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340

small r\

> 0. Due to

< 0, the

ODEs

(8)

imply

84 vi{\)4-v

cf -z4 f

> owi-*"*

(

-<°^

(A)

- ^,

^"(A)4 ZjCfo

(A)

Z+

) ^max^max

j _

4ax)

-D^'(A)

* d?>

- d?

{

hence

_a»c|

with o?

max

msx

, V

+,*•»"•'.

Consequently,

cf(x)

c[(-6

)e-

(z)

cf{Q)e-

aMx

[0,8

By means

of (12) we

obtain

(0)>e-°

aTI

-77—

hence there

such that

-a

whenever

r\ < rj

. To

pass

to the

membrane,

need

estimate from above

for

Exploitation

(6),

(3) and c^

eg*

(A)

well

> 0

yields

^k(0)e

aZk

l(0)^(e

azt

cf(0)z

(e°*'

1)for any I.

(14)

To obtain an appropriate index

notice that

cf (0)

< £

-cf (0)

= J2

4cf(0)

n^maxcf

(0).

Zk<0 z

Thus there

is an

index

such that

0 and

cf (0)

(Oj/nz+^j.

Therefore

341

due to (14), hence if we start with n < n

then e

< K by (13) with some K > 0

independent of

[0,1].

Consequently,

cf (0) = cf (OK" > cf (0)K->™* >

"_ K--.

(l +

max

)

Since

> 0, this finally implies

V>c?> cf(S

) >

K->™ -

r,~M

for all 0 < r, <

zn^i -t-

max

a contradiction for sufficiently small n > 0.

Therefore the claim holds, hence

d(/-G,n,o) = d(/-if(-,i),n,o) = d(7-if(-,o),n,o).

The case A = 0 corresponds to D\ — 1, iP = v

and c^ = 0, where the solution is

given by

F P M P R P I A V y.

C = C , C =

, C" =

(= : : C

This yields H(c

. 0) = c

/a, hence (eventually after adjustment of

rj)

d(I - H(-,0), ft, 0) = d((l + -)/ - c, ft, 0) = 1.

Consequently there is at least one fixed-point of G, i.e. a solution of the stationary

problem. •

Although uniqueness is to be expected from the physical viewpoint, this has not yet

been proven mathematically.

4 The instationary problem

Electroneutrality again allows for elimination of the electrical field which leads to a

strongly coupled quasilinear parabolic system with nonlinear transmission and dynam-

ical boundary conditions. Of course we now also assume inital conditions

c«(0)=c« c

(0,-) = 4on[-5

,0], c

(0, •) = c

[0,5

<=» = c

. (15)

Theorem 4.1 Fix p > 3/2. Suppose that all initial values are strictly positive satisfying

F g

2-2/

P([

SFI

„

]; RB)J C

^-2/p

([0i

SM]

the electroneutrality conditions

(z,c£) = <z,c

) = 0, (z,c

) = 0 a.e. on [S

,0], (z,c^) = c^ a.e. on [0,S

342

as well as the compatibility conditions corresponding to (4), (5) and (6).

Then there is a unique solution {c

) of the instationary problem (1), (3),

(4),

(5), (6) on a time interval [0,T] for some T > 0. The solution is continuous with

values in the phase space

R" x

W%-

2/p

([-6

,0];R

)

x W

2/p

([0,<5

];R

) * »",

stays electroneutral and depends continuously on the data. The differential equations are

satisfied in the strong IP-sense and the boundary as well as transmission conditions hold

pointwise.

The proof of this result is based on I^-maximal regularity (cf. [4]) of an appropriately

linearized system and will be given in a forthcoming paper.

Acknowledgement. The subject of this paper was brought to our attention by

Jakobs and H.-J. Warnecke, University of Paderborn. We appreciate many helpful

discussions and are grateful for the continuing cooperation.

References

[1] H. Amann, M. Renardy, Reaction-Diffusion problems in electrolysis. NoDEA 1 (1994),

91-117.

[2] Y.S. Choi, R. Lui, Multi-dimensional electrochemistry model. Arch. Rational Mech.

Anal. 130 (1995), 315-342.

[3] K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, Berlin, 1985.

[4] R. Denk, M. Hieber, J. Prtiss, iJ-Boundedness, Fourier Multipliers and Problems

of Elliptic and Parabolic Type. Reports of the Institute of Analysis, Univ. Halle-

Wittenberg 2001.

[5] D. Jakobs, Stofftransport durch Nanofiltrationsmembranen unter Berflcksichtigung

von Biofilmen. PhD-Dissertation Univ. Paderborn 2001.

[6] M. Mulder, Basic Principles of Membrane

Technology,

edition. Kluwer, Dordrecht,

1997.

[7] J.S. Newman, Electrochemical Systems, 2

edition. Prentice Hall 1991.

[8] S.I. Pohozaev, On the reaction-diffusion electrolysis nonlinear elliptic equations. Lect.

Notes Pure Appl. Math. 194, 255-288 (1998).

[9] J. Wiedmann, An electrolysis model and its solutions. PhD-Dissertation Univ. Zurich

1997.

The Hopf solution of Hamilton

Jacobi equations*

Italo Capuzzo Dolcetta

Dipartimento di Matematica,

Universita di Roma - La Sapienza,

P.le A. Moro 2, 00185 Roma, Italy

Email : capuzzo@mat.uniromal.it

1 Introduction

The Hopf formula

u(x,t)

= inf

«,(„)

+«r(^)]

(i.i)

provides

simple representation for the solution of the Hamilton

Jacobi equation

+ H(D

u) = 0

(x,t) eJR

(0,+oo)

(1.2)

equipped with the initial condition

u(x,0)=g{x) .leR". (1.3)

Indeed, formula (1.1) expresses the solution

of the Cauchy problem (1.2), (1.3) as the

optimal value of an unconstrained minimization problem on R", parametrized by

(x,t),

which involves the initial datum g and the Legendre

Fenchel transform H* of the given

(convex) Hamiltonian

The first part of this paper comprises

quick review of selected well

known results

about the interpretation of the Hopf function as

global generalized solution

(1.2),

(1.3),

either

the weak sense

Kruzkhov

well

as a

viscosity solution

a la

Cran-

dall

Lions. The first section is completed by some remarks concerning the asymptotic

behaviour of the classical integral representation formula for the solution of

standard

parabolic regularization of the Cauchy problem above: using the Varadhan's Large Devi-

ations Principle (see [16]), the Hopf function is interpreted as the limit of the solutions of

the regularized problem as the diffusion coefficient vanishes.

The second part is devoted to the presentation of a new result, due to the collaboration

with H. Ishii [7], which shows that the Hopf representation formula still holds for state

dependent Hamiltonians

= H(x,p), provided

has an associated stationary equation

'Work partially supported by the TMR Network Viscosity Solutions and Applications

343

344

which is of eikonal type and, more precisely, which exhibits a metric character in terms

either of a Riemannian distance on R.^ or, when some degeneracies in the x - dependence

are present, in terms of a Carnot - Caratheodory type distance.

For the sake of simplicity the results will not be reported at the highest possible level

of generality; the interested reader is rather referred to the bibliography (see [1], [2], [7],

[8],

[9], [11]).

2 The Hopf function as a generalized solution

Two preliminary simple observations are that for affine initial datum

g(x) = q

•

x + c

the smooth solution of (1.2), (1.3) is

v(x,t) = g{x)-tH(Dg(x))

and that, for general g, the functions

v>*(x,t)=g(y) + q-(x-y)-tH(q)

solve (1.2) for any choice of (y,

e TR

x IR" but do not satisfy (1.3).

It is not hard to realize that the envelope procedure proposed by E. Hopf [9], namely

to take

inf sup v

(x,t) ,

yem."

6]R

which defines indeed the function u in (1.1), preserves, at least at points of differentiability

of u, the fact that each

satisfies (1.2) and also enforces the matching of the initial

condition in the limit as t tends to 0

. A precise statement is as follows (see, for example,

[8]) =

Theorem 2.1 Assume that

H is convex (2-1)

,. H(p) ,

hm -+Y- = +oo (2.2)

|p|-»+o° \p\

there exists G > 0 : \g{x)

—

g(y)\ < G\x

—

y\ for all x, y. (2.3)

Then,

the Hopf function

u(x,t)=M

{g(y)

tH<(^l)

(2.4)

is Lipschitz continuous on IR

x (0,+oo), satisfies the equation (1.2) almost everywhere

and

lim u(x,t) = g(x) at any x € IR" .

345

By a well - known result of S. N. Kruzkhov [10], uniqueness for problem (1.2), (1.3)

holds in the class of Lipschitz continuous functions which are semiconcave, that is

u(x + h,t)- 2u{x,t) + u{x -h,t)<(c + A \h\

(2.5)

holds for some C > 0 and any x, t, h. It is easy to check that if the initial datum satisfies

g(x + h)-2g(x) + g(x-h)<C\h\

for some constant C > 0 , then the Hopf function satisfies condition (2.5) uniformly

in t > 0 and, consequently, is the unique weak solution of (1.2), (1.3) in the sense of

Kruzkhov. An alternative condition for (2.5) to hold is the uniform convexity of H (see

([8])-

The analysis of the Hopf function took later a new impulse with the work of P.L. Lions

[11] and M. Bardi - L.C. Evans [2]. The use of the notion of viscosity solution which makes

sense even for merely continuous functions and of the comparison results available in that

theory allow to interpret the Hopf function as the unique global solution of (1.2), (1.3) in

a larger class of functions.

A typical result in this direction (note that condition (2.2) is not assumed there) is

the following one taken from [2]:

Theorem 2.2 Assume (2.1) and (2.3). Then, the Hop} function (2.4) is the unique uni-

formly continuous function on ]R

x (0, +oo) which satisfies equation (1.2) in the viscosity

sense and

lim u(x,t) = g(x) at any x € TR

A classical method to construct solutions of first order fully nonlinear partial differen-

tial equations is through parabolic regularization. Let us discuss briefly this issue in the

special case H(p) =

||p|

Assume g bounded and consider the parabolic regularization

of the Cauchy problem (1.2), (1.3), that is

u\ - eA

u< + i|I>

uf = 0 , u

(x, 0) = g(x) , (2.6)

where e is a positive parameter. The Hopf - Cole transform of u

, namely

= e~^i

satisfies then the linear heat problem

w\ - eA

= 0 , w

{x, 0) = e"

. (2.7)

By classical linear theory, w

has the integral representation

w (x, t) = (4iret)

<< e

ay ;

346

hence,

(x, t) = -2e log ((47ret)-f J e-^SV^dy) (2.8)

is a solution of the quasilinear problem (2.6).

It is natural to expect that the solutions u

of (2.6) should converge, as e

—>

, to

the Hopf function of problem (1.2), (1.3) which in the present case is given by

u(x, t) = inf

9(y)

(2.9)

This can indeed be proved as a simple application of a general large deviations result by

S.N. Varadhan. Making reference to the notations in [16] Theorem 2.2, it is enough to

consider the family of probability measures P

x (

defined on Borel subsets of IR^ by

{B) = (tort)"* / e-^dy

and the rate function

i*Av)

-^f •

The above mentioned theorem states in fact that

lime log (f e^dPZ/yj) = sup

[F(y)

- I(y)\

£-+0+ Vfft" / ycTuN

for any bounded continuous function F; for F =

— §

the above gives (2.9).

For a direct asymptotic analysis (i.e. not making use of the explicit representation (2.8))

see [11], [3].

A further remark about the approach described above is that the Hopf

Cole transform

can be also used in a similar way to deal with the parabolic regularization of more general

equations such as

+ -\a(x)D

= 0

where a is a M x N given matrix, provided the regularizing second order operator is

chosen appropriately. Indeed, if one looks at the regularized problem

u\ - e div (a*{x)a(x)D

€

) + -\a(x)D

= 0 ,

then the Hopf - Cole transform w

—

solves the linear equation

—

e div (a*(x)a(x)D

) = 0 .

This observation will be developed at the purpose of asymptotic analysis of Hamilton

- Jacobi equations in a forthcoming work.

347

3 An Hopf type formula for state dependent Hamil-

tonians

In this section we present a new Hopf type formula, obtained in collaboration with H.

Ishii (see [7]), for the viscosity solution of the state - dependent Cauchy problem

+ H(x, D

u) = 0 , (x,t)&JR

x (0, +oo) (3.1)

u{x,0) = g(x) , x€B.

. (3.2)

It is easy to realize that the Hopf envelope method does not work if H depends on x.

However, as we shall show below, an Hopf type formula can be proved even in this more

general case under the basic structural assumption that the Hamiltonian H : JR

i-> 1R

is of the form

H(x,p) = 9(H„{z,p)) (3.3)

where H

is a continuous function on JR

2JV

satisfying the following conditions

p t-> H

(x,p) convex

(x, \p) = \H

(x,p) ,

(3.4)

{x,p) > 0 , \H

(x,p) -

(y,p)\

< u(\x - »|(1 + |p|))

for all x, y, p, for all

> 0 and for some modulus u such that lim

+ w(s) = 0 . Concerning

function

we assume

$ :

[0,

+oo)

—>

[0, +oo) , convex , non decreasing , $(0) = 0 . (3.5)

The next result shows that the validity of an Hopf type formula for the solution of

problem (3.1), (3.2) is guaranteed if the associated stationary eikonal problems

{x,D

d) = l,xeJR

\{y} (3.6)

d{y) = 0 (3.7)

have a solution d(x) = d(x; y) for any value of the parameter y e IR". We shall describe

below a setting in which this condition can be enforced.

Theorem 3.1 Assume (3.3), (3.4), (3.5) and

g lower semicontinuous , g(x) > —C(l + \x\) for some C > 0 . (3.8)

Assume also that for each y G IR^ problem (3.6), (3.7) has a unique nonnegative contin-

uous viscosity solution d(x) = d(x;y). Then, the function

u(x, t) = inf

sent"

g(y)

d{x

(3.9)

is the unique lower semicontinuous viscosity solution of (3.1) which is bounded below by

a function of linear growth and such that

}

(y>

) = 9{x) . (3.10)

348

In order to understand why the Hopf function (3.9) solves (3.1), let us proceed heuris-

tically by assuming that (3.6), (3.7) has a smooth solution d(x) and look for special

solutions of the form

,t)=g(y)

ty(^±)

(3.11)

where y e IR" plays the role of a parameter and $ is a smooth function to be appropriately

selected. Set T = ^

' > 0 and compute

y = *(

) + <*'(r)~ = *(r) - r^'M

= W(

)^ = V(T)D

Imposing that v

solves (3.1) gives

$(r) - TV(T) + $(H

{x,

V'(T)D

d))

= 0 .

By choosing a strictly increasing \&, the positive homogeneity of H

yields

0 = ^(T)-T^'(T)+^'{T)H

(x,D

d))

= tf(r) - rtf'(r) +

*(W(T))

, (3.12)

using the fact that d solves the eikonal equation. Since the solution of the Clairaut's

differential equation (3.12) is * = $*, the above heuristics lead then to formula (3.9).

The rigorous proof of Theorem 3.1 is made up of three basic steps. The first one is

to show that the functions v

defined in (3.11) are indeed viscosity solutions of (3.1) for

each y, even when d is assumed to be just a continuous viscosity solution of (3.6), (3.7)

and $ is a general convex nondecreasing function. This requires, in particular, to work

with regular approximations of $, namely

(

)

|«

and the use of the standard reciprocity formula

M(*:)'(

))+*:(

)

7oo.

The other fundamental tool in the proof is the use of the stability properties of semicon-

tinuous viscosity solutions with respect to inf and sup operations in order to show that

u is a lower semicontinuous viscosity solution of (3.1) in the sense of Barron - Jensen [4].

The second step is to check the initial condition in the weak sense (3.10) and the fact

that u is bounded below by a function of linear growth; this and the final step concerning

uniqueness are performed by suitable adaptations of the methods of [1], [4]. We refer to

[7] for details.

The assumption that the eikonal equation has a unique continuous viscosity solution

made in Theorem 3.1 is trivially satisfied when H

(x,p) = \p\.

349

In this case, the viscosity solution of (3.6), (3.7) is d(x;y) = \x

—

y\ and the Hopf

formula becomes

~y\.

u(x,t) = inf g{y) + t$*

a simple argument using the monotonicity of $ shows that

*.(l£^)

*oji.)-(]£^l

and our formula (3.9) reduces then to the classical one (2.4).

It is easy to check that the simplest state - dependent case covered by Theorem 3.1 is

{x,p) = h

{x)\p\ , /i

(z)>7>0.

More generally, one can deal with homogeneous Hamiltonians of the form

{x,p) = \A(x)p\

where A{x) is a symmetric positive definite N x N matrix. The associated eikonal equa-

tions are solved in this case by Riemannian metrics (see below and also [11], [15] for

previous results of this kind).

In all the cases mentioned above, the coercivity condition

lim HJx,p) = +oo (3.13)

|p|->+oo

obviously holds true. Let us discuss now the issue of finding sufficient conditions for the

validity of the eikonal assumption in Theorem 3.1 even for degenerate situations when

(3.13) fails. We will show next that a solution d of (3.6), (3.7) can indeed be constructed

in a quite general setting by a control theory approach.

By standard convex analysis arguments it is seen that the mapping

x^dH

{x,0)

is upper semicontinuous with closed convex values. Consider now the differential inclusion

X(i) e dH

{X{i),0) (3.14)

and, for i,y6 K", the set

x>y

of all trajectories X(-) of (3.14) such that

X(0) = x, X(T) = y

for some T = T (X{-)) > 0 . Define then

d{x;y)= inf T(X(-)) . (3.15)

and assume that there exist e > 0 and, for each x G IR^, anMxJV (with M < N) matrix

a(x) 6 C°°(fl

) such that

a(x) satisfies the Chow - Hormander rank condition of order k , (3.16)