A stability result for dirichlet problem of the first-order Hamilton-Jacobi equations

A STABILITY RESULT FOR DIRICHLET PROBLEM OF THE FIRST-ORDER

HAMILTON-JACOBI EQUATIONS

BUM IL HONG

1. Introduction

It is well known that Dirichlet problem of the first-order Hamilton- Jacobi equations

(H-J) u(x)

+

does not have a classical solution even though the Hamiltonian H is smooth. Therefore it is worth for us to deal with non-smooth solu- tions if we want a solution of (H-J) which satisfy the equations almost everywhere. The theory of first-order partial differential equations of Hamilton-Jacobi type has substantially developed with the introduc- tion by Crandall and Lions [1] ofthe class of viscosity solutions, which turns out to be the correct class of generalized solutions for such type of equations. They also showed the uniqueness of generalized solutions that satisfy a so-called "viscosity" condition. The book by Lions [6]

and papers by Jensen and Souganidis [5] and Souganidis [7] provided a view of the scope of the references to much of the recent literature.

Cauchy problem of Hamilton-Jacobi equations was studied by Crandall and Lions [2]. Hong [3], [4] showed some regularity results for Cauchy problem of Hamilton-Jacobi equations.

This paper is organized as follows. In chapter 2, we give the defi- nition of viscosity solutions of (H-J) in several space dimensions. We also review both uniqueness and stability of 'the viscosity solutions.

In chapter 3, we prove the following theorem that is the main result of this paper.

Received August 12, 1994.

1991 AMS Subject Classification: 70H20.

Key words and phrases: Hamilton-Jacobi equations.

This research was supported by Hoseo University.

THEOREM 1.1. Let bounded and continuous functions u and v be the viscosity solutions of

u(x)

+

respectively, whereHI andH² areLipscbitz continuous. Then

This stability result gives us an error estimate if we approximate the viscosity solutions of. (H-J),

2. Viscosity solutions of (H-J)

The general reference for this section is [6]. To repeat, one cannot in general find a classical solution of (H-J) onJR., while bounded Lipschitz continuous "generalized solutions" in the almost-everywhere sense exist but are not unique. For example,

u(x)

+

has two solutions that satisfy the equation almost everywhere, namely,

u = 1and

if x ::;Xo, otherwise,

which satisfies the equation classically except on the lines x = Xofor all Xo E JR.. Moreover, if u and v are generalized solutions of (H-J), then so are mineu, v) and max(u, v). In fact, if the problem is nonlinear, one can expect infinitely many generalized solutions. Crandall and Lions [1] resolved the uniqueness problem by introducing a notion of viscosity.

DEFINITION 2.1. A viscl!sity subsolution (respectively, supers~lu­

tion) of (H-J) with HE C(lR^N) is a bounded function u E C(lR^N) such that for every 4> E Cl(lRN ):

(2.1.1)

(respectively,

(2.1.2)

Ifx0 is &local maximum point ofu - 4> onlRN

textthen u(xo)

+

Ifx0 is a local minimum point ofu - 4>on lRN ,

then u(xo)

+

DEFINITION 2.2. A viscosity solution of (H-J) is a bounded function

u E C(lR^N) for which both (2.1.1) and (2.1.2) hold (i.e. u is both a viscosity subsolution and a viscosity supersolution).

REMARK. Ifu is a bounded classical solution of (H-J), then it is a viscosity solution, and ifuis a viscosity solution of (H-J), then u(xo)+

H("Vu(xo» = f(xo) at any point (xo) whereu is differentiable.

LEMMA 2.3. Suppose that u and v are viscosity solutions of

and

w(x)

+

respectively. Then

Proof. See [1] and [6] .

. 3. Stability of two viscosity solutions

We prove that viscosity solutions are stable under changes in the nonlinear Hamiltonians H as well as changes in the functions f. To prove that, we prepare two lemmas.

LEMMA 3.1. Let u andw be the viscosity solutions of u(x)+HI(Vu(x» = f(x), x E]RN, and

respectively, whereHI and H₂ axeLipscbitz continuous. Then

Theorem1.1 follows by combining Lemma 2.3 and Lemma 3.l.

LEMMA 3.2. Assume that u and w axe in Lemma 3.1. Let 1](z) be a smooth nonnegative function on ]R such that 1](-z) = 1](z),

O~ 17(Z) ~ 1,17(0)= land 1](Z)

=

u:= sup(u(x) - w(x» >

o.

m.^N

For anyf >0, define

'I/J(x,y) = u(x) - w(y)+(3M+"2)!3e(X - y),u

where!3e(x) is defined on]RN by!3e(x) = II~I1](7). If (3.2.1) sup lu(x)1 and

Ixl~R

sup Iw(x)1 --+ 0 as R ^{--+ 00,}

Ixl~R

then there exists a point (xo, Yo) E ]RN X ]RN such that 'I/J(xo, yo) ~

'I/J(x,y) on]RN X ]RN.

Proof. Fix f > O. Ifthere is a sequence {(Xi, Yi)}i>1 in ]RN X ]RN such that

(3.2.2)

then (Xi, Yi) remains bounded by the following arguments.

First,

sup t/J ~u(x) - w(x) +(3M+-2u)(3E(X - x)

Jl',NxJl',N

= u(x) - w(x)+3M+

i

Therefore,

(3.2.3)

sup t/J ~ sup(u(x) - w(x))

+

+-

V x V V 2

=u+3M+2"u

=3M+2"u.3

If(3E(X - y) = 0, then

t/J(x,y) = u(x) - w(y)

~2M.

Hence, (3.2.2) implies that (3E(Xi-Yi) > 0 for large i, whence /Xi-Yil <

e. If IXil ^{--+ 00} and IYil ^--+ 00, then

This contradicts (3.2.2) and (3.2.3). Therefore, {(Xi, Yi)

h>l

Let (xo, Yo) be the limit of the above subsequence. This completes the proof. 0

Proof of Lemma 3.1. We will prove that u, defined in Lemma 3.2, satisfies

By symmetry inu and w, we see that this implies

We first assume that

(A) sup lu(x)1 and sup Iw(x)l-t0 as R -t ^00.

Ixl~R Ixl~R

If^(J' = 0, then we are done. Otherwise, by Lemma 3.2, for any € > 0 we can find a point (xo, Yo) ERN x RN such that

u(x) - (w(yo) - (3M

+ i

attains a local maximum at (xo), whence (3.1.1)

Similarly

attains local maximum at Y = Yo and therefore

attains local minimum at Y = Yo. By the definition of the viscosity solution,

(3.1.2)

Combining (3.1.1) and (3.1.2) gives (3.1.3)

u(xo) - w(yo) S H2 (-(3M

+

- Hl ((3M

+ i

For all x ERN,

(3.1.4)

u( x) - w( x)+3M+

"2

:5t/J(xo, Yo)

:5u(xo) - w(yo) +3M+ 2'u

Therefore, by 3.1.3 and 3.1.4,

U:5H1(^-(3M+

i)V

i)V

+f(xo) - f(yo)

:5¹¹HI (-(3M

+ i)V

- H2(-(3M +

i)V

wherewI(f) is the modulus of continuity off. Since f is arbitrary,

We now drop the assumption (A). For R > 0, let p(x) be a smooth function having support in the ballB(O, R+l) = {x E ]RN Ilxl :5R+l}

such that p(x)

=

=

wP(x) = p(x)w(x). Then the corresponding viscosity solutions uP(x) and w^P(x) have the following properties:

uP(x) = u(x) on Ixl <R-IH1ILip and wP(x)= w(x) on Ixl < R-IH2ILip;

see [6]. Let L = ma.x{IH1ILip,IH2ILip}, ^Then

max lu(x) - w(x)1 = max luP(x) - wP(x)1 Ixl<R-L Ixl<R-L

:5I1H1 - H2I1LOO(ll\N) by the previous argument.

Hence, letting R ^-+ 00, we have

This completes the proof. 0 We now prove Theorem 1.1.

Proof of Theorem 1.1. In addition to the equation in the statement of Theorem 1.1, consider

w(x)

+

Then, by Lemma 3.1 and Lemma 2.3,

lI^{u -} vIlLoo(]iN)

:s

+

:s

+

This completes the proof. 0

References

1.M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equa- tions, Tran. Amer. Math. Soc., 211 (1983), 1-52.

2. M. G. Crandall and P. L. Lions, Two approximations of solutions of Hamil- ton-Jacobi equations, Math. Comp., 43 (1984), 1-19.

3. B.I. Hong, On Regularity and approximation for Hamilton-Jacobi equations, J. Korean Math. Soc. 301 (1993), 93-104.

4. B.I. Hong, Regularity for Hamilton-Jacobi equations via approximation, Bull.

Austral. Math. Soc. 51 (1995), 195-213.

5. R. Jensen and P.E.Souganidis, A regularity for Viscosity solutions of Hamil- ton-Jacobi equations in one space demension, Tran. Amer. Math. ·Soc., 301 (1987), 137-147.

6. P. L. Lions, Generalized solutions of Hamilton-Jacobi equations, Pitman Re- search Notes Series, Pitman, London, 1982.

7. P.E.Souganidis, Existence of Viscosity solutions of Hamilton-Jacobi equations in one space demension, J. differential Equations, 56 (1985), 345-390.

Department of Mathematics Hoseo University

Choongnam 336-795, Korea