Convergence of Rothe s Method for Fully Nonlinear Parabolic Equations

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1 Te Journal of Geometric Analysis Volume 15, Number 3, 2005 Convergence of Rote s Metod for Fully Nonlinear Parabolic Equations By Ivan Blank and Penelope Smit ABSTRACT. Convergence of Rote s metod for te fully nonlinear parabolic equation u t + FD 2 u,du,u,x,t) = 0 is considered under some continuity assumptions on F. We sow tat te Rote solutions are Lipscitz in time, Hölder in space, and tey solve te equation in te viscosity sense. As an immediate corollary we get Lipscitz beavior in time of te viscosity solutions of our equation. We consider te boundary value problem: 1. Introduction u t + F Dx 2u, D xu, u, x, t ) = 0 in D u = 0 on p D. 1.1) We assume tat ux, t) satisfies Equation 1.1) in te viscosity sense, tat D = [0,T], and tat F is uniformly elliptic. We recall tat a function u is a viscosity subsolution supersolution) of Equation 1.1) if for any ϕ C 2,1 D) and any ɛ>0 wic satisfies ϕ t + F D 2 ϕ,dϕ,ϕ,x,t ) ɛ>0 or ϕt + F D 2 ϕ,dϕ,ϕ,x,t ) ɛ<0 for te supersolution case ) 1.2) u ϕ cannot attain a local maximum minimum) of 0. In oter words, ϕ cannot touc u from above below).) u is a solution if it is bot a subsolution and a supersolution. Uniform ellipticity means tat tere are positive numbers λ and suc tat for any positive definite matrix N we ave or wat is equivalent: N FM + N, P, v, x, t) FM, P, v, x, t) λ N, 1.3) FM + N, P, v, x, t) FM, P, v, x, t)+ N λ N + 1.4) Mat Subject Classifications. 35J60; 65M20. Key Words and Prases. Rote s metod, te metod of lines, fully nonlinear, parabolic PDE, viscosity solutions. Acknowledgements and Notes. Te first autor was partly supported by te NSF grant No. DMS Te Journal of Geometric Analysis ISSN

2 364 Ivan Blank and Penelope Smit for all M and N. So te negative Laplacian would fit our structure conditions, and not te positive Laplacian. For linear F s, Rote s metod is commonly used as a numerical approximation. Rote s metod corresponds to doing a backward Euler approximation in Banac space, and is also known sometimes as te metod of lines. Kikuci and Kačur ave studied te convergence properties of Rote s metod in a series of recent articles in a variety of function spaces. See [5] and [6] and te references terein.) Pluscke as studied te quasilinear case see [7]), but te current work appears to be te first extension of Rote s metod to viscosity solutions in te fully nonlinear setting. We fix our mes size, 0 < 1, and define z n, x) recursively. We take z 0, x) 0, and for n 0 we let z n+1, be te solution of te elliptic problem: F D 2 z n+1,,dz n+1,,z n+1,,x,n+ 1) ) + z n+1, = z n, in z n+1, x) = 0 on. z n, x) is an approximation to ux, n), and we define te linear interpolation, U x, t), by U x, t) := [m + 1) t] z m, x) + 1.5) [t m] z m+1, x), 1.6) were m N is cosen so tat t [m, m + 1)). Wen Ux, t) := lim 0 U x, t) exists, we call it te Rote limit. Before we sow convergence and uniqueness, wen a subsequence converges we will still refer to it as a Rote limit. We make te following continuity assumption, wic corresponds to te structure condition SC) of [1]: We assume tat for every R>0, tere is a nondecreasing continuous function σ R suc tat ω R 0) = 0, and if X and Y are symmetric matrices and r, s R, ten P M N) γ P Q ω R s r) + ) FM, P, r, x, t) FN, Q, s, x, t) P + M N)+ γ P Q +ω R r s) + ) 1.7). We will define te Pucci operators P + and P in a moment.) Lastly, we assume tat for any compact subset of R n n R n R we ave FM, P, r,, ) C 0 C<. 1.8) D) 2. Elliptic preliminaries To study te elliptic problem in eac time we introduce te Pucci extremal operators wic we give by P + M) = λtr M +) + Tr M ) P M) = Tr M +) + λtr M ) and 2.1) were Tr M) is te trace of M. Lemma 2.1 First Bound). Tere exists a constant C wic is independent of suc tat z 1, L C. 2.2)

3 Convergence of Rote s Metod for Fully Nonlinear Parabolic Equations 365 Proof. By symmetry it suffices to sow tat tere exists a constant C suc tat z 1, x) C. z 1, satisfies 0 = F D 2 z 1,,Dz 1,,z 1,,x, ) + z 1, F D 2 z 1,,Dz 1,,z 1,,x, ) on {z 1, > 0} and terefore by Lemma 2.11 of [1] z 1, is also a solution of P D 2 z 1, ) γ Dz1, +F0, 0, z 1,,x,) 0 2.3) on te set were it is positive. By Proposition 2.12 of [1] we now ave z 1, C F0, 0, z 1,,x,)) + L n Ɣ + z 1, )), 2.4) were Ɣ + denotes te upper contact set. Now by using our structure conditions, we see tat σ R z1, ) +) F0, 0, z 1,,x,) F0, 0, 0, x,) σ R z1, ) +). Let fx):= F0, 0, 0, x,). Ten we see tat [ F0, 0, z1,,x,) ] + χ{z1, >0} fx). 2.5) By combining Equations 2.4), 2.5), and 1.8) we are done. Teorem 2.2 First Lipscitz Bound). suc tat Tere exists a constant C wic is independent of z 1, L C. 2.6) Proof. Let te absolute maximum of z 1, be attained at x 0, and let γ := z 1, x 0 ).Weave γ = z 1, x 0 ) = F D 2 z 1, x 0 ), 0, γ,x 0, ) in te viscosity sense. Since te plane x) γ touces z 1, from above at x 0 we know tat γ F 0, 0, γ,x 0,) 2.7) wic we know is bounded from above by a constant times by using te previous lemma along wit Equations 1.7) and 1.8). Teorem 2.3 Pucci Inequalities). Let fx) := F0, 0, 0, x,n + 1)). Te following inequalities are satisfied in te viscosity sense. P D 2 ) z n+1 γ Dzn+1 σ R zn+1 ) +) + z n+1 + f z n P + D 2 ) z n+1 + γ Dzn+1 + σ R z + ) z n+1 n f. 2.8) Proof. Let ϕ touc z n+1 from above at x 0. Since F D 2 z n+1,dz n+1,z n+1,x,n+ 1) ) + z n+1 = z n

4 366 Ivan Blank and Penelope Smit in te viscosity sense, by using Equation 1.7) we ave z n F D 2 ϕ,dϕ,ϕ,x,n+ 1) ) + ϕ P D 2 ϕ ) γ Dϕ σ R ϕ) + ) + ϕ + f. Te oter inequality is proven in te same fasion by toucing z n+1 from below. Now we introduce te sup-convolution zj ɛ of z j and te inf-convolution z j,ɛ of z j wic are defined by ) ) zj ɛ x y 2 x y 2 x) := sup z j y) z j,ɛ x) := inf z j y) ) y 2ɛ y 2ɛ Te next two results are taken from [3] and [4], respectively: Teorem 2.4 Basic Sup-Convolution Properties). Let R n be bounded, let u C ), and let u ɛ and u ɛ be its sup and inf-convolution, respectively. Ten i) ii) iii) u ɛ,u ɛ C 0,1 ). u ɛ u and u ɛ u as ɛ 0 uniformly on. For every ɛ>0, tere are measurable functions M ɛ,m ɛ : Sn) suc tat u ɛ y) = u ɛ x)+<du ɛ x), y x>+ 1 2 <Mɛ x)y x), y x>+o x y 2) iv) v) for a.e. x and similarly wit u ɛ and M ɛ ) were Sn) is te set of n n symmetric matrices. M ɛ x) 1/ɛ)I and M ɛ x) 1/ɛ)I for a.e x. If u η,ɛ and u η ɛ are standard mollifications of u ɛ and u ɛ, respectively, ten D 2 u η,ɛ 1/ɛ)I and D 2 u η,ɛ M ɛ x)a.e. x as η 0, and similarly wit D 2 u η ɛ. Teorem 2.5 Sup-Convolutions are Subsolutions). If u is a bounded viscosity subsolution of F D 2 u,du,u,x ) = fx) in B 1 and f, F are continuous, ten F M ɛ x), Du ɛ x), u x ɛ),x ɛ) f x ɛ) a.e. in B 1 2ɛ u ) 1/2 were x ɛ B 1 is any point wic satisfies u ɛ x) = u x ɛ) Obviously inf-convolutions are supersolutions. x ɛ x 2 Remark 2.6. Wat we ave denoted by x ɛ, is typically denoted by x, but we need to empasize te dependence on ɛ to avoid confusion later. For inf-convolutions we will use x ɛ. ) 2ɛ. Remark 2.7. Note tat x ɛ x 2 2ɛ = u ɛ x ɛ) u ɛ x) 2 u,

5 Convergence of Rote s Metod for Fully Nonlinear Parabolic Equations 367 so tat x ɛ x 2ɛ u ) 1/2. In particular, if x B 1 2ɛ u ) 1/2 ten xɛ B 1, explaining te appearance of tis set above. By using Equation 1.5) and te previous teorem, we get F Mn+1, ɛ x), Dzɛ n+1, x), zɛ n+1, x ɛ ),x ɛ,n+ 1) ) + zɛ n+1, xɛ ) for a.e. x. Similarly, we ave F M n,,ɛ x), Dz n,,ɛ x), z n,,ɛ x ɛ ), x ɛ,n ) + z n,,ɛx ɛ ) z n, x ɛ ) z n 1,x ɛ ) Hencefort we suppress te in te subscripts. By taking a difference we get: F M ɛ n+1 x), Dzɛ n+1 x), zɛ n+1 xɛ ),x ɛ,n+ 1) ) FM n,ɛ x), Dz n,ɛ x), z n,ɛ x ɛ ), x ɛ, n) + zɛ n+1 xɛ ) z n,ɛ x ɛ ) z n x ɛ ) z n 1 x ɛ ). Now we express te difference as a telescoping sum. F Mn+1 ɛ x), Dzɛ n+1 x), zɛ n+1 xɛ ),x ɛ,n+ 1) ) FM n,ɛ x), Dz n,ɛ x), z n,ɛ x ɛ ), x ɛ, n) = F Mn+1 ɛ x), Dzɛ n+1 x), zɛ n+1 xɛ ),x ɛ,n+ 1) ) F M n,ɛ x), Dzn+1 ɛ x), zɛ n+1 xɛ ),x ɛ,n+ 1) ) + F M n,ɛ x), Dzn+1 ɛ x), zɛ n+1 xɛ ),x ɛ,n+ 1) ) F M n,ɛ x), Dz n,ɛ x), zn+1 ɛ xɛ ),x ɛ,n+ 1) ) + F M n,ɛ x), Dz n,ɛ x), zn+1 ɛ xɛ ),x ɛ,n+ 1) ) F M n,ɛ x), Dz n,ɛ x), z n,ɛ x ɛ ), x ɛ,n+ 1) ) + F M n,ɛ x), Dz n,ɛ x), z n,ɛ x ɛ ), x ɛ,n+ 1) ) F M n,ɛ x), Dz n,ɛ x), z n,ɛ x ɛ ), x ɛ,n ). 2.10). 2.11) 2.12) 2.13) For te time being we will assume tat F is differentiable in te variables we need in order to define a ij x) := 1 0 F X ij tm ɛ n+1 x) + 1 t)m n,ɛ x), Dzn+1 ɛ x), zɛ n+1 xɛ ),x ɛ,n+ 1) ) dt b i x) := 1 0 F P i Mn,ɛ x), tdzn+1 ɛ x) + 1 t)dz n,ɛx), zn+1 ɛ xɛ ), x ɛ,n+ 1) ) 2.14) dt cx) := 1 0 F r Mn,ɛ x), Dz n,ɛ x), tzn+1 ɛ xɛ ) +1 t)z n,ɛ x ɛ ), x ɛ,n+ 1) ) dt. Now we use te fundamental teorem on eac pair of terms to get te following equalities: F Mn+1 ɛ x), Dzɛ n+1 x), zɛ n+1 xɛ ),x ɛ,n+ 1) ) F M n,ɛ x), Dzn+1 ɛ x), zɛ n+1 xɛ ),x ɛ,n+ 1) ) = a ij x) Mn+1,ij) ɛ x) M n,ɛ,ij)x) ) 2.15)

6 368 Ivan Blank and Penelope Smit F M n,ɛ x), Dzn+1 ɛ x), zɛ n+1 xɛ ),x ɛ,n+ 1) ) F M n,ɛ x), Dz n,ɛ x), zn+1 ɛ xɛ ),x ɛ,n+ 1) ) = b i x)d i z ɛ n+1 x) z n,ɛ x) ) F M n,ɛ x), Dz n,ɛ x), zn+1 ɛ xɛ ),x ɛ,n+ 1) ) FM n,ɛ x), Dz n,ɛ x), z n,ɛ x ɛ ), x ɛ,n+ 1) ) = cx) zn+1 ɛ xɛ ) z n,ɛ x ɛ ) ). By te continuity assumption 1.8) we also know FM n,ɛ x), Dz n,ɛ x), z n,ɛ x ɛ ), x ɛ,n+ 1)) FM n,ɛ x), Dz n,ɛ x), z n,ɛ x ɛ ), x ɛ, n) σ 1 ɛ) + σ 2 ) 2.16) 2.17) 2.18) were te σ i are moduli of continuity. By defining cx) := cx) + 1/), w ɛ n x) := zɛ n+1 x) z n,ɛ x), by defining D ij w ɛ n := Mɛ n+1 M n,ɛx), and by using te equations beginning wit Equation 2.12) we can conclude a ij x)d ij w ɛ n x) + b ix)d i w ɛ n x) + cx) [ w ɛ n xɛ ) + z n,ɛ x ɛ ) z n,ɛ x ɛ ) ] z n x ɛ ) z n 1 x ɛ ) + σ 1 ɛ) + σ 2 ) for a.e. x. 2.19) We define a semiconvex function to be a function wic becomes convex if γ x 2 is added to te function, and γ is sufficiently large. In particular, by Teorem 2.4 iv) we know tat sup-convolutions are semiconvex, as is te function wn ɛ in Equation 2.19). Now we need te following results see p. 56 Teorem A.2 and p. 58 Lemma A.3 of [2]). Teorem 2.8 Aleksandrov s Teorem). Semiconvex functions are twice differentiable a.e. Lemma 2.9 Jensen s Lemma). Let ϕ : R n R be semiconvex, and let ˆx be a strict local maximum of ϕ. Forp R n, set ϕ p x) = ϕx)+ <p,x>. Ten for any r, δ > 0, te set K := { ) x B r ˆx : tere exists p Bδ suc tat ϕ p as a local max at x } as positive measure. We also state for future use te following estimate. Lemma 2.10 Discrete Gronwall Inequality). Assuming {v i }, {B i }, and {D i } are sequences of nonnegative numbers wic satisfy v i+1 B i v i + D i,weave [ ] n 1 n 1 n 1 v n v 0 B i + B j. 2.20) i=0 i=0 D i j=i+1 Wit tese results we can turn to te estimate of te maximum of w n x) := z n+1 x) z n x). 3. Rote limits are Lipscitz in time Teorem 3.1 Inductive Estimation). If F satisfies 1.7), ten we ave w n L ) w n 1 L ) + σ 2 ). 3.1)

7 Convergence of Rote s Metod for Fully Nonlinear Parabolic Equations 369 Furtermore, if we assume tat σ 2 ) C, ten for any fixed T > 0, we ave te Lipscitz estimate: w [T/] C, 3.2) L ) were C is independent of, and [T/] is te greatest integer less tan or equal to T/. Proof. We will first prove Equation 3.1). To start we will need a few assumptions: First tat F is differentiable so tat Equation 2.19) is valid, second tat w n as a strict maximum at ˆx, and finally tat our approximating functions, wn ɛ are twice differentiable at teir corresponding maxima wic we call ˆx ɛ. We will sow ow to avoid tese assumptions after proving te estimate in tis case. Because ˆx is a maximum, and wn ɛ is twice differentiable at ˆxɛ, Equation 2.19) becomes: c ˆx )[ wn ɛ ˆx ɛ ) + z n,ɛ ˆx ɛ ) )] z n ˆx ɛ ) ) z n 1 ˆxɛ z n,ɛ ˆxɛ + σ 1 ɛ) + σ 2 ). 3.3) Now by using Remark 2.7we see tat ˆx ɛ ) ˆx ɛ 2 ɛ z ɛ 1/2 n+1 + 2ɛ zn,ɛ ) 1/2, 3.4) and ten by sending ɛ to zero and using te Hölder regularity of te z n in space we can conclude: c ˆx ) ) ) w n ˆx w n ˆx + = c ˆx ) ) ) w n 1 ˆx w n ˆx + σ 2 ). 3.5) By using Equation 1.7) we see tat F r 0 and terefore c 0. So, since w n ˆx) 0, we ave w n ˆx ) wn 1 L ) + σ 2 ). 3.6) Obviously we can argue similarly for a negative minimum, but it remains to sow ow to eliminate our regularity assumptions. First of all we need stability of our approximations as we take smoot approximators F m, to our original F. Te desired stability can be found in Section 6 of [2]. See Remark 6.3 in [2] in particular.) In terms of applying te estimates we conclude for our F m s for our original F tere is no problem, because no matter ow badly beaved te a ij, b i, and c are in a given approximation to F, tey get trown away at te maximum of w n by considerations wic depend only on te signs of te terms involved. Stated differently, even if te derivatives of te approximations are diverging, tey are necessarily diverging in te rigt direction. Equation 3.3) is obtained from Equation 2.19) by using te nonpositivity of te second derivative and te vanising of te gradient of w n at its maximum.) Next, if w n as a maximum at ˆx but it is not te unique maximum, ten we look at te equation satisfied by w n x) := w n x) ˆɛ x ˆx 2, were ˆɛ >0 is arbitrarily small. Here we need te continuity assumptions on our function F given in 1.7).) Finally, to ensure twice differentiability of te wn ɛ at ˆxɛ we appeal to Jensen s Lemma 2.9 and Aleksandrov s Teorem 2.8. Now we need to prove Equation 3.2). We apply Lemma 2.10 to Equation 3.1) to get wit n =[T/] and by using Teorem 2.2 we are done. w n L w 0 L + nσ 2 ) 3.7) Teorem 3.2 Existence of Rote Limits). If σ 2 ) C ten te Rote limit exists after coosing an appropriate subsequence, and tese limits are locally Lipscitz in time, and locally Hölder in space.

8 370 Ivan Blank and Penelope Smit Remark 3.3 Uniqueness of te Rote Limit). In fact, we will be able to deduce uniqueness of te Rote limit and convergence along te original sequence witout taking subsequences by te results of te next section. Proof. Te last teorem sows tat te sequence of approximators are uniformly Lipscitz in time. In order to sow tat te approximators are equicontinuous in space, we observe tat our uniform Lipscitz estimate implies FD 2 z n+1,dz n+1,z n+1,x,n+1)) is uniformly bounded in L in te viscosity sense. Ten we invoke te spatial Hölder estimates known for fully nonlinear elliptic equations see Teorem 5.1 Part III of [9]). Finally, we can invoke Arzela-Ascoli to guarantee te existence of te limit after coosing a subsequence. 4. Rote limits are viscosity solutions Teorem 4.1 Rote Limits Are Viscosity Solutions). Te Rote limits are viscosity solutions. Proof. By symmetry, it will suffice to prove tat Rote limits are viscosity subsolutions. Suppose not. Ten tere exists an ɛ>0and a supersolution ϕ C 2,1 D) wic touces a given Rote limit U from above at a point x 0,t 0 ) and wic satisfies: ϕ t + F D 2 ϕ,dϕ,ϕ,x,t ) ɛ>0, 4.1) pointwise in a neigborood of x 0,t 0 ). Now let H be te set of wic belong to te subsequence wic converges to U. By adding a very small multiple of t t x x 0 2 to ϕ, by using te continuity of F, and by allowing a sligtly smaller but still positive) ɛ in Equation 4.1), we can assume witout loss of generality tat at least for a small neigborood of x 0,t 0 ) wic we will call N te only contact between ϕ and U is at x 0,t 0 ). Because of te uniform continuity of ϕ t tere exists a neigborood Ñ of x 0,t 0 ) suc tat for a sufficiently small >0, and for all x, t) Ñ we ave te estimate ϕx,t) ϕx,t ) ϕ t x, t) ɛ 10 for all, wit H. For te remainder of tis proof we will abuse notation by not mentioning tat always belongs to H.) At tis point let N denote a compact set of te form B r x 0 ) [t 0 γ,t 0 + γ ] wic is contained in N Ñ. Let 4.2) S := N \{B r/2 x 0 ) t 0 γ/2,t 0 + γ/2)}. 4.3) Because ϕ>uon S, tere exists a ɛ >0 suc tat ϕ U + ɛ on S. By te uniform convergence of U to U, for any δ 0, ɛ/3) we can be sure tat by srinking if necessary we ave ϕ>u + ɛ δ on S wile ϕx 0,t 0 ) U x 0,t 0 ) <δ/4 4.4) for all. Coose δ sufficiently small to guarantee tat ω R δ)<ɛ/2, 4.5) were ω R is te modulus given in Equation 1.7) and ɛ is taken from Equation 4.1). Fix <min{, γ /10}, and sufficiently small to guarantee tat ϕx 0,t) U x 0,t) <δ/2 4.6)

9 Convergence of Rote s Metod for Fully Nonlinear Parabolic Equations 371 for any t [t 0, t 0 ]. Here we need te second alf of Equation 4.4) and uniform continuity in time of ϕx 0,t) U x 0,t)as 0. Te continuity is guaranteed by te previous teorem.) If we let ϕ = ϕ + c, for any c wit c <δ, ten by using Equations 1.7) and 4.5) we will ave ϕ t + F D 2 ϕ, D ϕ, ϕ, x, t ) ɛ ω R δ)>ɛ/2 > ) Now we coose c so tat te minimum of ϕ U on te set N {x, t) : t N} is equal to zero. In oter words, ϕ is allowed to be less tan U between te mes values of t, but wen considering only mes values of t, we can say tat ϕ touces U from above. t=m 1) t=m t=m+1) x ~ 0 U <_ ϕ ~ x, k) x, k) wen k is a wole number U ϕ ~ t U <_ ϕ ~ but old in general. does not FIGURE 1 x Because of Equation 4.6), we can be sure tat c δ/2. Let x 0, m) be a point were ϕ = U. By using Equation 4.4) we know tat x 0, m) B r/2 x 0 ) t 0 γ/2,t 0 + γ/2). If we return to our z n we now ave te following setting: a) ϕx, m) touces z m x) from above at x 0. b) ϕx,m 1)) z m 1 x) in B r/2 x 0 ). Because <γ/10 we know tat B r/2 x 0 ) {t = m 1)} N.) Because x 0, m) Ñ we ave [see Equation 4.2)] Recall tat z m is a viscosity solution of ϕ x 0,m) ϕ x 0,m 1)) ϕ t z m z m 1 x 0,m) ɛ ) + F D 2 z m,dz m,z m,x,m ) = ) Since ϕx, m) touces z m x) from above at x 0, we conclude tat ϕ z m 1 x 0 ) + F D 2 ϕ,d ϕ, ϕ, x 0,m ) )

10 372 Ivan Blank and Penelope Smit Now we combine Equations 4.7), 4.10), and 4.8) to get ɛ/2 < ϕ t x 0 ) + F D 2 ϕ,d ϕ, ϕ, x 0,m ) ϕ t x 0 ) + z m 1 x 0 ) ϕ x 0,m) ϕ t x 0 ) + ϕ x 0,m 1)) ϕ x 0,m) ɛ/10 wic is a contradiction. Now by invoking Jensen s uniqueness teorem for viscosity solutions we get te following corollary. Corollary 4.2 Te Rote Limit Exists and is Unique). Te Rote limit exists and is unique. Corollary 4.3 Lipscitz Regularity in Time). Lipscitz in time. Te viscosity solution to Equation 1.1) is Proof. Simply combine te tree previous results. References [1] Caffarelli, L. A., Crandall, M. G., Kocan, M., and Świȩc, A. On viscosity solutions of fully nonlinear equations wit measurable ingredients, Comm. Pure Appl. Mat. 494), , 1996). [2] Crandall, M. G., Isii, H., and Lions, P.-L. User s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Mat. Soc. 271), 1 67, 1992). [3] Jensen, R. Te maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arc. Ration. Mec. Anal. 101, 1 27, 1988). [4] Jensen, R., Lions, P.-L., and Souganidis, P. E. A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations, Proc. Amer. Mat. Soc. 102, , 1988). [5] Kikuci, N. Hölder estimates of solutions to difference partial differential equations of elliptic-parabolic type, J. Geom. Anal. 111), 77 89, 2001). [6] Kikuci, N. and Kačur, J. Convergence of Rote s metod in Hölder spaces, Appl. Mat. 485), , 2003). [7] Pluscke, V. Local solutions to quasilinear parabolic equations witout growt restrictions, Z. Anal. Anwendungen 15, , 1996). [8] Rektorys, K. Te Metod of Discretization in Time and Partial Differential Equations, D. Reidel, 1982). [9] Trudinger, N. S. Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Rev. Mat. Iberoamericana 43/4), , 1988). Received June 2, 2004 Department of Matematics, Worcester Polytecnic Institute, Worcester, MA blanki@wpi.edu Department of Matematics, Leig University, Betleem, PA Penny314@aol.com Communicated by David Jerison