Bemelmans J. et. al. (editor) Elliptic and Parabolic Problems: Rolduc and Gaeta 2001

Подождите немного. Документ загружается.

[5]

Bandle,

Brillard,

Flucher, Green's function, harmonic transplantation,

and

best Sobolev constant

spaces

constant curvature. Trans. Amer. Math.

Soc. 350,

(1998),1103-1128.

[6]

Bandle,

L.A.

Peletier, Best Sobolev constants

and

Emden equations

for the

critical

exponent

in S

Math.Ann.,

313

(1999),

83-93.

[7]

Bandle,

Benguria,

The

Brezis-Nirenberg problem

on S

. to

appear

in J.

Diff.Equ.

[8]

Brezis, Elliptic equations with limiting Sobolev exponents

-the

impact

topology.

Comm. Pure Appl. Math., XXXIX (1986), 17-39.

[9]

Brezis,

Nirenberg, Positive Solutions

Nonlinear Elliptic Equations Involving

Critical Sobolev Exponents. Comm. Pure Appl. Math., XXXVI (1983), 437-477.

[10]

C. J.

Budd,

A.R.

Humphries, Numerical

and

analytical estimates

existence

re-

gions

for

semi-linear elliptic equations with critical Sobolev exponents

cuboid

and

cylindrical domains. Mathematics Preprint, University

Bath 99/22 (1999).

[11]

C.J.

Budd,

Norbury, Semilinear elliptic equations

and

supercritical growth.

Diff.Equ.

(1987), 169-197.

[12]

Egnell, Semilinear elliptic equations involving critical Sobolev exponents. Arch.

Rat. Mech. Anal.

104

(1988), 27-56.

[13]

Esteban,

P.L.

Lions

compactness lemma. Nonlinear Analysis

TMA

(1983),

381-385.

[14]

P.L.

Lions,

The

concentration-compactness principle

in the

calculus

variations.

The limit case, part.2.

Rev. Mat.

Iber.l (1985),

45-121.

[15]

Reichel,

H. Zou,

Non-existence results

for

semilinear cooperative elliptic systems

via moving spheres.

J. Diff. Equ. 161

(2000), 219-243.

[16]

Reichel, Supercritical variational problems with critical points. This volume.

[17]

Schoen, Conformal deformation

of a

Riemarmian metric

constant scalar curva-

ture.

J. Diff.

Geom.

(1984), 479-495.

[18]

Stapelkamp,

The

Brezis-Nirenberg problem

on H^.

This volume.

Diffraction problems for quasilinear elliptic and

parabolic systems

G. Boyadjiev and N. Kutev

Institute of Mathematics, 1113 Sofia, Bulgaria

Email : kutev@math.bas.bg

Abstract

Existence, uniqueness and qualitative properties of the weak and piecewise clas-

sical solutions of the diffraction problem for weakly coupled quasilinear elliptic and

parabolic systems are considered. As a consequence of the comparison principle, the

continuous dependence of the solutions on the data as well as the monotonicity and

stabilization of the solutions with respect to t are obtained. Some applications for

concrete systems describing chemical reactions in electrochemistry are discussed.

1 Introduction

We investigate the existence, uniqueness and global behavior of the weak and piecewise

classical solutions of the diffraction problem for weakly coupled parabolic systems of the

type

u\-div(a! (x

t,u',Du

)) + F

(x,t,u\ ...,u

,Du

) =

f'(x,t),

in Q = ft x (0,T),

(x, t) = g'(x, t), on the parabolic boundary F = (<9ft x

[0,

T\) U (ft x {t = 0}),

(1)

as well as for the stationary (elliptic) problem

-div (a

(x,u

,Du

)) + F

(x,u\ ...u

,Du

) =

f'(x),

in ft,

(x) =

(x),

on 9ft,

(2)

for I = 1,...,

TV,

where ft is a bounded and smooth domain of R

, and D = (D\,..., D

with Di = d/dxi. Without loss of generality, we may assume that, for every I = 1,..., N

(x,t,z,0) = 0 V(i,t)eQ,Vzef,

F'(x,t,0,0) = 0 V(x,t)_& Q, Vz e R,

(x,z,0) = 0 Vi £ ft, Vz £ 1,

(x,0,0) = 0 Vzgft, VzeR,

We suppose that ft

C ft, 7 = 1,..., m

—

1, are strictly interior subdomains of ft with

smooth boundaries <9ft

and without intersection points, i.e. ft

nft« = 0, for 7 ^ S, as well

as they do not intersect dfl, i.e. dfi

n 50 = 0. For simplicity, we denote fi

= fi\

On dfi

some positive direction is fixed by means of a unit normal vector v{x) to

dCl~,

and the following diffraction conditions are prescribed on 5 = US

, S

= dfi,

[0,T],

7 = l,...,m-l

[«']|

= 0,

^o^i.u'.Dti'Ji/'ti)

= 0

(3)

where

[u']|

is the jump of the function u

through 5 in the direction of the normal i/(x).

The coefficients of the system (1) (resp. (2)) are smooth enough in Q , Q

= fi

x (0,T)

(resp.

), and are possibly jump discontinuous on S (resp. on U

3f2

The motivation to consider such problems is the natural interest to study quasilin-

ear elliptic and parabolic systems with discontinuous coefficients under the most general

structure conditions, as well as some real processes, for example in reaction-diffusion phe-

nomena in porous medium leading to the problem (1), (3). More precisely, the following

model example (see [4])

- 0(x)u

- 6(x)v

—

6(x)w

+ ki

V + K

ClM

+ fcl

9(x - a) = 0,

(4)

0(x - a)

in Q = (a,

(0,

T), with the initial and boundary conditions

( u{x,

= v(x,

= w{x,

= 0, Vz 6 [0,6]

i u{0,t) = C

(t), v(0,t) = w{0,t) = 0, Vt e

[0,T],

{ u

{b,t) = 0, v

(b,t) = -w

{b,t), w(b,t) = 0, Vt G

[0,T],

as well as the diffraction conditions on {x = a}

[u]

[v]

= H =

[9u

]

= [^

] = [e

] = 0,

\[0Ux]

= eu

(a + 0) - ^(a - 0)J ,

(5)

(6)

describes the concentration of phenol in the chemical industry. Here 0 is the Heaviside

function : 6(x) = d for 0 < x < a, 0(x) = e for a < x < b, where Cj, k

, d, e are positive

constants and C

(t) is the phenol concentration which should be controlled during the

technological process. Crossing the thin membrane (o,

6),

which is filled in with a special

enzyme, the phenol reacts with the enzyme and transforms into catechol and o-quinone.

This is the so-called Michaelis-Menten reaction, well described, for example, in [6]. The

o-quinone reacts on x = b, changing into catechol and producing electricity. In fact, only

the electricity measured on x = b indicates the concentration of phenol. This is why it

is important to prove the monotonicity and the continuous dependence of the solution of

(4)-(6) with respect to the boundary data, i.e. to prove a comparison principle.

As it is well-known, in general, there is no comparison principle for elliptic and

parabolic systems (see Examples 2.1, 2.2 below). In fact in [2, 8, 10, 11, 13, 14, 15],

the nonnegativity of the classical solutions of linear elliptic systems with nonnegative

boundary data has been proved, for the class of the so-called cooperative systems. The

same result was proved in [9] for parabolic systems such that the operators associated to

all these equations have the same linear principal part. The nonlinear lower order terms

satisfy the quasimonotonicity condition, which is the nonlinear analog of the cooperative-

ness condition in the linear case. Finally, the positivity of the weak solutions of general

nonlinear reaction-diffusion systems was proved in [1] under some special structure as-

sumptions for the coefficients, which allow to reduce the problem to a single parabolic

equation. Note that the positivity of the solutions is a weaker statement than the compar-

ison result for arbitrary sub- and super-solutions, which can be true without comparison

and uniqueness of the solutions.

As for the diffraction problem for linear elliptic and parabolic equations, the existence

of a piecewise classical solution was proved in [7]. In that paper, it was mentioned that

the result is also true for linear elliptic and parabolic systems. For more general nonlinear

elliptic and parabolic equations, the existence of a piecewise classical solution was proved

in [12]. The proofs in these papers are based on Schauder's fixed point argument and the

method of continuation of parameters and there is no comparison principle or uniqueness

results.

Let us recall that a comparison principle for the viscosity sub- and super-solutions of

general fully nonlinear elliptic systems G

(x,u

,...u

,Du

) = 0, / =

1,...,JV,

was

proved in [5] (see also the references therein). The systems considered in [5] are degenerate

elliptic ones and satisfy the same structure-smoothness condition as for a single equation.

The first main assumption in [5] guarantees the quasimonotonicity of the system, but

is more general than the quasimonotonicity condition. Unfortunately, the second main

assumption in [5] is satisfied only for systems with continuous coefficients. This is why

this comparison principle is not applicable for diffraction problems of the type (1), (3).

As shown in Theorem 2.3 below, the comparison principle only holds under the quasi-

monotonicity condition for the weak Lipschitz sub- and supersolutions for uniformly

parabolic systems (1), without any additional structure conditions. For the stationary

problem (2), the comparison result is true under some additional conditions which are the

same as in the case of a single elliptic equation (see [3, Th. 9.5]). Since we do not suppose

any regularity assumptions for the coefficients of (1), (2) with respect to the x, t variables,

the comparison principle in Theorems 2.3 and 2.4 is also valid for the diffraction problem

(3).

By the way, even in the case of systems with smooth coefficients, our result is new.

As a consequence, we get uniqueness and continuous dependence on the data of the weak

Lipschitz solutions, as well as existence, monotonicity and stability of the solutions.

2 Main results and proofs

In order to formulate our results, we follow the notations indicated in [7] and [12] and for

simplicity we use the vector form for the system (1) as well as the vector notation u > v,

which means that v! > v

for every I = 1, ...N.

We define the weak solutions of (1), (3) in the Hilbert spaces Wl'°(Q), Wl'

{Q) as in

[7] and [12].

Furthermore, we suppose that (1) is uniformly parabolic, i.e.

HH)\e\

<J2~(x,ty,...,u

j)^<A(\u\)\e\

(7)

for every v! and £' = (f[, ...£,) 6l", I =

1,2,...,

N, with A(|u|) > 0.

The minimal smoothness of the coefficients of (1) which is required in order to prove

the uniqueness and the comparison principle is that a

(x,t,u,p), F

(x,t,u,p),

(x,t),

(x,t) are measurable functions with respect to the x, t variables and locally Lipschitz

continuous with respect to «', u and p, i.e.

(x,t,u,p)

—

(x,t,v,q)

a!(x,t,u\p)

—

a!(x,t,v

,q)

<C(K)(\u-v\ +

\p-q\),

<C(K)(\u

-v'\ +

\p-q\),

(S)

for every (z,t) e Q, \u\ + \v\ +

\p\

+ \q\<K,l =

1,...N.

The conditions (7), (8) are enough for the validity of the comparison principle for

weak Lipschitz solutions for quasilinear parabolic system (1). However, in general, under

the same assumptions there is no comparison principle for systems (see Examples 2.1

and 2.2 below). This is why we need the following condition for the quasimonotonicity

(cooperativeness) of (1):

{x,t,u},

...,u

,p) are non increasing functions with respect to u

, . .

for

l^k,l,k

l,...,

N and every (x,t) e Q

, u G E

, p £ W. ^ '

The following examples show that the condition (9) is, in general, necessary for the

validity of the comparison principle.

Example

2.1 Let Q =

(0,TT)

X (0,T) and u\ - u\

~ u

+ u

= 0, u\ - u\

= 0 in Q,

uHM) = u

(0,t) = uV,*) = y?(-K,t) =0forte [0,T], u\x,0) = u

{x,0) = 0 for

x e

[0,7r].

Since F

(x,t,u,p) = —u

+ u

and |^- = 1, condition (9) fails. The above system

has one trivial solution v

= v

= 0 (which is also a sub-solution) and a nontrivial super-

solution w

= —tsinx, w

= sin x. Since — isinz = w

< v

= 0 in Q the comparison

principle does not

hold.

For more general (strongly coupled) systems of the type (1), for which F

depends

on Du

or a

depend on u

for k ^ I, the comparison principle, in general, fails as the

following variant of Example 2.1 shows.

Example 2.2 Let Q = (0, n) x

(0,

T) and u\ - u\

- u

- u\ = 0, u

- u\

= 0mQ,

with zero initial and boundary data as in Example 2.1.

As before, v

= v

= 0 is a trivial solution (sub-solution) while w

= -tsinx, w

2cos

(x/2) is a nontrivial super-solution. Conditions (7)-(9)

hold,

but

—

tsinx = w

= 0 in Q, violating the comparison principle. Since the system can be rewritten as

t ~ (

l ~

)x - u

= 0, u

—

= 0 in Q with F

, F

independent of Du but a

depending on u

, the previous example shows that the structure assumptions for (1) are

sharp.

Theorem 2.3 Suppose u, v e

(Q)

0 C(Q) are, respectively, weak sub- and super-

solutions of (1), (3). If (7)-(9) are satisfied and u < v onT, then u < v in Q.

As for the elliptic system (2) with the diffraction conditions (3) on (J^dQ.i, we need

the same additional structure conditions for (2) as in the case of a single equation with

smooth coefficients (see [3, Th. 9.5]).

Theorem 2.4 Let u, v 6

W^,(Q)

n C(Q) be, respectively, weak sub- and super-solutions

of (2), (3) and suppose that (7)-(9) are satisfied. Suppose at least one of the assumptions

(10)-(12)

hold,

for l =

l,2,...,N:

Qpj

(x,u,,p) are independent of p and ]T -^-j(x,v},...,u

) > 0, . .

for xeQ

,u€ R

Qpl

(x,u',p) are independent of u

and J2

TT~i~(^i"

•••i

,p) > 0, , ,

for

x e fi

, u e R

, v e R

;

dF'

dp'

dp?

the (n + 1)JV x (n + l)N matrix A

••

Soil

d£i

for i,j = 1, ...n, I, s = 1, ...N is a nonnegative one.

(12)

If u < v on <9fi, then u < v in Q.

Sketch of the

proof.

The proof is based on the appropriate choice of a test-function

depending on ui

= max jj(it

—

v, 0) for the difference of the equations for u and v.

If (12) holds, by means of the test function n =

and conditions of Theorem

2.4, we get the inequality Y^L\ l-^+l

< C\ Y^i=\

{

which gives the final estimate

In (1 + \

YM=I

o>+

) < C2 in f2, for some constant C

> 0 independent of e and for every

e > 0. When e

—>

0, the above estimate is impossible.

When (11) holds and sup

jw'

= M

> 0, by means of the test-function

= max(w- M),

0 < M < M

, simple computations prove that the set fl

= Ufl

— {x e Q

: to

= Mo}

has nonzero measure. Then Poincare's inequality implies the estimate

o^E

(n'

)

2/n

mes

where :

= {x 6 fl

> M} and the constants C3, C

are independent on e. Letting

—>

Mo and using the fact that mes (fi'

\f2

)

—>

0, mes (fi+)

—>

mes (fi

), mesfio > 0,

we get a contradiction.

Suppose (10) holds. Then the test-function rf =

/{u>'

for every e > 0 gives the

estimate

ln[l + ^

with some constant C5 independent of e. The above inequality is impossible when e

—>

•

In order to prove Theorem 2.3, we choose a test-function n = u)

, L = const > 0,

= max(u

—

v, 0). We get a contradiction using the same a priory estimates as in the

beginning of the proof of Theorem 2.4. •

Corollary 2.5 Under the assumptions of Theorem 2.3 (resp. Theorem 2-4) problem (1),

(3) (resp. (2), (3)) has at most one weak

W^{Q)nC(Q)

(resp. W^(Q,)nC(Q,)) solution.

Corollary 2.6 Suppose u,u& W]^{Q)

C\C(Q)

are weak solutions of (1), (3) with right-

hand sides

f(x,t), f(x,t),

respectively, and boundary data

g(x,t),

~g{x,t), respectively. If

the conditions of Theorem 2.3 are satisfied then the estimate

\u - u\ < M (sup \g-g\+ sup |/ - /|),

holds, where M =

(

)

for K = sup (|u| + |u| + \Du\ + \Du\) and C{K) is defined

in (8).

In order to prove the existence of a piecewise classical solution of (1), (3) (resp. (2),

(3)),

we need some additional regularity and the so-called natural restrictions on the

nonlinearity of the coefficients of the system at infinity. More precisely, we suppose that

€

1+a

x R

_), F

1+a

x R

{x,0) G C

(0) nC

a+2

), g\x,t) £ C

, for x e dQ, (13)

dD,,dn

a+2

ioi i = 1, ...n,l = 1, ...N,k = I, ...m, and some a e (0,1).

Only in the existence part of the paper, we will assume that (1) has one and the same

principal part which is a potential, i.e. for I = 1, ...N

(x,t,u',p) = A^XjtV.p) = —-(x,t,u

,p) for i = 1,

OPi

(14)

and the following structure conditions are satisfied

for (a;,i,u,p)

€

xl"

R" and

i,j =

1, ...n,

fc,Z

1, ...iV, s

l,...m.

As for the initial data g'(x, 0) we suppose that the compatibility conditions

£^(x,o,<?'M),zVM))^

= 0,

1,..

.N,

(16)

hold.

We also need some conditions guaranteeing

priori estimates

the solutions

(1),

(3).

For example, the simplest and easily checkable conditions are the following ones

F\x,t,u\...u

,Q)

>-C

\u\

-C

, (17)

for

(x,t) e

u e

1, ...N and some nonnegative constants C

, C

Theorem 2.7 Suppose

(7),

(9),

(13)-(17)

hold.

Then the diffraction problem

(1),

(3)

has

unique piecewise classical solution u(x,t)

€

,/3

(Q),

J-^

G C

,3,l3

®.g

f €

(Q),

£^ e

(Q)

for

some

0 <

< a

and

for

i,j = 1,

'...n,

...771.

Idea

the

proof.

The proof is the same as

the case

single equation (see [12])

and is based on the method of continuation of parameters and Leray-Schauder' principle.

Remark 2.8 The same result

true

the elliptic case, without the

ditions (16) and the potential condition (14).

In order to investigate some qualitative properties of the weak solutions of problem (1),

(3) we will draw our attention to the class of systems with time independent coefficients,

i.e.

— diva

(x,u

,Du') + Fix,^,,, ,u

,Du

)

f'(x)

in Q

u(x,t)

(x,t)

on T,

for

1, ...N, including the model problem (4).

Theorem 2.9 Suppose (7), (9)

hold,

g{x,t) G

W^)(Q)r\C(Q)

nondecreasing vector

function with respect to

for every

x €

dQ, and g(x,0)

a sub-solution

(2), (3). Then

any weak solution u(x,t)

W^)(Q)

C(Q)

(18), (3)

is a

nondecreasing function

time

for

every

x £

£1.

It is not difficult, combining Theorem 2.9 with the

priori estimates given in the proof

of Theorem 2.7,

show that every solution

(18), (3) tends (in

monotone increasing

way in time)

for

—>

to a

solution of the corresponding stationary problem, provided

(18),

(3) has a time independent super-solution. However, we will formulate this result

only for weakly nonlinear systems (19), (3), including the model example (4), i.e.

4-lX

aHx)d

+Fl{x

'---'

uN)=f,{x)AnQ

(i9)

Theorem 2.10 Suppose (13), (16) and all conditions of Theorem 2.9 hold andg'(x,t) =

g'(x)

for t sufficiently large and x E 9fi. Moreover, assume that (19), (3) has a weak

(°>

°°)) n C (f2 x

[0,

oo)) time independent super-solution satisfying the

bound-

ary data g

(x) for x € dQ. Then (19), (3) has an unique globally defined in Q x [0, oo)

piecewise classical solution which tends, in a monotone increasing way in time, for t

—>

to the smallest piecewise classical solution v(x) greater than

(x,0).

References

[1] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion

systems.

Diff.

Int. Eq. 3 (1990), 13-75.

[2] D. De Figueiredo, E. Mitidieri : A maximum principle for an elliptic system and

applications to semilinear problems, SIAM J. Math. Anal. 17 (1986), 836-849.

[3] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order.

2nd ed., Springer-Verlag, New York.

[4] I. Iliev, N. Tzonkov, Mathematical modeling of enzyme gas-diffusion electodes (in

print).

[5] Ishii, Sh. Koike, Viscosity solutions for monotone systems of second order elliptic

PDEs.

Comm. Part.

Diff.

Eq. 16 (1991), 1095-1128.

[6] T. Keleti, Basic enzyme kinetics. Akademiai kiado, Budapest.

[7] O. Ladyzhenskaya, V. Rivkind, N. Ural'tseva, Classical solvability of diffraction prob-

lem for elliptic and parabolic equations with discontinuous coefficients. Trudy Mat.

Inst. Steklov, 92 (1996), 116-146 (In Russian).

[8] J. Lopez-Gomez, M. Molina-Meyer, The maximum principle for cooperative weakly

coupled elliptic systems and some applications.

Diff.

Int. Eq. 7 (1994), 383-398.

[9] R. Martin, Jr., A maximum principle for semilinear parabolic systems. Proc. Amer.

Math. Soc. 74 (1979), 66-70.

[10] E. Mitidieri, G. Sweers, Weakly coupled elliptic systems and positivity. Math. Nachr.

173 (1995), 259-286.

[11] M. Protter, H. Weinberger, Maximum Principle in Differential Equations. Prentice

Hall, 1976.

[12] V. Rivkind, N. Ural'tseva, Classical solvability and linear schemes for the approxi-

mated solutions of the diffraction problem for quasilinear elliptic and parabolic equa-

tions.

Matem. Zametki, 69-111 (1973) (In Russian).

[13] G. Sweers, Strong positivity in C(Q) for elliptic systems. Math. Z. 209 (1992), 251-

271.

[14] W. Walter, The minimum principle for elliptic systems. Appl. Anal. 47 (1992), 1-6.

[15] H. Weinberger, Some remarks on invariant sets for systems. In Proceedings Con-

ference on maximum principle and eigenvalue problems in PDE. P. Schaefer Ed.,

Harlow, Longman, 189-207, 1987.