Good solutions of fully nonlinear parabolic equations

Applied Mathematics

Good solutions of fully nonlinear parabolic equations

?

Tran Duc Van

1

and Tran Van Bang

2

1 Hanoi Institute of Mathematics, P.O. Box 631 Bo Ho, 10.000 Hanoi, Vietnam

e-mail:tdvan@thevinh.ncst.ac.vn

2 Hanoi Pedagogical University, Xuan hoa, Hanoi, Vietnam

Received: April 02, 2002

Summary.

In this paper, we introduce the notion of a "good" so-lution of a fully nonlinear parabolic equation and show that "good" solutions are equivalent to Lp_{-viscosity solutions of such equations.}

The results here generalize the ones in 8] about "good" solutions of fully nonlinear elliptic equations. We give here an explicit construc-tion of parabolic equaconstruc-tions withLp_{-strong solutions that approximate}

some nonlinear parabolic equation and itsLp_{-viscosity solution.}

Key words:

Lp_{-viscosity solutions, good solutions, strong}

solu-tions, fully nonlinear parabolic equations

Mathematics Subject Classication (1991): 35J60, 35J65, 49L25

1. Introduction

The notion of viscosity solutions which was introduced in 1981 by M. G. Crandall and P. L. Lions for the general Hamilton { Jacobi equations plays an important role in the theory of partial dieren-tial equations 2]. For convex Hamiltonians, the viscosity solution characterized by a semiconcave stability condition was rst intro-duced by S. N. Kruzkov 9]. There is an enormous activity which is based on these studies. Many fundamental results about viscosity solutions for partial dierential equations of rst and second order have been obtained by M. G. Crandall, L. C. Evans, P. L. Lions,

? _{This research was supported in part by National Council on Natural Science,}

H. Ishii, R. Jensen, P. E. Souganidis, M. Bardi, G. Barles, E. N. Bar-ron, M. Kocan, A. Swiech and so on. There results concern existence and uniqueness theorems, properties of solutions and applications.

We note that a comprehensive treatment of Lp_{-theory for fully}

nonlinear parabolic equations is presented in recent paper by M. G. Crandall, M. Kocan and A. Swiech 5].

Following 8] we introduce the notion of a good solution of a fully nonlinear parabolic equation and also prove some results that are sim-ilar to the ones for good solutions of fully nonlinear elliptic equations. The reader is referred to 1-8] for the modern theory of nonlinear el-liptic and parabolic equations.

In this paper, Rn(n2) is always assumed to be a bounded domain and satisfy an uniform exterior cone condition, Q =  (0T], (T >0), S(n) be the set of symmetricnnreal matrices. For

X2S(n) we callX

+,X;respectively the positive, negative parts of

X and also dene kXk= tr(X

+) + tr(X;), where tr(X) is the trace of X.

Let X 2 S(n), 0 <   . The Pucci extremal operators are dened as follows P +(X) = ;tr(X +) +tr(X;) P ;(X) = ;tr(X +) +tr(X;): Consider the nonlinear parabolic partial dierential equations of the form

(1:1) ut+G(xtuDuD2u) = 0 in Q where G : QRRnS(n) ! R, Du = (ux

1:::uxn), D 2u = (uxixj) correspond to the spatial gradient and Hessian matrices of u

respectively.

We will always assume that G(xtrpX) is continuous in (rp,

X) with modulus independent of (xt)2Qand is jointly measurable in all variables (xtrpX).

We also require that G of (1.1) satisfy the following structure conditions (1:2) jG(xtrpX);G(xtrqX)j jp;qj (1:3) P ;(X ;Y)G(xtrpX);G(xtrpY)P +(X ;Y) for all (xt)2 Q,r2 R,p, q 2Rn, X,Y 2S(n), and , are positive constants which are xed for all time. Thus (1.3) amounts to (1:4) G(xtrpX)G(xtrpY) whenever XY:

As for the dependence onr, we assume that (1:5) 8 < : r!G(xtrpX) is uniformly continuous, uniformly for (xt)2Q jrj+jpj+kXkR(R >0) (1:6) r!G(xtrpX) is nondecreasing:

We sometimes rewrite (1.1) as follows

(1:7) ut+F(xtuDuD2u) =f(xt) by setting

f(xt) =;G(xt000) F(xtrpX) =G(xtrpX)+f(xt) so that

(1:8) F(xt000)0: Regarding the behavior off, we require that (1:9) f 2Lp(Q) p > p

0

where p0 < n+ 1 is a constant such that the generalized maximum principle for parabolic equations holds forp > p0.

Clearly, F of (1.7) will satisfy versions of (1.2){(1.6) whenever G

of (1.1) does, and vice versa.

The notions of solutions which are used in this work are Lp

-viscosity subsolutions,Lp_{-viscosity supersolutions,L}p_-viscosity

solu-tions,Lp_{-strong solutions, pointwise a. e. solutions (see 5]).}

2. Good solutions

Denition 1.

We say that the functionsG1,G2,:::,Gm,:::satisfy

structure conditions uniformly inm if (1.2), (1.3), (1.5) are satised uniformly inmwith the same xed,, and if jGm(xt000)j

g(xt) for someg2Lp(Q).

Denition 2.

We say that u 2 C(Q) is a good solution of (1.1) if

there existGm satisfying structure conditions uniformly inm and an

Lp_{-strong solution um of ut+Gm = 0 in Q such that um} ! u in

C(Q) and Gm(xtrpX)!G(xtrpX) for a. e. (xt)2 Q and

First, we are going to construct anLp_{-strong solution of the}

Cauchy { Dirichlet problem (2:1) ut;u+G(xtuDuD

2u) = 0 in Q u= on @

pQ

whenG is bounded. This will turn out to be an important construc-tion in proving that anLp_{-viscosity solution of (1.1) is a good solution}

of (1.1) as well.

Theorem 1.

LetG:QRRnS(n)!Rbe measurable,

bound-ed and satisfy (1.2), (1.4), and (1.5),  2 C(@pQ). Then

the Cauchy { Dirichlet problem (2.1) has an Lp_{-strong solution u} 2

C(Q)\W 21p loc (Q).

Proof. We are going to solve (2.1) by the xed point method. For any given v2C(Q) we consider the Cauchy { Dirichlet problem

(2:2) ut;u+G(xtv(xt)DuD

2u) = 0 inQ u=on @

pQ:

Since the equation in (2.2) is independent ofu, from 3, Theorem 4.5 and Remark 4.8] it follows that (2.2) has an Lp_{-viscosity solution}

in C(Q) which we denote by Tv. By 5, Propositions 3.2 and 4.1],

Tv is parabolically twice pointwise dierentiable a. e., a pointwise a. e. solution and, therefore, anLp_{-viscosity solution of the Cauchy {}

Dirichlet problem

(2:3) ut;u=;g(xt) inQ u= on@pQ whereg(xt) =G(xtv(xt)D(Tv)(xt)D2(Tv)(xt)) (g

2L 1(Q) since G is bounded). However, it is clear that (2.3) has a unique

Lp_{-strong solution which must coincide with Tv, and it follows that}

Tv 2 W 21p

loc (Q). In particular, Tv is a unique Lp-strong solution of (2.3).

On the other hand, sincegis bounded it follows 3, Proposition 4.6 and Remark 4.8] that if Ris large enough then T is a compact map-ping in the ball with the center at 0 and the radiusRinC(Q). Thus, by Schaefer's xed point theorem,T has a xed point or (2.1) has a

Lp_{-strong solution and the proof is complete.} ut

Theorem 2. (Existence of good solutions)

Let 2C(@pQ), let

F be measurable and satisfy (1.2), (1.3), (1.5), (1.6), (1.8), andf 2

Lp_{(Q). Then the Cauchy { Dirichlet problem}

(2:4) ut+F(xtuDuD2u) =f(xt) in Q u= on@

pQ

Proof. Noting that under the same conditions we know that (2.4) also has anLp_{-viscosity solution. We will use Theorem 1 to construct}

good solutions of (2.4). For this we rewrite the dierential equation in (2.4) as follows ut;u+G(xtuDuD 2u) = 0 where G(xtrpX) =tr(X) +F(xtrpX);f(xt): Clearly,Gsatises (1.4).

Without loss of generality we may replace (2.4) by

ut;u+G(xtuDuD

2u) = 0 inQ u= on @

pQ:

For m= 12:::, consider truncating functions

m(r) = max(min(rm);m) and an approximating Cauchy { Dirichlet problem (2:5) ut;u+ m(G(xtuDuD

2u)) = 0 inQ u=on@

pQ:

Since m(G(xtuDuD2u)) satises the condition of Theorem 1 and G does (1.4), by similar argument, we assert that (2.5) has a unique Lp_{-strong solution um} 2 C(Q) \W

21p

loc (Q). The family of equations satises structure conditions uniformly in m, and by 3, Proposition 4.6 and Remark 4.8] the setfumgis precompact inC(Q). Therefore, passing to a subsequence if necessary, we can assume that

um !u in C(Q). The function u2 C(Q) is a desired good solution since the approximations in (2.5) obviously converge toG a. e. and the proof is complete. ut

Next, we consider the Cauchy { Dirichlet problem (2:6) ut+F(xtDuD2u) =f(xt) inQ u= on@

pQ:

We will prove that every Lp_{-viscosity solution of (2.6) is a good}

solution. The fact that good solutions are Lp_{-viscosity solutions is}

obvious from the denition of good solution. First, we remind the

sup-convolution w" _{and inf-convolutionw" of a function w:Q}!R,

" >0, as w"_{(xt) = sup} y2  w(yt); 1 2"jx;yj 2   w"(xt) = inf_y 2  w(yt_{) + 12"}jx;yj 2   (xt)2Q: Now for" >0, >0, we put

w"_{= (w}"+)

 w" = (w"+)

Theorem 3.

LetF be measurable and satisfy (1.2), (1.3), (1.8), letf

satisfy (1.9),2C(@pQ). Then everyLp-viscosity solution of (2.6) is

a good solution, i. e. there are a sequence of operatorsFm independent

of u, satisfying (1.2), (1.3), and (1.8), a sequencefm 2Lp(Q) and a

sequence um 2C(Q)\W 21p

loc (Q) of Lp-strong solutions of

(um)t+Fm(xtDumD2u

m) =fm(xt) in Q

such that

(2:7) um !u in C(Q)

(2:8) Fm(xtpX)!F(xtpX)

for a. e. (xt)2Q and for all(pX)2RnS(n), and

(2:9) fm !f in L

p_(Q):

Proof. We will use the method proposed in 8] in order to prove The-orem 3. Indeed we repeat all steps of the proof of TheThe-orem 3.1 in 8] for elliptic equations. Assume thatu is anLp_{-viscosity solution of}

(2.6).

Step 0.

Firstly we x a countable, dense in RnS(n) sequence (piXi)2RnS(n),i= 12:::.

Step 1.

Choose a sequence ~fm 2 C(Q)\L

1(Q) such that 

m =

kf ;f~mkLp (Q)

!0. By 5, Theorem 2.8], the following problem

Wt+P

;(D2W)

; jDWj=f ;f~m inQ W = 0 on@pQ has a uniqueLp_{-strong solutionW and}kWkL1

(Q)

Cm. Letum =

u;W. Hence um is an Lp-viscosity solution of (2:10) (um)t+F(xtDumD2u m)f~m(xt) in Q and (2:11) ku;umkL1 (Q)= kWkL1 (Q) Cm: Similarly, solving Wt+P +(D2W) + jDWj=f ;f~m inQ W = 0 on@pQ and setting um = u;W, we conclude that um is an Lp-viscosity solution of

(um)t+F(xtDumD2u

and

(2:12) ku;umkL1 (Q)

Cm: Moreover,um=u==um on@pQ:

Step 2.

Letm b be a subdomain ofwith a smooth boundary and such that

(2:13) if x2nm then d(x@) 1

m:

Let Qm = m (0T]. It follows from the proof of Lemma 1.2 in 6] and 8] that, for every suciently small " > 0, one can choose a suitable =(") such that u+

m = (um)" 2W

211(Q

m) and it is an

Lp_{-strong solution of a perturbed version of (2.10), namely, for a. e.}

(xt)2Qm, (2:14) (u+ m)t+Fm(xtDu+ m(xt)D2u+ m(xt))fm(xt) where (2:15) Fm(xtpX) =F(T+ mxtpX) fm(xt) = ~fm(T+ mxt) withT+ mx=x+"Du+ m.

Similarly, we conclude that u;

m = (um)"2W 211(Q m), and (2:16) (u; m)t+Fm(xtDu; m(xt)D2u; m(xt))f m(xt) for a. e. (xt)2Qm, where (2:17) Fm(xtpX) =F(T; mxtpX) f_m(xt) = ~fm(T; mxt) withT; mx=x;"Du ; m.

Also there is m>0 independent of"such that

(2:18) DT

m mI for a. e. x2m whereI is the (nn) identity matrix.

It follows from (2.18) thatFm andFm are measurable. Moreover,

decreasing"if necessary, we can achieve that (2:19) kfm;f~mkLp (Qm)  1 m kf m;f~mkLp (Qm)  1 m and (2:20) Z Qm jF(T  mxtpiXi);F(xtpiXi)jdxdt 1 m i= 12:::m:

Further, without loss of generality we may also assume that (2:21) kum;u + mkL1 (Qm) Cm kum;u ; mkL1 (Qm) Cm: Finally, redening u; m = u; m;3Cm and u + m = u+ m+ 3Cm we see

that (2.14) and (2.16) still hold, while by (2.11), (2.12), and (2.21) (2:22) u;5Cmu

;

m u;Cm u+Cmu +

m u+5CmonQm:

Step 3.

We now establish some limiting properties of the approxi-mations constructed in Step 2, which will be needed later in Step 6.

From (2.20), for every (piXi) and Q0

bQ, we have Z Q0 jF(T  mxtpiXi);F(xtpiXi)jdxdt!0 asm!1: Thus, there exists a subsequence mk and a null setN Qsuch that

for all (xt)2QnN and i

F(T

mkxtpiXi)!F(xtpiXi) ask!1:

Using structure conditions (1.2) and (1.3) we can generalize this con-vergence, namely, for all (xt)2QnN and (pX)2RnS(n) (2:23) F(T

mkxtpX)!F(xtpX) ask!1:

Step 4.

Next choose a constant Mmm such that for a. e. (xt)2

Qm (2:24) jFm(xtDu + m(xt)D2u+ m(xt)) +u+ m(xt)jMm jFm(xtDu ; m(xt)D2u; m(xt)) +u; m(xt)jMm: Let Gm(xtpX) =;tr(X)+ Mm(Fm(xtpX)+tr(X)): It follows from (2.14) and (2.24) that

(u+ m)t+Gm(xtDu+ m(xt)D2u+ m(xt))fm(xt) for a. e. (xt)2Qm:Similarly, (2:25) (u; m)t+Gm(xtDu; m(xt)D2u; m(xt))f m(xt) for a. e. (xt)2Qm, where Gm(xtpX) =;tr(X)+ Mm(Fm(xtpX)+tr(X)):

Let:QmR!01] be a continuous function such that (xts) =  0 if u+ m(xt)s 1 if u; m(xt)s: This can be done due to (2.22). Dene

Hm(xtrpX) =(xtr)Gm(xtpX) +(1;(xtr))Gm(xtpX) =;tr(X) +(xtr) Mm(Fm(xtpX)+tr(X)) +(1;(xtr)) Mm(Fm(xtpX)+tr(X)) (2:26) hm(xtr) =(xtr)fm(xt) + (1;(xtr))f m(xt)

and consider the Cauchy { Dirichlet problem (2:27) vt+Hm(xtvDvD2v) = h

m(xtv)inQm v=uon@pQm:

All assumptions of Theorem 1 are satised and thus (2.27) has an

Lp_{-strong solution vm}2C(Qm)\W 21p

loc (Qm). Now we dene

Gm(xtpX) =Hm(xtvm(xt)pX)

gm(xt) =hm(xtvm(xt))

so thatvm solves

(vm)t+Gm(xtDvm(xt)D2v

m(xt)) =gm(xt)

for a. e. (xt)2Qm,vm =u on@pQm. Observe that by construction (2:28) kgmkL1

(Qm)

kf~mkL1 (Q)

jtr(X)+Gm(xtpX)jMm: Moreover, from (2.19) and (2.26) we have

(2:29) kgm;f~mkLp (Qm) kfm;f~mkLp (Qm)+ kf m;f~mkLp (Qm)  2 m:

Next we claim that

(2:30) vmu + m+ 8Cm onQm: To show (2.30) we considerQ+ m=f(xt)2Qm:vm(xt)> u + m(xt)g. Then (xtvm(xt))0 onQ + m. Therefore, gm(xt) =f_m(xt) andGm(xtpX) =Gm(xtpX)

for (xt) 2 Q +

m. It follows that vm is an Lp-strong solution of ut+

G_m =f_m on Q+

m. Since by (2.25)u;

m is a supersolution of the same

equation, by the minimum principle and (2.22) we have inf Q+ m(u ; m;vm) inf @pQ+ m(u ; m;vm) min( inf @pQ+ mn@pQm (u; m;u + m) inf_@pQm(u; m;u));10Cm: Use of (2.22) again yields (2.30).

A symmetric argument shows that

(2:31) vmu

;

m;8Cm onQm: Summarizing (2.30), (2.31), and (2.22) together gives (2:32) u;13Cmvm u+ 13Cm on Qm:

Step 5.

We now extend gm and Gm to the whole Qas follows

fm(xt) =  gm(xt) for (xt)2Qm 0 for (xt)2QnQm Fm(xtpX) =  Gm(xtpX) for (xt)2Qm ;tr(X) for (xt)2QnQm:

Taking into account (2.28), by Theorem 1, the Cauchy { Dirichlet problem

ut+Fm(xtDuD2u) =f

m inQ u=on @pQ

has a unique Lp_{-strong solution um} 2 C(Q)\W 21p

loc (Q). Moreover, it is similar to Remark 4.3 in 3], there exists a modulus of continuity

determined only by the parameters of the cone condition , ,n,

p, ,kfmkLp

(Q), the modulus of continuity of, diam(Q) such that (2:33) jum(xt);(ys)j(jx;yj+jt;sj)

(xt)2Q, (ys)2@pQ.

Step 6.

Denote by u the modulus of continuity of u on Q. From

(2.13) and (2.33) we conclude that (2:34) jum(xt);u(xt)j( 1

m) +u( 1m) (xt)2QnQm: Since bothum andvm solve the same equationut+Gm=gm inQm

and vm=u on@pQm, by the maximum principle and (2.34) we have

sup (xt)2Qm jum;vmj sup (xt)2@pQm jum;uj( 1 m) +u( 1m):

By (2.32) this yields sup (xt)2Qm

jum;uj( 1

m) +u( 1m) + 13Cm:

Using (2.34) again we conclude that sup

(xt)2Q

jum;uj( 1

m) +u( 1m) + 13Cm

and therefore (2.7) follows. From (2.29) kfm;fkLp (Q) kfkLp (QnQm)+ kgm;f~mkLp (Qm) +kf;f~mkLp (Q) kfkLp (QnQm)+ 2 m +m

which, together with (2.13), establishes (2.9) along a subsequence. Finally we show that the convergence in (2.8) holds along a sub-sequence mk constructed in Step 3. Since Fm is just a convex

com-bination of truncations ofFmFm dened in (2.15) and (2.17), from

(2.23) it follows easily that for all (xt)2QnN, (pX)2RnS(n),

Fmk(xtpX)!F(xtpX) as k!1 or (2.8) holds and the proof is complete. ut

Acknowledgment.

The rst author thanks Professor A. Swiech for sending him the preprint 8] before its publication in Proceedings of AMS.

References

1. Caarelli, L. A. and Cabre, X.(1995):FullyNonlinearEllipticEquations,

Prov-idence, RI: American Mathematical Society.

2. Crandall, M. G. and Lions, P. L. (1983): Viscosity solutions of Hamilton { Jacobi equations, Trans. Amer. Math. Soc.,277, 1{42.

3. Crandall, M. G, Kocan, M., Lions, P. L., and Swiech, A. (1999): Existence re-sults for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electronic J. Dierential Equations, 24, 1{20.

4. Crandall, M. G., Ishii, H., and Lions, P. L. (1992): User's guide to viscosity solutions of second order partial dierential equations, Bull. Amer. Math. Soc.,

27, 1{67.

5. Crandall, M. G., Kocan, M., and Swiech, A. (2000):Lp-theory for fully

non-linear uniformly parabolic equations, Commun. in PDEs,25, 1997{2053.

6. Crandall, M. G., Fox, K., Kocan, M., and Swiech, A. (1998): Remarks on nonlinear uniformly parabolic equations, Indi. Univ. Math. Jour., 47, 1293{

7. Evans, L. C. (1998):PartialDierentialEquations, Providence, RI: American

Mathematical Society.

8. Jensen, R., Kocan, M., and Swiech, A. (2002): Good and viscosity solutions of fully nonlinear elliptic equations, Proc. Amer. Math. Soc.,130, 533-542.

9. Kruzkov, S. N. (1964): The Cauchy problem in the large for nonlinear equations and for certain quasilinear systems of the rst order with several variables, Soviet. Math. Dokl.,5, 493{496.