have D u ′′ ¯ F ∈ S δ ;(iii ) For any u ′′ ∈ S with | u ′′ | = 1, we have Z ˆ C r ( z ) sup τ>τ τ − | F ( u ′ , τ u ′′ , z ) − ¯ F ( τ u ′′ ) | dz ≤ θ | ˆ C r ( z ) | , (1.3)where ˆ C r ( z ) = ( t , t + r ) × (Ω ∩ B r ( x ));(iv) There exists a continuous increasing function ω F ( τ ), τ ≥
0, such that ω F (0) = 0 and for any u ′ , v ′ ∈ R , ( t, x ) ∈ Π, and u ′′ ∈ S we have | F ( u ′ , u ′′ , t, x ) − F ( v ′ , u ′′ , t, x ) | ≤ ω F ( | u ′ − v ′ | ) | u ′′ | . Remark . Assumptions 1.2 and 1.3 (iv) imply that | H ( u ′ , , t, x ) | ≤ K | u ′ | + ¯ G ( t, x ) ∀ u ′ , ( t, x ) ∈ R d +1 . (1.4) Assumption 1.5.
We are given a function g ∈ W , p (Π).If z i = ( t i , x i ) ∈ R d +1 , i = 1 ,
2, we set ρ ( z , z ) = | t − t | / + | x − x | . Definition 1.6.
For a function u ∈ C ( ¯Π) set ω u (Π , ρ ) = sup {| u ( z ) − u ( z ) | : z , z ∈ Π , ρ ( z , z ) ≤ ρ } ,ω F,u, Π ( ρ ) = ω F ( ω u (Π , ρ )) . We will sometimes say that a certain constant depends only on A,B,...,and the function ω F,u, Π . This is to mean that it depends only on A,B,..., andon the maximal solution of an inequality like N ω F,u, Π ( ρ ) ≤ /
2, where therange of ρ and the value of N depending only on A,B,... could be alwaysfound out from our arguments.In the following theorem about a priori estimates there is no ellipticityassumption on H . If Q is a subdomain in R d +1 , by ∂ ′ Q we denote itsparabolic boundary. Theorem 1.7.
Let p > d + 1 . Then there exist constants θ, ˆ θ ∈ (0 , ,depending only on d , p , δ , K F , and M (Ω) ( ρ (Ω) and M (Ω) are introducedlater ) , such that, if Assumptions 1.2 and 1.3 are satisfied with these ˆ θ and θ , respectively, then for any u ∈ W , d +1 (Π) that satisfies (1.1) in Π ( a.e. ) andequals g on ∂ ′ Π we have k u k W , p (Π) ≤ N k ¯ G k L p (Π) + N k g k W , p (Π) + N τ + N sup Π | u | , (1.5) N.V. KRYLOV where the constants N depend only on K , K F , d , p , δ , R , ρ (Ω) , M (Ω) , diam(Ω) , T , and the functions ω F,u, Π and ω F,g, Π . In the literature, interior W p , p > d , a priori estimates for a class of fullynonlinear uniformly elliptic equations in R d in the framework of viscositysolutions were first obtained by Caffarelli in [2] (see also [3]). Adapting histechnique, similar interior a priori estimates were proved by Wang [15] forparabolic equations. In the same paper, a boundary estimate is stated butwithout proof; see Theorem 5.8 there. By exploiting a weak reverse H¨older’sinequality, the result of [2] was sharpened by Escauriaza in [7], who obtainedthe interior W p -estimate for the same equations allowing p > d − ε , with asmall constant ε > d .The above cited works are quite remarkable in one respect–they do notsuppose that H is convex or concave in D u . But they only show that toprove a priori estimates it suffices to prove the interior C –estimates for“harmonic” functions. However, up to now, these estimates are only knownunder convexity assumptions.Also obtaining boundary W p estimate by using the theory of viscositysolutions turned out to be extremely challenging and only in 2009, twentyyears after the work of Caffarelli, Winter [16] proved the solvability in W p (Ω)of equations with Dirichlet boundary condition in Ω ∈ C , .It is also worth noting that a solvability theorem in the space W , p, loc (Π) ∩ C ( ¯Π) is given in M. G. Crandall, M. Kocan, A. ´Swi¸ech [5] for the boundary-value problem for fully nonlinear parabolic equations. The above mentionedexistence results of [5] and [16] are proved under the assumption that H is convex in D v and in all papers mentioned above a small oscillation as-sumption in the integral sense is imposed on the operators. In the case oflinear equations this small oscillation assumption is equivalent to requiringthe main coefficients to be uniformly close to uniformly continuous ones.Our Assumption 1.3 is satisfied in this case if the main coefficients are justin VMO. The above cited works are performed in the framework of viscositysolutions.To the best of the author’s knowledge the only paper treating the solv-ability in the global Sobolev spaces for parabolic equations is [6], where theassumptions are much heavier than here.To have the solvability we need ellipticity and more regularity of H . Assumption 1.8.
For any ( t, x ) ∈ R d +1 , the function H ( u , t, x ) is continu-ous with respect to u , is Lipschitz continuous with respect to u ′′ , and at allpoints of differentiability of H with respect to u ′′ we have D u ′′ H ∈ S δ .In the following theorem we need higher values of p than in Theorem 1.7because in the proof we need to use the embedding W , p ⊂ C , . Theorem 1.9.
Let p > d + 2 and suppose that Assumptions 1.5 and 1.8 aresatisfied and ¯ G ∈ L p (Π) . Then there exist constants θ, ˆ θ ∈ (0 , , depending ULLY NONLINEAR PARABOLIC EQUATIONS 5 only on d , p , δ , K F , and M (Ω) , such that, if Assumptions 1.3 and 1.2are satisfied with these θ and ˆ θ , respectively, then there exists u ∈ W , p (Π) satisfying (1.1) in Π ( a.e. ) and such that u = g on ∂ ′ Π .Remark . Observe that generally there is no uniqueness in Theorem 1.9.For instance, in the one-dimensional case the (quasilinear) equation ∂ t u + D u − (1 − t ) p | Du | + 2 p (1 − | x | ) u = 0in Π = [0 , × ( − ,
1) with zero boundary data on ∂ ′ Π has two solutions:one is identically equal to zero and the other one is (1 − t ) (1 − | x | ).Uniqueness of solutions can be investigated by using the results in [9]. Remark . In case of linear equations Theorem 1.9 contains (apart fromthe restrictions on p ) the corresponding result of [1] proved for equationswith VMO main coefficients.In Theorem 5.9 of Wang [15] one can find an a priori estimate for any viscosity solution in case H is independent of u ′ and Π = C .By the way, it can be seen from our proofs that, if H is independent of[ u ′ ] := ( u ′ , ..., u ′ d ), we can take p > d + 1 in Theorem 1.9. Example . For τ > H ( u ) = (1 + τ cos p | ln | u ′′ || ) trace u ′′ , and choose τ so small that D u ′′ H ∈ S δ for a δ ∈ (0 , H isneither convex nor concave with respect to u ′′ and our assumptions are sat-isfied perhaps with a further reduced τ for ¯ F ( u ′′ ) = trace u ′′ . An interestingfeature of this example is that, for generic u , the limit of (1 /λ ) H ( λ u ) as λ → ∞ does not exist. Example . Let A and B be some countable sets and assume that for α ∈ A , β ∈ B , ( t, x ) ∈ R d +1 , and u ′ ∈ R d +1 we are given an S δ -valued func-tion a α ( u ′ , t, x ) (independent of β ) and a real-valued function b αβ ( u ′ , t, x ).Assume that these functions are measurable in t, x , a α and b αβ are contin-uous with respect to u ′ uniformly with respect to α, β, t, x , and (cid:12)(cid:12) b αβ ( u ′ , t, x ) (cid:12)(cid:12) ≤ K | u ′ | + ¯ G ( t, x ) , where ¯ G ∈ L p (Π), p > d + 2.Consider equation (1.1), where H ( u , t, x ) := inf sup β ∈ B α ∈ A h d X i,j =1 a αij (cid:0) u ′ , t, x (cid:1) u ′′ ij + b αβ ( u ′ , t, x ) i . Our measurability, boundedness, and countability assumptions guaranteethat H is measurable in t, x and Lipschitz continuous in u ′′ . One can alsoeasily check that at all points of differentiability D u ′′ H ∈ S δ . Next assume N.V. KRYLOV that there is an R ∈ (0 , ∞ ) such that for any z ∈ Π, r ∈ (0 , R ], and u ′ ∈ R one can find ¯ a α ∈ S δ (independent of t, x ) such that – Z ˆ C r ( z ) sup α ∈ A | a α ( u ′ , z ) − ¯ a α | dz ≤ θ, (cid:16) – Z Γ h dz := | Γ | − Z Γ h dz (cid:17) , where θ is taken from Theorem 1.9.Then we claim that the assertions of Theorem 1.9 hold true and estimate(1.5) holds with τ = 0.To prove the claim introduce F ( u ′ , u ′′ , t, x ) = sup α ∈ A d X i,j =1 a αij ( u ′ , t, x ) u ′′ ij , G = H − F. Notice that Assumption 1.3 is satisfied with τ = 0 and¯ F ( u ′′ ) := sup α ∈ A d X i,j =1 ¯ a αij u ′′ ij because these functions are convex, positive homogeneous of degree one withrespect to u ′′ and, for | u ′′ | = 1, – Z ˆ C r ( z ) (cid:12)(cid:12) F ( u ′ , u ′′ , z ) − ¯ F ( u ′′ ) (cid:12)(cid:12) dz ≤ – Z ˆ C r ( z ) sup α ∈ A (cid:12)(cid:12)(cid:12) d X i,j =1 (cid:2) a αij (cid:0) u ′ , z (cid:1) − ¯ a α (cid:3) u ′′ ij (cid:12)(cid:12)(cid:12) dz ≤ – Z ˆ C r ( z ) sup α ∈ A (cid:12)(cid:12) a α (cid:0) u ′ , z (cid:1) − ¯ a α (cid:12)(cid:12) dz ≤ θ. On can easily check that the remaining item (iv) in Assumptions 1.3 andAssumption 1.2 (with ˆ θ = 0) are satisfied as well and this proves our claim.Thus Theorem 1.9 is applicable.As a result we have a solvability theorem for (1.1), which covers (apartfrom the restriction on p ), as A and B are singletons, the first result aboutsolvability of linear parabolic equations with VMO coefficients obtained byBramanti and Cerutti in [1]. In this singleton case we also consider quasi-linear equations.In the following theorem Assumption 1.3 is not used. Theorem 1.14.
Let p > d + 2 and suppose that Assumptions 1.1, 1.8, and1.5 are satisfied, ¯ G ∈ L p (Π) , and (1.4) holds true. Let P ( u ′′ ) be a convexfunction on S such that at each point of its differentiability D u ′′ P ∈ S δ ′ ,where δ ′ ∈ (0 , δ ] . Also assume that for any a ∈ S δ and u ′′ ∈ S we have a ij u ′′ ij ≤ P ( u ′′ ) + K, where K is a constant. Then the equation ∂ t u + max (cid:0) H [ u ] , P [ u ] (cid:1) = 0( a.e. ) in Π with boundary condition u = g on ∂ ′ Π has a solution u ∈ W , p (Π) . ULLY NONLINEAR PARABOLIC EQUATIONS 7
Proof. Introduceˆ H ( u , t, x ) = max (cid:0) H ( u , t, x ) , P ( u ′′ ) (cid:1) , ˆ F ( u ′′ , t, x ) = P ( u ′′ ) − P (0) , ˆ G = ˆ H − ˆ F .
Obviously Assumptions 1.3 and 1.8, are satisfied for ˆ H , ˆ F , and ˆ F in placeof H , F , and ¯ F , respectively, with a K F , τ = θ = 0, and δ ′ in place of δ .Finally, for any u , t, x ,ˆ G ( u , t, x ) = max (cid:0) H ( u , t, x ) − P ( u ′′ ) + P (0) , P (0) (cid:1) ≥ P (0) , where for an a ∈ S δ H ( u , t, x ) − P ( u ′′ ) = H ( u , t, x ) − H ( u ′ , , t, x ) − P ( u ′′ ) + H ( u ′ , , t, x )= a ij u ′′ ij − P ( u ′′ ) + H ( u ′ , , t, x ) ≤ K + H ( u ′ , , t, x ) , which together with (1.4) shows that Assumption 1.2 is also satisfied withˆ θ = 0 and ¯ G + K + (cid:12)(cid:12) P (0) (cid:12)(cid:12) in place of ¯ G .Hence, Theorem 1.9 is applicable and our theorem is proved. (cid:3) Interior estimates of integral oscillations of D u Let F ( u ′′ ) be a convex function of u ′′ ∈ S (independent of ( t, x )) such that(i) F (0) = 0,(ii) at all points of differentiability of F we have D u ′′ F ∈ S δ , where δ ∈ (0 ,
1] is a fixed number.The following theorem is a particular case of the results in [14].
Theorem 2.1.
There exists and ¯ α = ¯ α ( d, δ ) ∈ (0 , such that for any α ∈ (0 , ¯ α ] and g ∈ C ( ∂ ′ C ) there exists a unique v ∈ C ( ¯ C ) ∩ C α loc ( C ) satisfying ∂ t v + F ( D v ) = 0 in C , v = g on ∂ ′ C . (2.1) Furthermore, | D v ( z ) − D v ( z ) | ≤ N ρ α ( z , z ) sup ∂ ′ C | g | as long as z , z ∈ C , where N depends only on δ, α , and d . Below in this section we fix α ∈ (0 , ¯ α ]. Recall that for a measurable setΓ ⊂ R d +1 we denote by | Γ | its Lebesgue measure, and if | Γ | 6 = 0 and u isintegrable over Γ we set u Γ = – Z Γ u dxdt = 1 | Γ | Z Γ u dxdt. Lemma 2.2.
Let r ∈ (0 , ∞ ) , ν ≥ and let φ ∈ C ( ∂ ′ C νr ) . Then there existsa unique v ∈ C ( ¯ C νr ) ∩ C α loc ( C νr ) such that ∂ t v + F ( D v ) = 0 in C νr , v = φ on ∂ ′ C νr . N.V. KRYLOV
Furthermore, – Z C r – Z C r | D v ( z ) − D v ( z ) | dz dz ≤ N ( d, α, δ ) ν − − α r − sup ∂ ′ C νr | φ | . Proof. Scalings show that it suffices to concentrate on r = 2 /ν . In thatcase the existence of solution follows from Theorem 2.1, which also impliesthat for z , z ∈ C /ν ⊂ C | D v ( z ) − D v ( z ) | ≤ N ν − α sup ∂ ′ C | φ | . It only remains to observe that – Z C /ν – Z C /ν | D v ( z ) − D v ( z ) | dz dz ≤ sup z ,z ∈ C /ν | D v ( z ) − D v ( z ) | . The lemma is proved. (cid:3)
Here is Theorem 1.9 of [10] combined with Theorem 2.3 of [10] (see also[6]).
Theorem 2.3.
Let u ∈ C ( ¯ C ) ∩ W , d +1 ,loc ( C ) . Then there are constants ¯ γ = ¯ γ ( d, δ, K ) ∈ (0 , and N , depending only on δ, d , and K , such that forany γ ∈ (0 , ¯ γ ] and any operator L = a ij D ij + b i D i , with measurable S δ -valuedcoefficients a ij and b i , such that | ( b i ) | ≤ K , given in C , we have Z C (cid:0) | D u | γ + | Du | γ (cid:1) dx dt ≤ N sup ∂ ′ C | u | γ + N (cid:18)Z C | ∂ t u + L u | d +1 dx dt (cid:19) γ/ ( d +1) . (2.2)Below we take γ ∈ (0 , ¯ γ ]. Lemma 2.4.
Let r ∈ (0 , ∞ ) and ν ∈ [2 , ∞ ) . Then for any u ∈ W , d +1 ( C νr ) we have (cid:0) – Z C r – Z C r | D u ( z ) − D u ( z ) | γ dz dz (cid:1) /γ ≤ N ν ( d +2) /γ (cid:0) – Z C νr | ∂ t u + F [ u ] | d +1 dz (cid:1) / ( d +1) + N ν − α (cid:0) – Z C νr | D u | d +1 dz (cid:1) / ( d +1) , (2.3) where N depends only on d, α , and δ . Proof. Define v to be a unique C ( ¯ C νr ) ∩ C α loc ( C νr )-solution of equation ∂ t v + F [ v ] = 0 in C νr with boundary condition v = u on ∂ ′ C νr . Such afunction exists by Lemma 2.2. Furthermore, v ( x ) − b i x i − c satisfies thesame equation for any constant b i , c . Hence by Lemma 2.2 and H¨older’sinequality I r := (cid:16) – Z C r – Z C r | D v ( z ) − D v ( z ) | γ dz dz (cid:17) /γ ULLY NONLINEAR PARABOLIC EQUATIONS 9 ≤ N ν − − α r − sup z =( t,x ) ∈ ∂ ′ C νr | u ( z ) − ( D i u ) C νr x i − u C νr | . By Poincar´e’s inequality (see, for instance, Corollary 5.3 in [6]) the lastsupremum is dominated by a constant times ν r (cid:16) – Z C νr | D u | d +1 dz (cid:17) / ( d +1) . It follows that I r ≤ N ν − α (cid:16) – Z C νr | D u | d +1 dz (cid:17) / ( d +1) . (2.4)Next, the function w := u − v is of class W , d +1 , loc ( C νr ) ∩ C ( ¯ C νr ) and foran operator L = a ij D ij we have ∂ t u + F [ u ] = ∂ t u + F [ u ] − ( ∂ t v + F [ v ]) = ∂ t w + L w. Moreover, w = 0 on ∂ ′ C νr . Therefore, by Theorem 2.3, there exists N = N ( d, δ ) < ∞ such that – Z C r | D w | γ dz ≤ ν d +2 – Z C νr | D w | γ dz ≤ N ν d +2 (cid:16) – Z C νr | ∂ t u + F [ u ] | d +1 dz (cid:17) γ/ ( d +1) . Upon combining this result with (2.4) we come to (2.3) and the lemma isproved. (cid:3) A priori estimates in W , p, loc Here we suppose that Assumptions 1.2 and 1.3 are satisfied. Thus, weassume that all assumptions on H and F stated before Theorem 1.7 aresatisfied. Take α ∈ (0 , ¯ α ] and γ ∈ (0 , ¯ γ ]. First we note the following. Lemma 3.1.
For any q ∈ [1 , ∞ ) and µ > there is a θ = θ ( d, δ, K F , µ, q ) > such that, if Assumption 1.3 is satisfied with this θ , then for any u ′ ∈ R , r ∈ (0 , R ] and z ∈ Π such that C r ( z ) ⊂ Π – Z C r ( z ) sup u ′′ ∈ S , | u ′′ | >τ (cid:12)(cid:12) F ( u ′ , u ′′ , z ) − ¯ F ( u ′′ ) (cid:12)(cid:12) q | u ′′ | q dz ≤ µ q , where ¯ F = ¯ F z ,r, u ′ . The proof of this lemma is practically identical to that of Lemma 5.1 of[12] given there for the elliptic case.
Lemma 3.2.
Let u ∈ W , d +1 , loc (Π) . Then there exist an S δ -valued function a ( t, x ) , R d -valued functions b ( t, x ) , and real-valued function f ( t, x ) , suchthat they are measurable, | b | ≤ K , | f | ≤ ¯ G + K | u | , and in Π ( a.e. ) a ij D ij u + b i D i u + f = H [ u ] . (3.1)This is a simple consequence of the fact that there is an S δ -valued function a such that H [ u ]( t, x ) − H ( u, Du, , t, x ) = a ij D ij u, and | H ( u, Du, , t, x ) | ≤ K ( | u | + | Du | ) + ¯ G. Lemma 3.3.
Let r ∈ (0 , ∞ ) and ν ≥ satisfy νr ≤ R . Take µ ∈ (0 , ∞ ) , β ∈ (1 , ∞ ) , and suppose that Assumption 1.3 is satisfied with θ = θ ( d, δ, K F , µ, βd + β )( see Lemma 3.1 ) . Take a function u ∈ W , d +1 (Π) and for z ∈ Π such that C νr ( z ) ⊂ Π ( if such z ’s exist ) denote I r ( z ) = (cid:0) – Z C r ( z ) – Z C r ( z ) | D u ( z ) − D u ( z ) | γ dz dz (cid:1) /γ . Then I r ( z ) ≤ N ν ( d +2) /γ (cid:18) – Z C νr ( z ) (cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) d +1 dz (cid:19) / ( d +1) + N τ ν ( d +2) /γ + N h(cid:0) µ + ω F,u, Π ( νr ) (cid:1) ν ( d +2) /γ + ν − α i(cid:18) – Z C νr ( z ) | D u | β ′ ( d +1) dz (cid:19) / ( β ′ ( d +1)) , (3.2) where β ′ = β/ ( β − and N depends only on d, K F , α , and δ . Proof. Set ρ := νr . Since ρ ≤ R , ¯ F = ¯ F z ,ρ,u ( z ) is well defined and byLemma 2.4 I r ( z ) ≤ N ν ( d +2) /γ (cid:18) – Z C ρ ( z ) (cid:12)(cid:12) ∂ t u + ¯ F [ u ] (cid:12)(cid:12) d +1 dz (cid:19) / ( d +1) + N ν − α (cid:18) – Z C ρ ( z ) | D u | d +1 dz (cid:19) / ( d +1) . (3.3)By setting ˆ F [ u ]( z ) = F ( u ( z ) , D u ( z )) we find – Z C ρ ( z ) (cid:12)(cid:12) ∂ t u + ¯ F [ u ] (cid:12)(cid:12) d +1 dz ≤ N – Z C ρ ( z ) (cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) d +1 dy + N J + N J , where J = – Z C ρ ( z ) (cid:12)(cid:12) ˆ F [ u ] − ¯ F [ u ] (cid:12)(cid:12) d +1 dz is dominated by – Z C ρ ( z ) I | D u | >τ (cid:12)(cid:12) ˆ F [ u ] − ¯ F [ u ] (cid:12)(cid:12) d +1 | D u | d +1 | D u | d +1 dz + N τ d +10 , ULLY NONLINEAR PARABOLIC EQUATIONS 11 which in turn owing to Lemma 3.1 and H¨older’s inequality is less than
N µ d +1 (cid:18) – Z C ρ ( z ) | D u | β ′ ( d +1) dz (cid:19) /β ′ + N τ d +10 , and J = – Z C ρ ( z ) (cid:12)(cid:12) ˆ F [ u ] − F [ u ] (cid:12)(cid:12) d +1 dz ≤ ω d +1 F ( osc C ρ ( z ) u ) – Z C ρ ( z ) | D u | d +1 dz. It follows that (cid:18) – Z C ρ ( z ) (cid:12)(cid:12) ∂ t u + ¯ F [ u ] (cid:12)(cid:12) d +1 dy (cid:19) / ( d +1) ≤ N (cid:18) – Z C ρ ( z ) (cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) d +1 dy (cid:19) / ( d +1) + N µ (cid:18) – Z C ρ ( z ) | D u | β ′ ( d +1) dy (cid:19) / ( β ′ d + β ′ ) + N τ + N ω
F,u, Π ( ρ ) (cid:18) – Z C ρ ( z ) | D u | d +1 dz (cid:19) / ( d +1) . This and (3.3) yield (3.2) since (cid:18) – Z C ρ ( z ) | D u | d +1 dz (cid:19) / ( d +1) ≤ (cid:18) – Z C ρ ( z ) | D u | β ′ ( d +1) dz (cid:19) / ( β ′ ( d +1)) by H¨older’s inequality. The lemma is proved. (cid:3) Lemma 3.4.
Take p > d + 1 , R ∈ (0 , , and u ∈ W , p ( C R ) . Then thereexist constants ˆ θ, θ ∈ (0 , , depending only on d , p , δ , and K F , such that, ifAssumptions 1.2 and 1.3 are satisfied with these ˆ θ and θ , respectively, thenthere is a constant N , depending only on R , d , p , K , K F , δ , and ω F,u,C R ,such that k D u k L p ( C R ) ≤ N (cid:13)(cid:13) ∂ t u + H [ u ] (cid:13)(cid:13) L p ( C R ) + N k ¯ G k L p ( C R ) + N τ + N R ( d +2)(1 /p − /γ ) (cid:13)(cid:13) | D u | γ (cid:13)(cid:13) /γL ( C R ) + N k u k L p ( C R ) , (3.4) k D u k L p ( C R ) ≤ N τ + N R ( d +2) /p − sup C R | u | + N (cid:0)(cid:13)(cid:13) ∂ t u + H [ u ] (cid:13)(cid:13) L p ( C R ) + k ¯ G k L p ( C R ) (cid:1) . (3.5)Proof. For ρ >
0, and z ∈ Q := R + × R d introduce I r ( h, z ) = (cid:18) – Z C r ( z ) – Z C r ( z ) | h ( z ) − h ( z ) | γ dz dz (cid:19) /γ ,h = k Q,γ,ρ ( z ) = sup { I r ( h, z ) : z ∈ Q, r ∈ (0 , ρ ] , C r ( z ) ∋ z } , M h ( z ) = sup r> ,C r ( z ) ∋ z – Z C r ( z ) | h ( ζ ) | dζ, (3.6) whenever these definitions make sense. Note that h = k Q,γ,ρ is well defined in C R for measurable h even defined only in C R +2 ρ .Then take ε ∈ (0 ,
1] to be specified later and take R < R ≤ R suchthat R − R ≤ εR , R ≤ R . (3.7)Next, take ν ≥ r = ( R − R ) / ( ν + 1) . Observe that νr ≤ εR and R − νr = R + r . It follows that, if r ≤ r , z ∈ C R , and z ∈ C r ( z ), then C νr ( z ) ⊂ C R , which by Lemma 3.3 appliedwith Π = C R implies that I r ( z ) ≤ N ν ( d +2) /γ M / ( d +1) (cid:0)(cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) d +1 I C R (cid:1) ( z ) + N τ ν ( d +2) /γ + N h(cid:0) µ + ω F,u,C R ( νr ) (cid:1) ν ( d +2) /γ + ν − α i M / ( β ′ ( d +1)) (cid:0) | D u | β ′ ( d +1) I C R (cid:1) ( z )with N depending only on d, K F , and δ . It follows that in C R ( D u ) = k Q,γ,r ≤ N ν ( d +2) /γ M / ( d +1) (cid:0)(cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) d +1 I C R (cid:1) + N τ ν ( d +2) /γ + N h(cid:0) µ + ω F,u,C R ( εR ) (cid:1) ν ( d +2) /γ + ν − α i M / ( β ′ ( d +1)) (cid:0) | D u | β ′ ( d +1) I C R (cid:1) . By Theorem 7.1, with κ = r /R ≤ / , χ = ( d + 2) /γ, χ = ( d + 2)(1 /γ − /p )and the Hardy-Littlewood maximal function theorem, by taking β so that p > β ′ ( d + 1), we obtain k D u k L p ( C R ) ≤ N ν ( d +2) /γ (cid:13)(cid:13) F [ u ] (cid:13)(cid:13) L p ( C R ) + N τ ν ( d +2) /γ | R | ( d +2) /p + h N (cid:0) µ + ω F,u,C R ( εR ) (cid:1) ν ( d +2) /γ + N ν − α i k D u k L p ( C R ) + N ν χ ( R − R ) − χ R − χ + χ (cid:13)(cid:13) | D u | γ (cid:13)(cid:13) /γL ( C R ) , (3.8)where and below the constants N , N i depend only on d , p , K F , and δ .Now we take and fix ν ≥ N ν − α ≤ / . Then (3.8) becomes k D u k L p ( C R ) ≤ N (cid:13)(cid:13) F [ u ] (cid:13)(cid:13) L p ( C R ) + N τ | R | ( d +2) /p + h N (cid:0) µ + ω F,u,C R ( εR ) (cid:1) + 1 / i k D u k L p ( C R ) + N ν χ ( R − R ) − χ R − χ + χ (cid:13)(cid:13) | D u | γ (cid:13)(cid:13) /γL ( C R ) , (3.9)Next, we use the fact that (cid:12)(cid:12) F [ u ] (cid:12)(cid:12) ≤ (cid:12)(cid:12) H [ u ] (cid:12)(cid:12) + K | u | + K | Du | + ¯ G + ˆ θ | D u | ULLY NONLINEAR PARABOLIC EQUATIONS 13 and that by interpolation inequalities K N k Du k L p ( B R ) ≤ (1 / k D u k L p ( B R ) + N k u k L p ( B R ) . Then we take ˆ θ and µ so small that N ˆ θ ≤ / , N µ ≤ / , and, finally, take the largest ε ≤ N ω F,u,C R ( εR ) ≤ / . This ε will appear later in our arguments and this is the way how theconstant N in the statement of the lemma depends on ω F,u,C R .Then we require that Assumptions 1.2 and 1.3 be satisfied with the abovechosen ˆ θ and θ = θ ( d, δ, K F , µ, βd + β ) (see Lemma 3.1), respectively. Bycombining the above we get k D u k L p ( C R ) ≤ N (cid:13)(cid:13) ∂ t u + H [ u ] (cid:13)(cid:13) L p ( C R ) + N τ R d/p + (5 / k D u k L p ( C R ) + N ( R − R ) − χ R − χ + χ (cid:13)(cid:13) | D u | γ (cid:13)(cid:13) /γL ( C R ) + N k u k L p ( C R ) + N k ¯ G k L p ( C R ) . (3.10)Now we are going to iterate this estimate by defining R = R and for k ≥ R k +1 = R k + cR ( n + k ) − , where the constant c = O ( n ) is chosen so that R k ↑ R as k → ∞ , that is c ∞ X k =1 ( n + k ) − = 1 . and n is chosen so that for k ≥ R k +1 − R k = cR ( n + k ) − ≤ Rcn − ≤ R ≤ R k , which is satisfied if n is just an absolute constant, and (this time we need n − = o ( εR ) as εR → R k +1 − R k = cR ( n + k ) − ≤ cn − ≤ εR . Also observe that R ≤ R k ≤ R and( R k +1 − R k ) − χ R − χ + χ k ≤ N ( n + k ) χ R − χ . Then for k ≥ k D u k L p ( C Rk ) ≤ N (cid:13)(cid:13) ∂ t u + H [ u ] (cid:13)(cid:13) L p ( C Rk +1 ) + N τ R d/p + (5 / k D u k L p ( C Rk +1 ) + N ( n + k ) χ R − χ (cid:13)(cid:13) | D u | γ (cid:13)(cid:13) /γL ( C R ) + N k u k L p ( C R ) + N k ¯ G k L p ( C R ) . We multiply both parts of this inequality by (5 / k and sum up the resultsover k = 1 , , ... . Then we cancel like terms ∞ X k =2 (5 / k k D u k L p ( C Rk ) , which are finite since u ∈ W , p ( B R ), and finally take into account that ∞ X k =2 (5 / k ( n + k ) χ ≤ N n χ ∞ X k =2 (5 / k + N ∞ X k =2 (5 / k k χ ≤ N. Then we come to (3.4).Next, by using equation (3.1) and performing scaling in Theorem 2.3(here we need R ≤ d + 1 to p ), anddenoting I = k ¯ G k L p ( C R ) + (cid:13)(cid:13) ∂ t u + H [ u ] (cid:13)(cid:13) L p ( C R ) we infer that in (3.4) (cid:13)(cid:13) | D u | γ (cid:13)(cid:13) /γL ( C R ) ≤ N R χ (cid:16) k ¯ G + K | u | k L p ( C R ) + (cid:13)(cid:13) ∂ t u + H [ u ] (cid:13)(cid:13) L p ( C R ) (cid:17) + N R χ sup C R | u | ≤ N R χ I + N R χ sup C R | u | , where χ = ( d +2) /γ −
2. After that it suffices to roughly estimate k u k L p ( C R ) in (3.4) by the last term above. The lemma is proved. (cid:3) Boundary a priori estimates in the simplest case
Introduce R d +1+ = { ( t, x ) : t ∈ R , x = ( x , ..., x d ) ∈ R d , x > } ,B + r ( x ) = B r ( x ) ∩ { x > } , C + τ,r ( t , x ) = [ t , t + τ ) × B + r ( x ) ,∂ x C + τ,r ( t , x ) = ¯ C + τ,r ( t , x ) ∩ { x = 0 } , where τ, r ≥
0, ( t , x ) ∈ ¯ R d +1+ . If t = 0 , x = 0, we drop ( t , x ) in thearguments above. Also, if τ = r we write r in place of τ, r in the subscripts,for instance, C + r ( t , x ) := C + r ,r ( t , x ) . Take γ from Section 3 and α ∈ (0 ,
1) to be determined later. Let F bethe function from Section 2. Lemma 4.1. If r > , z ∈ ¯ R d +1+ , ν ≥ , u ∈ \ ρ<νr W , d +1 ( C + ρ ( z )) ∩ C ( ¯ C + νr ( z )) , and u vanishes on ∂ x C + νr ( z ) if this set is nonempty, then we have (cid:16) – Z C + r ( z ) – Z C + r ( z ) | D u ( z ) − D u ( z ) | γ dz dz (cid:17) /γ ≤ N ν ( d +2) /γ (cid:16) – Z C + νr ( z ) (cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) d +1 dz (cid:17) / ( d +1) + N ν − α (cid:16) – Z C + νr ( z ) | D u | d +1 dz (cid:17) / ( d +1) , (4.1) where N depends only on d and δ . ULLY NONLINEAR PARABOLIC EQUATIONS 15
Proof. Scalings show that it suffices to prove the lemma only for νr = 3.Furthermore, without loss of generality we may assume that z = (0 , x )and x = ( | z | , , ..., ∈ R d . Then we consider two cases. Case 1: | z | > / . In this case, we have B + r ( x ) = B +3 /ν ( x ) = B r ( x ) ⊂ B ν ′ r ( x ) ⊂ R d + , C ν ′ r ( z ) ⊂ R d +1+ , where ν ′ = ν/ ≥ Case 2: | z | ∈ [0 , / . Since r = 3 /ν ≤ /
2, we have B + r ( x ) ⊂ B +1 ⊂ B +2 ⊂ B +3 ( x ) = B + νr ( x ) . Let v be the classical solution of ∂ t v + F [ v ] = 0 in C +2 with boundarycondition v = u on ∂ ′ C +2 . Such a solution exists due to the results in [14],which also provide an estimate on D v , so that (for α ∈ (0 , α ( d, δ )]) I := – Z C + r ( z ) – Z C + r ( z ) | D v ( z ) − D v ( z ) | dz dz ≤ N r α [ D v ] C α ( C +1 ) ≤ N r α sup C +2 | v | = N r α sup ∂ ′ C +2 | u | where the last equality is a consequence of the maximum principle and thefact that F (0) = 0. By employing Poincar`e’s inequality ( u = 0 on ∂ x C +2 ),we see that I ≤ N r α (cid:16) – Z B +2 (cid:0) | ∂ t u | d +1 + | D u | d +1 (cid:1) dz (cid:17) / ( d +1) . Here r α = N ν − α and | ∂ t u | ≤ (cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) + (cid:12)(cid:12) F [ u ] (cid:12)(cid:12) ≤ (cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) + N | D u | . Therefore, I ≤ N ν − α (cid:16) – Z B +2 | D u | d +1 dz (cid:17) / ( d +1) + N (cid:16) – Z C +2 (cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) d +1 dz (cid:17) / ( d +1) Next, recall that γ ∈ (0 , – Z C + r ( z ) – Z C + r ( z ) | D v ( z ) − D v ( z ) | γ dz dz ≤ N ν − γα (cid:16) – Z C + νr ( z ) | D u | d +1 dz (cid:17) γ/ ( d +1) + N (cid:16) – Z C + νr ( z ) (cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) d +1 dz (cid:17) γ/ ( d +1) . (4.2)Next, use again that f := ∂u + F ( D u ) = ∂ ( u − v ) + F ( D u ) − F ( D v ) = ∂w + a ij D ij w in C +2 and w = 0 on ∂ ′ C +2 , where ( a ij ) is an S δ -valued function and w = u − v .We extend f and w to all of C as odd functions of x and adjust a ij appropriately so as to have equation f = ∂w + a ij D ij w in C , to which weapply Theorem 2.3 and get (recall that νr = 3) – Z C + r ( z ) | D w | γ dz ≤ N r − d − Z C +2 | D w | γ dz ≤ N ν d +2 (cid:16) – Z C + νr ( z ) | f | d +1 dz (cid:17) γ/ ( d +1) and – Z C + r ( z ) – Z C + r ( z ) | D w ( z ) − D w ( z ) | γ dz dz ≤ N ν d +2 (cid:16) – Z C + νr ( z ) | f | d +1 dz (cid:17) γ/ ( d +1) . Combining this with (4.2) and observing that D u = D v + D w yield (4.1)in Case 2 as well. The lemma is proved. (cid:3) Coming back to our domain Ω recall that we say that Ω is a C , -domainif there exists ρ = ρ (Ω) ∈ (0 ,
1] for which at any point x ∈ ∂ Ω there is anorthonormal system of coordinates Ψ( x ) with the origin at x such that inthe new coordinates ˜ x = (˜ x , ˜ x ′ ) there exists a function ψ ∈ C , ( { ˜ x ′ ∈ R d − : | ˜ x ′ | ≤ ρ } )with the C , ( B ρ )-norm majorated by a constant M (Ω) independent of x and such that ψ (0) = 0 , ψ ˜ x i (0) = 0 , i = 2 , ..., d, | D x ′ ψ (˜ x ′ ) | ≤ | ˜ x ′ | ≤ ρ , { ˜ x : | ˜ x ′ | ≤ ρ , ψ (˜ x ′ ) + 8 ρ ≤ ˜ x ≤ ψ (˜ x ′ ) + 8 ρ } ∩ Ω= { ˜ x : | ˜ x ′ | ≤ ρ , ψ (˜ x ′ ) < ˜ x ≤ ψ (˜ x ′ ) + 8 ρ } . Below in this section we assume that0 ∈ ∂ Ωand that the original system of coordinates in R d coincides with the onedescribed above for x = 0. Lemma 4.2.
Introduce
Γ := { x : | x ′ | ≤ ρ (Ω) , ψ ( x ′ ) < x ≤ ψ ( x ′ ) + 8 ρ (Ω) } ( ⊂ Ω) , ˆΓ := { y : | y ′ | ≤ ρ (Ω) , < y ≤ ρ (Ω) } . Also introduce a mapping x → y ( x ) of Γ onto ˆΓ by x → y = y ( x ) = x − ψ ( x ′ ) , x ′ → y ′ = y ′ ( x ) = x ′ . (4.3) Then this mapping has an inverse y → x ( y ) . Furthermore, the Jacobians ofboth mappings are equal to one. ULLY NONLINEAR PARABOLIC EQUATIONS 17
This lemma is obvious.It is convenient to extend ψ ( x ′ ) for | x ′ | ≥ ρ (Ω), so that the extension issmooth and has the magnitude of the gradient bounded by one and define y ( x ) by the same formula (4.3) for all x ∈ R d . Of course, by x ( y ) we meanthe inverse of y ( x ). Obviously, the assertions of Lemma 4.2 hold true forsuch extensions. Remark . For r ∈ (0 , ∞ ) and z ∈ R d defineˇ B + r = x ( B + r ) , ˇ B + r ( z ) = x ( B + r ( y ( z ))) . Then, as is easy to see(i) ˇ B + r ⊂ Γ ⊂ Ω if r ≤ ρ (Ω);(ii) ˇ B + r ( z ) ⊂ ˇ B +4 ρ (Ω) if ρ > ρ + r ≤ ρ (Ω), and z ∈ ˇ B + ρ . Lemma 4.4.
Take z ∈ ˇ B +2 ρ (Ω) . Then ( i ) for r ≤ ρ (Ω) we have ˇ B + r ( z ) ⊂ B r ( z ) ∩ Ω , B r/ ( z ) ∩ Ω ⊂ ˇ B + r ( z ); (4.4)( ii ) if ν ≥ and νr ≤ ρ (Ω) , we have | ˇ B + νr ( z ) | ≤ N ( d ) ν d | ˇ B + r ( z ) | . (4.5)Proof. (i). First notice that ˇ B + r ( z ) ⊂ Γ. Then, since | D x ′ ψ | ≤
1, for any x , x ∈ Γ we have | y ( x ) − y ( x | ≤ | x − x | and | x ( y ) − x ( y ) | ≤ | y − y | if y , y ∈ ˆΓ. In particular, if | y − y ( z ) | ≤ r , then | x ( y ) − z | ≤ r , so thatˇ B + r ( z ) ⊂ B r ( z ) and ˇ B + r ( z ) ⊂ B r ( z ) ∩ Γ ⊂ B r ( z ) ∩ Ω . which proves the first inclusion in (4.4).Furthermore, if | x − z | ≤ r/ ≤ ρ (Ω) and x ∈ Ω, then, since z ∈ B ρ (Ω) , x ∈ B ρ (Ω) ∩ Ω ⊂ Γ . Then | y ( x ) − y ( z ) | ≤ r and y ( x ) ∈ ˆΓ, that is, y ( x ) ∈ B + r ( y ( z )) so that x ∈ x ( B + r ( y ( z ))), which yields the second inclusion in (4.4).To prove (ii), it suffices to note that | ˇ B + νr ( z ) | = | B + νr ( y ( z ) | ≤ N ν d | B + r ( y ( z ) | = N ν d | ˇ B + r ( z ) | . The lemma is proved. (cid:3)
Corollary 4.5. If z ∈ ˇ B +2 ρ (Ω) and r ≤ (1 / ρ (Ω) , then for any measurablefunction g – Z ˇ B + r ( z ) | g ( x ) | dx ≤ N ( d ) – Z B r ( z ) ∩ Ω | g ( x ) | dx. (4.6)Indeed, the domain of integration on the right is wider than the one theleft owing to (4.4), and N ( d ) | ˇ B + r ( z ) | ≥ | ˇ B +4 r ( z ) | ≥ | B r ( z ) ∩ Ω | in light of (4.5) and (4.4). Next, set ˇ C + R = [0 , R ) × ˇ B + R and for ρ + r ≤ ρ (Ω) and z = ( t, x ) such that x ∈ ˇ B + ρ and t ∈ R defineˇ C + r ( z ) = [ t, t + r ) × ˇ B + r ( x ) . Lemma 4.6.
There exist ¯ γ = ¯ γ ( d, δ ) ∈ (0 , and α = α ( δ, d ) ∈ (0 , such that for any γ ∈ (0 , ¯ γ ] and α ∈ (0 , α ) , whenever ( i ) r, ρ > , ν ≥ , ρ + νr ≤ ρ (Ω) , z ∈ ˇ C + ρ , ( ii ) u ∈ W , p ( ˇ C + ρ + νr ) and u ( t, x ) = 0 if x ∈ ∂ Ω ,we have I r ( z ) := (cid:16) – Z ˇ C + r ( z ) – Z ˇ C + r ( z ) | D u ( z ) − D u ( z ) | γ dz dz (cid:17) /γ ≤ N ν ( d +2) /γ (cid:16) – Z ˇ C + νr ( z ) ( | ∂ t u + F ( D u ) | d +1 + | Du | d +1 ) dz (cid:17) / ( d +1) + N ( ν d +2) /γ r + ν − α ) (cid:16) – Z ˇ C + νr ( z ) | D u | d +1 dz (cid:17) / ( d +1) , (4.7) where the constants N depend only on d, α , M (Ω) , and δ . Proof. By the change of variables formula we see that I r ( z ) equals (cid:16) – Z C + r ( t ,y ( x )) – Z C + r ( t ,y ( x )) | ( D u )( x ( z )) − ( D u )( x ( z )) | γ dz dz (cid:17) /γ . Then with A ( y ) := ∂x ( y ) /∂y we define A = A ( y ( z )) , ˇ F ( u ′′ ) = F (( A − ) ∗ u ′′ A − ) . As is easy to see, D u ′′ ˇ F ∈ S ˇ δ , where ˇ δ = ˇ δ ( d, δ ) ∈ (0 , u ( t, y ) = u ( t, x ( y )) , which belongs to W , p ( C + ρ + νr ), and, since | y ( x ) | < ρ , it also belongs to W , p ( C + νr ( t , y ( x ))) and vanishes on ∂ x C + νr ( t , y ( x )) if this set in nonempty.By Lemma 4.1, since ν ≥
12, we have (cid:16) – Z C + r ( t ,y ( x )) – Z C + r ( t ,y ( x )) | D ˆ u ( z ) − D ˆ u ( z ) | γ dz dz (cid:17) /γ ≤ N ν ( d +2) /γ (cid:16) – Z C + νr ( t ,y ( x )) | ∂ t ˆ u + ˇ F ( D ˆ u ) | d +1 dz (cid:17) / ( d +1) + N ν − α (cid:16) – Z C + νr ( t ,y ( x )) | D ˆ u | d +1 dz (cid:17) / ( d +1) . (4.8)Observe also that for y = y ( x ) and x = x ( y ) D ˆ u ( t, y ) = ( Du )( t, x ) A ( y ) , ULLY NONLINEAR PARABOLIC EQUATIONS 19 where the D ’s are row vectors, and D ˆ u ( t, y ) = A ∗ ( y )[ D u ( t, x )] A ( y ) + [ D k u ( t, x )] D x k ( y ) . (4.9)Since | A − A ( y ) | ≤ N | y − y ( x ) | , where N depends only on d and the bound on | D ψ | , for z i = ( t , y i ) ∈ C + r ( t , y ( x )), i = 1 ,
2, we have | D ˆ u ( z ) − D ˆ u ( z ) | ≥ (1 /N ) | D u ( t , x ) − D u ( t , x ) |− N r ( | D u ( t , x ) | + | D u ( t , x ) | ) − N ( | Du ( t , x ) | + | Du ( t , x ) | ) , where x i = x ( y i ) and N depends only on M (Ω) and d . Hence, the left-handside of (4.8) is greater than or equal to(1 /N ) I r ( z ) − N (cid:16) – Z ˇ C + r ( z ) ( r | D u | + | Du | ) γ dz (cid:17) /γ ≥ (1 /N ) I r ( z ) − N r (cid:16) – Z ˇ C + r ( z ) | D u | d +1 dz (cid:17) / ( d +1) − N (cid:16) – Z ˇ C + r ( z ) | Du | d +1 dz (cid:17) / ( d +1) ≥ (1 /N ) I r ( x ) − N νr (cid:16) – Z ˇ C + νr ( z ) | D u | d +1 dz (cid:17) / ( d +1) − N ν (cid:16) – Z ˇ C + νr ( z ) | Du | d +1 dz (cid:17) / ( d +1) , (4.10)where the first inequality follows by H¨older’s inequality and the second oneis true owing to (4.5).In what concerns the first term on the right-hand side of (4.8), observethat, owing to the Lipschitz continuity of F , the fact that | A ( y ) A − − ( δ ij ) | ≤ N | y − y ( x ) | , and (4.9), we have (with x = x ( y )) | ˇ F ( D ˆ u ( y )) − F ( D u ( x )) | ≤ | F (( A − ) ∗ A ∗ ( y )[ D u ( x )] A ( y ) A − ) − F ( D u ( x )) | + N | Du ( x ) | ≤ N | y − y ( z ) || D u ( x ) | + N | Du ( x ) | . This and an easy estimate of the last term in (4.8) shows that its right-handside is less than
N ν ( d +2) /γ (cid:16) – Z ˇ C + νr ( z ) | ∂ t u + F ( D u ) | d +1 dx (cid:17) / ( d +1) + N ( ν d +2) /γ r + ν − α ) (cid:16) – Z ˇ C + νr ( z ) | D u | d dx (cid:17) /d + N ( ν ( d +2) /γ + ν − α ) (cid:16) – Z ˇ C + νr ( z ) | Du | d dz (cid:17) /d . Upon combining this result with what was said about (4.10) we come to(4.7). The lemma is proved. (cid:3)
Change of variables help derive Lemma 4.6 from its “flat” counterpart.We also allude to it in the following remark.
Remark . Suppose that Assumptions 1.1 , and 1.8 and condition (1.4)are satisfied. Let r ≤ ρ (Ω), p ≥ d + 1, and u ∈ W , p ( ˇ C + r ) be such that u ( t, x ) = 0 if x ∈ ∂ Ω. Then Z ˇ C + r ( | D u | γ + | Du | γ ) dz ≤ N r d +2 − γ ( d +2) /p k ∂ t u + H [ u ] k γL p ( ˇ C + r ) + N r d +2 − γ ( d +2) /p k ¯ G k γL p ( ˇ C + r ) + N r d +2 − γ sup ∂ ′ ˇ C + r | u | γ , where N depend only on δ , K , d , p , and M (Ω) and the range of γ isspecified below.Indeed, by using the notation from the above proof and using equation(3.1) in Lemma 3.2 introduce the operators L u ( t, x ) = a ij ( t, x ) D ij u ( t, x ) + b i ( t, x ) D i u ( t, x ) , ˆ L ˆ u ( t, y ) = [ L u ]( t, x ( y )) . The operator ˆ L can be written as a differential operator with derivativeswith respect to y . Clearly, its matrix of second-order derivatives will belongto S ˆ δ for a ˆ δ = ˆ δ ( δ, M (Ω)) ∈ (0 ,
1) and the drift term by magnitude will bedominated by N = N ( K , d, M (Ω)). Since | ∂ t ˆ u ( t, y )+ ˆ L ˆ u ( t, y ) | ≤ (cid:12)(cid:12) ∂ t u ( t, x ( y ))+ H [ u ]( t, x ( y )) (cid:12)(cid:12) + ¯ G ( t, x ( y ))+ K | u ( t, x ( y )) | in C +4 ρ (Ω) , by Theorem 2.3 for an appropriate ¯ γ = ¯ γ ( d, δ, K , M (Ω)) ∈ (0 , γ ∈ (0 , ¯ γ ], after using scalings and H¨older’s inequality (to replace d + 1with p ) we get, Z C + r ( | D y ˆ u | γ + | D y ˆ u | γ ) dydt ≤ N r ( d +2)(1 − γ/p ) (cid:16) Z C + r (cid:12)(cid:12) ∂ t u + H [ u ] (cid:12)(cid:12) p ( t, x ( y )) dydt (cid:17) γ/p + N r ( d +2)(1 − γ/p ) (cid:16) Z C + r (cid:12)(cid:12) ¯ G ( t, x ( y )) (cid:12)(cid:12) p dydt (cid:17) γ/p + N r d +2 − γ sup ∂ ′ ˇ C + r | u | γ . Now our assertion follows after changing variables.5.
A priori estimates in W , p near the boundary and the proofof Theorem 1.7 We assume that p > d + 1 , ∈ ∂ Ωand take ρ = ρ (Ω), ˇ C + r , ˇ C + r ( z ) from Section 4 and suppose that theassumptions of Theorem 1.7 are satisfied with ˆ θ and θ which are yet to bespecified.First we note the following. Lemma 5.1.
For any q ∈ [1 , ∞ ) and µ > there exists θ = θ ( d, δ, K F , µ, q ) > such that, if Assumption 1.3 is satisfied with this θ , then for any u ′ ∈ R , ULLY NONLINEAR PARABOLIC EQUATIONS 21 z ∈ ˇ C +2 ρ (Ω) and r ≤ ρ (Ω) ∧ R , we have – Z ˇ C + r ( z ) sup u ′′ ∈ S , | u ′′ | >τ | F ( u ′ , u ′′ , z ) − ¯ F ( u ′′ ) | q | u ′′ | q dz ≤ µ q , where ¯ F = ¯ F z,r, u ′ is taken from Assumption 1.3. For the proof of this lemma note that, in light of Corollary 4.5 and As-sumption 1.3, for any u ′′ ∈ S with | u ′′ | = 1 we have – Z ˇ C + r ( z ) sup τ>τ τ − | F ( u ′ , τ u ′′ , z ) − ¯ F ( τ u ′′ ) | dz ≤ N ( d ) θ if 2 r ≤ ρ (Ω) ∧ R . After that, as in the case of Lemma 3.1, the assertion ofthe current lemma is obtained by repeating the proof of Lemma 5.1 of [12].Recall that ω u (Π , ρ ) is introduced in Definition 1.6. Lemma 5.2.
Let r, ρ ∈ (0 , ∞ ) and ν ≥ satisfy ρ + νr ≤ ρ (Ω) and νr ≤ R . Take µ ∈ (0 , ∞ ) , β ∈ (1 , ∞ ) , and suppose that Assumption 1.3 is satisfied with θ = θ ( d, δ, K F , µ, βd + β )( see Lemma 5.1 ) . Assume that we are given a function u ∈ W , p ( ˇ C + ρ + νr ) and u ( t, x ) = 0 if x ∈ ∂ Ω . Use I r ( z ) introduced in (4.7) .Then, for γ and α from Lemma 4.6, for z ∈ ˇ C + ρ , we have I r ( z ) ≤ N η (cid:0) – Z ˇ C + νr ( z ) | D u | β ′ ( d +1) dz (cid:17) / ( β ′ ( d +1)) + N ν ( d +2) /γ (cid:16) – Z ˇ C + νr ( z ) ( (cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) d +1 + | Du | d +1 ) dz (cid:17) / ( d +1) + N τ ν ( d +2) /γ , where η = (cid:0) µ + νr + ω F,u, ˇ C + ρ + νr ( νr ) (cid:1) ν ( d +2) /γ + ν − α , and the constants N depend only on d, p, K F , δ , and M (Ω) . The proof of this lemma is based on Lemma 5.1 and, in light of Lemma4.6, is practically identical to that of Lemma 3.3.We now come to the main a priori estimate near the boundary for non-linear parabolic equations with VMO “coefficients”.
Theorem 5.3.
Take p > d + 1 , let
R > satisfy R ≤ ρ (Ω) ∧ R , and let u ∈ W , p ( ˇ C +2 R ) be such that u ( t, x ) = 0 if x ∈ ∂ Ω . Then thereexist constants ˆ θ, θ ∈ (0 , , depending only on d, p, δ , and K F , such thatif Assumptions 1.2 and 1.3 are satisfied with these ˆ θ and θ , respectively, then there exist constants N , depending only on R , d, p, K , K F , δ , ρ (Ω) , M (Ω) , and the function ω F,u, ˇ C +2 R ( see Definition 1.6 ) , such that k D u k L p ( ˇ C + R ) ≤ N k ∂ t u + H [ u ] k L p ( ˇ C +2 R ) + N k ¯ G k L p ( ˇ C +2 R ) + N τ + N k u k L p ( ˇ C +2 R ) + N R − χ k | D u | γ k /γL ( ˇ C +2 R ) , (5.1) k D u k L p ( ˇ C + R ) ≤ N k ∂ t u + H [ u ] k L p ( ˇ C +2 R ) + N k ¯ G k L p ( ˇ C +2 R ) + N τ + N R ( d +2) /p − sup ˇ C +2 R | u | , (5.2) where χ = ( d + 2)(1 /γ − /p ) and γ is the same as in Lemma 5.2. Proof. Whenever it makes sense, for ρ ≤ ρ (Ω), and z ∈ ˇ C +2 ρ (Ω) introduce h ≍k Ω ,γ,ρ ( z ) = sup { I r ( h, z ) : z ∈ R + × Ω , r ∈ (0 , ρ ] , ˇ C + r ( z ) ∋ z } , where I r ( h, z ) = (cid:18) – Z ˇ C + r ( z ) – Z ˇ C + r ( z ) | h ( z ) − h ( z ) | γ dz dz (cid:19) /γ . The reader should pay attention to the above curved sharp symbol, remind-ing of curved boundaries.Observe that, if r ≤ ρ ≤ ρ (Ω) and z ∈ ˇ C +2 ρ (Ω) ∩ ˇ C + r ( z ), then ˇ C + r ( z ) ⊂ ˇ C +4 ρ (Ω) , so that h ≍k Ω ,γ,ρ ( z ) is well defined on ˇ C +2 ρ (Ω) even if h is given only onˇ C +4 ρ (Ω) ( ⊂ Ω).Then take ε ∈ (0 ,
1] to be specified later, take R < R ≤ R such that R ≤ R , R − R ≤ εR , take ν ≥
12, and set r = ( R − R ) / ( ν + 1) , κ = r /R ( ≤ ( R − R ) / (2 R ) ≤ / . We are going to use Theorem 7.2 according to which, if h ∈ L p ( ˇ C + R ),then k h k L p ( ˇ C + R ) ≤ N k h ≍k Ω ,γ,r k L p ( ˇ C + R ) + N ν χ ( R − R ) − χ R χ − χ k | h | γ k /γL ( ˇ C + R ) , (5.3)where χ = ( d + 2) /γ , χ = ( d + 2)(1 /γ − /p ), and the constants N dependonly on d, γ , and p .Next for z ∈ ˇ C +2 R ( ⊂ ˇ C + ρ (Ω) ) define M Ω h ( z ) = sup n – Z ˇ C + r ( z ) | h ( y ) | dy : 2 r ≤ ρ (Ω) , z ∈ R + × Ω , ˇ C + r ( z ) ∋ z o . ULLY NONLINEAR PARABOLIC EQUATIONS 23
Observe that, owing to the fact that Ω ∈ C , and to Corollary 4.5, if z ∈ ˇ C +2 ρ (Ω) and r ≤ (1 / ρ (Ω), – Z ˇ C + r ( z ) | h | dy ≤ N – Z C r ( z ) | h | I ˇ C +2 R dy, where N depends only on d , ρ (Ω), and M (Ω). Therefore, for z ∈ ˇ C +2 R M Ω h ( z ) ≤ N M hI ˇ C +2 R ( z ) . (5.4)The above conclusion (5.4) is, actually, also based on the fact similar to thefollowing. For r ≤ r , z ∈ ˇ C + R , and z ∈ R + × Ω, such that ˇ C + r ( z ) ∋ z , wehave z ∈ ˇ C + ρ , where ρ = R + r . In this situation also ρ + νr ≤ R < ρ (Ω)and νr ≤ εR and it follows from Lemma 5.2 that I r ( z ) ≤ N ν ( d +2) /γ M / ( d +1)Ω (cid:0)(cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) d +1 I ˇ C + R (cid:1) ( z ) + N τ ν ( d +2) /γ + N η M / ( β ′ ( d +1))Ω (cid:0) | D u | β ′ ( d +1) I ˇ C + R (cid:1) ( z )+ N ν ( d +2) /γ M / ( d +1)Ω (cid:0) | Du | d +1 I ˇ C + R (cid:1) ( z ) , where η = (cid:0) µ + νr + ω F,u, ˇ C + R ( εR ) (cid:1) ν ( d +2) /γ + ν − α , By definition and (5.4) we obtain that on ˇ C + R ( D u ) ≍k Ω ,γ,r ≤ N ν ( d +2) /γ M / ( d +1) (cid:0)(cid:12)(cid:12) ∂ t u + F [ u ] (cid:12)(cid:12) d +1 I ˇ C + R (cid:1) + N τ ν ( d +2) /γ + N η M / ( β ′ ( d +1)) (cid:0) | D u | β ′ ( d +1) I ˇ C + R (cid:1) + N ν ( d +2) /γ M / ( d +1) (cid:0) | Du | d +1 I ˇ C + R (cid:1) . Thanks to (5.3) and the Hardy-Littlewood maximal function theorem, bytaking β so that p > β ′ d , we obtain k D u k L p ( ˇ C + R ) ≤ N ν ( d +2) /γ (cid:13)(cid:13) ∂ t u + F [ u ] (cid:13)(cid:13) L p ( ˇ C + R ) + N τ ν ( d +2) /γ + h N (cid:0) µ + νr + ω F,u, ˇ C + R ( εR ) (cid:1) ν ( d +2) /γ + N ν − α i k D u k L p ( ˇ C + R ) + N ν ( d +2) /γ k Du k L p ( ˇ C + R ) + N ν χ ( R − R ) − χ R χ − χ (cid:13)(cid:13) | D u (cid:12)(cid:12) γ k /γL ( ˇ C +2 R ) , where the constants N , N depend only on d , p , K F , and δ .This estimate looks almost like (3.8). Then we repeat the argument after(3.8) and choose and fix ε , ν , ˆ θ , and µ , recall what r is, and conclude that k D u k L p ( ˇ C + R ) ≤ N (cid:13)(cid:13) ∂ t u + H [ u ] (cid:13)(cid:13) L p ( ˇ C + R ) + N τ ν d/γ +(5 / N ( R − R )) k D u k L p ( ˇ C + R ) + N k u k L p ( ˇ C +2 R ) + N k ¯ G k L p ( ˇ C +2 R ) + N ( R − R ) − χ R χ − χ (cid:13)(cid:13) | D u (cid:12)(cid:12) γ k /γL ( ˇ C +2 R ) . After that, to prove (5.1), it suffices to repeat almost literally what follows(3.10) (only replacing C with ˇ C ). By using Remark 4.7 we estimate the lastterm in (5.1) and then finish the proof of the theorem in the same way asin the case of Lemma 3.4. The theorem is proved. (cid:3) Proof of Theorem 1.7 . To start, assume that g ≡
0. Observe that inthat case we may assume that u ( t, x ) is defined for t ≥ T , x ∈ ¯Ω, as zeroand still satisfies there (1.1). It suffices for the latter that H (0 , t, x ) = 0 if t ≥ T , which is easy to accommodate without altering our assumptions justby replacing G ( u , t, x ) and ¯ G with G ( u , t, x ) I t
0, find ˆ R ≤ R such that ω F,g, Π ( ˆ R ) ≤ θ / R and θ / R and θ , respectively.Then we see that the above result is applicable to w , and along with theembedding inequality: | g | ≤ N k g k W , p (Π) , lead to (1.5) in the general case.The theorem is proved. (cid:3) Proof of Theorem 1.9
The proof of Theorem 1.9 is based on the following.
Theorem 6.1.
Suppose that Assumption 1.8 is satisfied, the number ¯ H := sup u ′ ,t,x (cid:0) | H ( u ′ , , t, x ) | − K | u ′ | (cid:1) ( ≥ is finite, and g ∈ W , ∞ ( R d +1 ) .Then there exists a convex positive homogeneous of degree one function P ( u ′′ ) such that at all points of its differentiability D u ′′ P ∈ S ¯ δ , where ¯ δ =¯ δ ( d, δ ) ∈ (0 , δ ) , and for P [ u ] = P ( D u ) and any K > the equation ∂ t v + max( H [ v ] , P [ v ] − K ) = 0 (6.1) in Π with boundary condition v = g on ∂ ′ Π has a solution v ∈ W , p (Π) forany p ≥ . This theorem follows from Theorem 2.1 of [11], proved there under theadditional conditions that Ω ∈ C and that there is an increasing continuousfunction ω ( r ), r ≥
0, such that ω (0) = 0 and | H ( u ′ , u ′′ , t, x ) − H ( v ′ , u ′′ , t, x ) | ≤ ω ( | u ′ − v ′ | )for all u , v , t , and x . That these additional conditions can be dropped willbe proved elsewhere. Step 1 . We take P ( u ′′ ) from Theorem 6.1, and first we assume that g ∈ W , ∞ (Π) and there exists constants N , ¯ H such that, for all t, x, u ′ , | H ( u ′ , , t, x ) | ≤ N | u ′ | + ¯ H. (6.2)By Theorem 6.1 for any K > v K which is in W , p (Π) for any p >
1, such that v K = g on ∂ ′ Π, and it satisfies ∂v K + H K [ v K ] = 0 in Π (a.e.) , (6.3) where H K ( u , t, x ) = max( H ( u , t, x ) , P ( u ′′ ) − K ) . Set F K ( u ′ , u ′′ , t, x ) = max( F ( u ′ , u ′′ , t, x ) , P ( u ′′ ) − K ) , ¯ F K,z,r, u ′ ( u ′′ ) = max( ¯ F z,r, u ′ ( u ′′ ) , P ( u ′′ ) − K ) ,G K ( u , t, x ) = H K ( u , x ) − F K ( u ′ , u ′′ , t, x ) . It is not hard to see that Assumptions 1.3, 1.2, and 1.8 are satisfied for H K , F K , and G K in place of H, F , and G , respectively, with the same K ,¯ G , R , θ , ˆ θ , ω F , with ¯ δ in place of δ and ¯ δ − + K F in place of K F . ByTheorem 1.7 there exist constants ˆ θ, θ ∈ (0 , d , p , δ , K F , ρ (Ω), and M (Ω), such that, if Assumptions 1.3 and 1.2 are satisfiedwith these θ and ˆ θ , respectively, then for any K >
0, we have k v K k W , p (Π) ≤ N (cid:0) k ¯ G k L p (Π) + k g k W , p (Π) + k v K k C (Π) (cid:1) + N τ , where the constants N depend only on K , K F , d , p , δ , R , diam(Ω), ρ (Ω), M (Ω), and the function ω F,v K , Π (independent of N and ¯ H ).Since H K satisfies (1.4), formula (3.1) is valid with v K and H K in placeof u and H . This converts equation (6.3) into a linear equation and by thewell-known results from the linear theory allows us to estimate | v K | and themodulus of continuity of v K through that of g , sup | g | , and k ¯ G k L d +1 (Π) withconstants independent of K .Thus, k v K k W , p (Π) ≤ N (cid:0) k ¯ G k L p (Π) + k g k W , p (Π) (cid:1) + N τ , (6.4)where the constants N are independent of K .In this way we completed a crucial step consisting of obtaining a uniformcontrol of the W , p (Π)-norms of v K .Next, we let K → ∞ . Estimate (6.4) guarantees that there is a sequence K n → ∞ as n → ∞ and v ∈ W , p (Π) such that v K n → v weakly in W , p (Π)and v K n → v uniformly in ¯Π. Then, of course, v = g on ∂ ′ Π. The said weakconvergence implies pointwise convergence Dv n → Dv in Π in light of thecompactness of the embedding W , p ⊂ C , ( p > d + 2).Next, for m = 1 , , ... define H m ( u ′′ , t, x ) = sup n ≥ m max( H ( v K n ( t, x ) , Dv K n ( t, x ) , u ′′ , t, x ) , P ( u ′′ ) − K n ) . Observe that H m ( u ′′ , t, x ) are Lipschitz continuous in u ′′ and at all pointsof differentiability satisfy D u ′′ H m ∈ S ¯ δ . Also | H m (0 , t, x ) | ≤ K max n ≥ m (cid:16) | v K n ( t, x ) | + | Dv K n ( t, x ) | (cid:17) + ¯ G ( t, x ) , which is in L p, loc (Π). Therefore, the operators H m [ u ] fit into the scheme ofSection 3.5 of [9]. Furthermore, for n ≥ m obviously ∂ t v K n + H m ( v K n , t, x ) ≥ ULLY NONLINEAR PARABOLIC EQUATIONS 27 (a.e.) in Π. By Theorem 3.5.9 of [9] we conclude that for any m∂ t v + sup n ≥ m max( H ( v K n , Dv K n , D v, t, x ) , P ( D v ) − K n ) ≥ t, x ) at which (6.5) holds for all m (that is, we fix almostany ( t, x )) and since H ( u ′ , u ′′ , t, x ) is continuous in u ′ , we have that | H ( v K n ( t, x ) , Dv K n ( t, x ) , D v ( t, x ) , t, x ) − H ( v ( t, x ) , Dv ( t, x ) , D v ( t, x ) , t, x ) | → n → ∞ . Then, in light of (6.5), ∂ t v ( t, x )+max( H ( v ( t, x ) , Dv ( t, x ) , D v ( t, x ) , t, x ) , P ( D v ( t, x )) − K m ) ≥ o (1) , which for m → ∞ yields ∂ t v ( t, x ) + H ( v ( t, x ) , Dv ( t, x ) , D v ( t, x ) , t, x ) = ∂ t v ( t, x ) + H [ v ]( t, x ) ≥ . The inequality ∂ t v + H [ v ] ≤ n ≥ m max( H ( v K n ( t, x ) , Dv K n ( t, x ) , u ′′ , t, x ) , P ( u ′′ ) − K n ) . Owing to (6.4), of course, v ∈ W , p (Π) and (6.4) holds with v in place of v K .This proves the theorem if condition (6.2) is satisfied and g ∈ W , ∞ (Π). Step 2 . Assume that g ∈ W , ∞ ( R d +1 ) and abandon (6.2). Let η ( t ) = t for | t | ≤ η ( t ) = sign t for | t | ≥
1. For n = 1 , , ... define η n ( t ) = nη ( t/n )and ˆ H n ( u , t, x ) = H ( u , t, x ) − H ( u ′ , , t, x ) + η n ( H ( u ′ , , t, x )) , ˆ G n ( u , t, x ) = ˆ H n ( u , t, x ) − F ( u ′ , u ′′ , t, x ) . Then | ˆ G n ( u , t, x ) | = | G ( u , t, x ) + η n ( H ( u ′ , , t, x )) − H ( u ′ , , t, x ) |≤ ˆ θ | u ′′ | + 2 K | u ′ | + 2 ¯ G ( t, x ) , so that Assumption 1.2 is satisfied for ˆ H n with 2 K and 2 ¯ G in place of K and ¯ G . Assumptions 1.3 and 1.8 are also valid for ˆ H n with the sameparameters.Furthermore | ˆ H n ( u ′ , , t, x ) | = | η n ( H ( u ′ , , t, x )) | , which is bounded.Hence there are ˆ θ and θ as in Step 1, for any n , there exists u n ∈ W , p (Π) ∩ C ( ¯Π) satisfying ∂ t u n + ˆ H n [ u n ] = 0in Π (a.e.) and such that u = g on ∂ ′ Π. Estimate (1.5), applicable to v n bythe above again guarantees that the W , p (Π)-norms of v n are bounded and v n are equicontinuous in ¯Π. This enables us to find a subsequence v n ′ anda function v ∈ W , p (Π) such that v n ′ → v weakly in W , p (Π) and v n ′ → v uniformly in ¯Π. Then, of course, v = g on ∂ ′ Π. After that we repeat the rest of Step 1 by takingsup n ′ ≥ m h H ( v n ′ , Dv n ′ , u ′′ , t, x ) − H ( v n ′ , Dv n ′ , , t, x ) + η n ′ ( H ( v n ′ , Dv n ′ , , t, x )) (cid:3) in place of H m ( u ′′ , t, x ). One thing which makes the argument here easier isthat for any ( t, x ) ∈ Π − H ( v n , Dv n , , t, x ) + η n ( H ( v n , Dv n , , t, x )) = 0if n is large enough.In this way we finish Step 2. Finally, to treat the general g ∈ W , p (Π) itsuffices to use approximations and very simple arguments about passing tothe limit, which we have seen already above. This step is left to the reader.The theorem is proved. (cid:3) Appendix
Fix γ ∈ (0 ,
1] and for r ∈ (0 , ∞ ) and z ∈ R d +1 define I r ( h, z ) = (cid:18) – Z C r ( z ) – Z C r ( z ) | h ( z ) − h ( z ) | γ dz dz (cid:19) /γ (7.1)whenever the right-hand side makes sense.For ρ > h by the formula h = k Q,γ,ρ ( z ) = sup (cid:8) I r ( h, z ) : z ∈ Q, r ∈ (0 , ρ ] , C r ( z ) ∋ z (cid:9) (7.2)whenever it makes sense. Note that, if Q = R + × R d , h = k Q,γ,ρ is well definedin C R for measurable h even defined only in C R +2 ρ . Theorem 7.1.
Let p ∈ (1 , ∞ ) , κ ∈ (0 , , R ∈ (0 , ∞ ) , and h ∈ L p ( C R (1+2 κ ) ) .Let Q = R + × R d . Then k h k L p ( C R ) ≤ N (cid:13)(cid:13) h = k Q,γ,κR (cid:13)(cid:13) L p ( C R ) + N κ − χ R − χ (cid:13)(cid:13) | h | γ (cid:13)(cid:13) /γL ( C R ) , (7.3) where χ = ( d + 2) /γ , χ = ( d + 2)(1 /γ − /p ) and the constants N dependonly on d, γ , and p . This theorem will be proved elsewhere by closely following the proof ofTheorem 7.1 of [12] given there in the elliptic framework.The remaining results of this section treat smooth cylinders or smoothdomains. If Ω ∈ C , and 0 ∈ ∂ Ω, we assume that the original system ofcoordinates in R d coincides with the one described before Lemma 4.2 andwith the help of the mappings x ( y ) and y ( x ) introduced in that lemma, for r > z ∈ R d , we constructˇ B + r = x (cid:0) B + r (cid:1) , ˇ B + r ( z ) = x (cid:0) B + r ( y ( z ) (cid:1) . By Remark 4.3 we have ˇ B + r ⊂ Ω for r ≤ ρ (Ω) and ˇ B + r ( z ) ⊂ ˇ B +4 ρ (Ω) if ρ > ρ + r ≤ ρ (Ω), and z ∈ ˇ B + ρ . Generally, these are objects in R d .Then set ˇ C + R = [0 , R ) × ˇ B + R ULLY NONLINEAR PARABOLIC EQUATIONS 29 and for ρ, r ≤ ρ (Ω) and z = ( t, x ) such that x ∈ ˇ B + ρ and t ∈ R defineˇ C + r ( z ) = [ t, t + r ) × ˇ B + r ( x ) . Finally, whenever it makes sense, for ρ ≤ ρ (Ω), and z ∈ ˇ C +2 ρ (Ω) introduce h ≍k Ω ,γ,ρ ( z ) = sup n I r ( h, z ) : z ∈ R + × Ω , r ∈ (0 , ρ ] , ˇ C + r ( z ) ∋ z o , (7.4)where I r ( h, z ) = (cid:18) – Z ˇ C + r ( z ) – Z ˇ C + r ( z ) | h ( z ) − h ( z ) | γ dz dz (cid:19) /γ . The reader should pay attention to the above curved sharp symbol, remind-ing of curved boundaries.Observe that, if r ≤ ρ ≤ ρ (Ω) and z ∈ ˇ C +2 ρ (Ω) ∩ ˇ C + r ( z ), thenˇ C + r ( z ) ⊂ ˇ C +4 ρ (Ω) , so that h ≍k Ω ,γ,ρ ( z ) is well defined on ˇ C +2 ρ (Ω) even if h is given only on ˇ C +4 ρ (Ω) ( ⊂ [0 , ρ (Ω) × Ω).
Theorem 7.2. If p ∈ (1 , ∞ ) , κ ∈ (0 , / , < R ≤ ρ (Ω) , and h ∈ L p (cid:0) ˇ C + R (1+2 κ ) (cid:1) , then k h k L p (cid:0) ˇ C + R (cid:1) ≤ N (cid:13)(cid:13) h ≍k Ω ,γ,κR (cid:13)(cid:13) L p (cid:0) ˇ C + R (cid:1) + N κ − χ R − χ (cid:13)(cid:13) | h | γ (cid:13)(cid:13) /γL (cid:0) ˇ C + R (cid:1) , (7.5) where χ = ( d + 2) /γ , χ = ( d + 2)(1 /γ − /p ) and the constants N dependonly on d, γ , and p . This theorem is derived from Theorem 7.1 by changing variables and evenextension of the functions involved across the plane { x = 0 } . Acknowledgment . Part of the work on the article was done during theauthor’s stay at the Center of Mathematical Analysis and Application ofHarvard University for one month in November 2015, and it is my greatpleasure to thank S.-T. Yau for his kind invitation. The author is alsosincerely grateful to the referee for very careful reading and many commentsand suggestions.
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