Annali di Matematica pura ed applicata (IV), Vol. CLIX (1991), pp. 211-227

An L p Semigroup Approach to Degenerate Parabolic Boundary Value Problems (*).

JEROME A. GOLDSTEIN - CHIN-YUAN LIN

Summary. - The Crandall-Liggett theorem is applied to

au/at = ~ ~(~i (x, Vu))/axi + f ( x , u) i = 1

with various boundary conditions. Moreover, the ellipticity of the operator on the right hand side is allowed to degenerate (mildly) on the spatial boundary.

1. - I n t r o d u c t i o n .

In this paper we develop an L p approach to the parabolic boundary value problem

(1) au/at = ~ ~ i= 1 ~ (~i (x, V~u)) + f ( x , u ) ,

for t > O, x in a smooth bounded domain • in R n, and with boundary conditions of Dirichlet or Neumann or periodic or nonlinear Robin type. We also allow the elliptici- ty to degenerate on the boundary. In case (1) specializes to

au = ~ a2~i x, + ~ a ~ x, au/~t = ~ ~ x, ~ , i = 1 i = 1 ~ X 2 i = 1

this degeneracy means that as ~i (x; ~) is positive on ~ x R but can tend to zero as x ap- proaches a boundary point x0 �9 aD. A more general condition will be discussed in the sequel. Our goal is to solve (1) by applying nonlinear semigroup theory in the real spaces L p (~) for all p, 1 ~ p <~ oo.

There is a large literature devoted to problems like (1). Here we indicate briefly some of the papers using the semigroup approach. BURCH and GOLDSTEIN [3] treated

(*) Entrata in Redazione il 6 marzo 1989. Indirizzo degli AA.: J. A. GOLDSTEIN; Department of Mathematics, Tulane University, New

Orleans, LA 70118; C.-Y. LIN: Department of Mathematics, Texas A&M University, College Station, TX 77843 and National Central University, Chung-Li 320, Republic of China.

212 J. A. GOLDSTEIN - C . -Y . LIN: A n L p semigroup approach etc.

the uniformly parabolic cases

(2) au/at = ~(x, au/ax) a2u/~x 2 in C[O, 1],

(3) au/at = ~(au/ax) a2u/ax 2 in L p (0, 1),

with linear boundary conditions. Note that equation (3) is in divergence form since it can be written as

au/at = a(r

where ~ '= ~. SERIZAWA [16] treated (3) under nonlinear Robin conditions. All of these one dimensional results in C [0, 1] were extended and unified by GOLDSTEIN and LIN [10], who dealt with a variety of boundary conditions and allowed ~(x, ~) to ap- proach zero slowly as x-~ 0 or 1. The higher dimensional version

au/~t = ~(x, Vu)Au + f (x, u, Vu),

was treated by Lin [14]. It is by now a well-established principle that parabolic problem in divergence

form are well-posed in L p for all p, 1 ~<p ~< ~. Our purpose here is to establish this for equation (1).

In [11] GOLDSTEIN and LIN established well-posedness for the initial value prob- lem for

(4) au/at = ~(x, Vu)Au ,

in C(~) where (0 < )~(x, ~ ) ~ 0 arbitrarily fast as x-~ x0 e at~. This necessitated using a special boundary condition, namely the Wentzel boundary condition. This work was inspired by a result of CLI~MENT and TIMMERMANNS [4] in the linear one dimensional case. Their results interpreted analytically the competition between the drift and dif- fusion coefficients of the Markov process described by their analogue of (4). But in (4) there is no drift, so we get a clean result. In the case of equations in divergence form, there is a drift-like term to worry about. Thus, for instance, for the equation

~u/at= ~ X, T x =~l~(x ,u~)+a2~(x ,u~)U~x,

how fast a2 ~(x, ~) can go to zero as x o 0 (or 1) will necessarily be related to properties of al ~(x, ~). Problems of this sort are open and we hope to return to them in a future paper.

Here is an outline of this paper. The semigroup approach is sketched in Section 2. Sections 3 and 4 contain the statements of our results in the one and higher dimen- sional cases, and Sections 6 and 7 contain the proofs.

J. A. GOLDSTEIN - C.-Y. LIN: An L p semigroup approach etc. 213

2. - The semigroup a p p r o a c h .

Rewrite (1) as

du(t)/dt = A(u(t)) ,

together with u(O)= u0; this is an initial value problem for an ordinary differential equation in a Banach space X. Here A acts as

n

(A(u))(x) = ~ ~ (~i (x, Vu)) +f(x , u) i l l ' i X i

on its domain 0~(A) which incorporates the boundary conditions. Then one checks that A is m-dissipative. This requires two steps. First, A is dissipative means that if u i - - LAui = hi for i = 1, 2 and ~ > 0, then Ilul - ur I] < ]lhl - h2 II. The <<m>> part of m-dissipa- tive is the range condition Ran (I - ),4) = X for some (hence all) ~ > 0. Actually it is sufficient to check Ran (I - ),4) is dense in X for some ;( > 0; in this case A is called es- sentially m-dissipative. Then, Letting A denote the closure of the graph of A in X • x X, A is m-dissipative and the Crandall-Liggett theorem says that our problem has a unique global solution (in a certain precise sense) which is given by the exponential formula

t - ~ u(t): lim ( I - - - ) uo,

for t i> 0. Moreover, if u, v both satisfy w' = Aw, then I]u(t) - v(t)l] is a nonincreasing function of t on [0, ~).

When X is reflexive and u0 e O~(A), then the right derivative d+u(t)/dt exists and satisfies d+u(t)/dt e A(u(t)) for all t ~> 0. Moreover, u is differentiable for [0, ~ ) \ N where N is an at most countable set. When X is not reflexive, the sense in which du/dt = Au holds is more general, cf. [1], [2], [5], [6], [9]. Equations such as (4) which are not in divergence form define dissipative operators on L p for p = :r only.

3. - R e s u l t s in o n e s p a c e d i m e n s i o n .

Of concern is the parabolic equation

a [x, au) + f ( x ,u ) aulat= Tx ~t -~x

for u -- u(x, t) with x e [0, 1], t i> 0, together with the initial condition u(x, O) = uo (x) and a variety of boundary conditions.

Let Xp =LP(0, 1) for l < p < ~ and X= = C[0, 1]. We take these to be Banach

214 J. A. GOLDSTEIN - C.-Y. LIN: A n L p semigroup approach etc.

spaces of real functions on [0, 1]. Define A formally by

d ( ( d u ) ) + f ( x , u ) A u = -~x 9 x,-~x x (5)

Let

D1 = (u e C 2 [0, 1]: A u e C[O, 1]},

D2 = {u e C2(0, 1): c~ C1[0, 1]: A u e C[O, 1]),

D8 = (u e C 2 (0, 1): n C[0, 1]: A u e C[0, 1]}.

Uniform and degenerate ellipticity are described by the following assump- tions.

(UE) 9 e C 1 ([0, 1] • R) and (a/a~)9(x , ~) >>. ~ > o for some constant ~ > 0 and all (x, ~) e [0,1] x R.

(DEq) 9 e C 1 ([0, 1] x R) and (a/aDg(x, ~) I> 9o (x) for all (x, ~) e [0, 1] x R, where 90 e C[0, 1], 9o > 0 on (0, 1), and 1/9o e L q (0, 1).

Here l ~ < q < ~ . Concerning f and the nonlinear Robin boundary conditions we introduce the fol-

lowing hypotheses.

(Hx) f e C([0, 1] • R), f (x, O) = O for all x e [0, 1], and f (x, ~) is monotone nonin- creasing in ~ e R for all x e [0, 1].

(NRBC) fl0 and ~1 are a maximal monotone grpahs in R x R with 0 e ~o (0) c~ ~ (0).

THEOREM 1. - Let (DE1) and (H/) hold. Then A equipped with the domain

(D(A) = D1 c~ {u e DI: u(0) = u(1) = 0}

is essentially m-dissipative on Xp, 1 ~ p <. ~.

This takes care of the Dirichlet boundary condition. For the nonlinear Robin con- dition we have

THEOREM 2. - Let (Hf) and (NRBC) hold. Assume either (UE) or (DE2) together with strict monotonicity of ~o and ill. Then A with domain

(D(A) = D2 n {u: u'(0) e 80 (u(0)), -u ' (1 ) e fl~ (u(1))},

is essentially m-dissipative on Xp, 1 <. p <- oo.

The next two results deal with periodic and Neumann boundary conditions.

J. A. GOLDSTEIN - C.-Y. LIN: A n L p semigroup approach etc. 215

THEOREM 3. - Let (DE1) and (Hf) hold. Give A the domain

(~(A) = D2 n {u: u(0) = u(1), u'(0) = u'(1)}.

Then A is essentially m-dissipative on X~ , 1 <. p <. oo. I f (UE) holds, then the same conclusion holds when D2 is replaced by D1, in the description of (~(A).

THEOREM 4. - Let (UE) and (H/) hold. Then A, with domain

(~(A) = D2 n {u: u'(0) = u'(1) = 0},

is essentially m-dissipative on Xp, 1 <. p <. ~.

R E M A R K S . - (i) Each of the A's in Theorems 1-4 is actually m-dissipative on Z~.

(ii) In Theorem 1, the conclusion remains valid if the degeneracy assumption of at the boundary is replaced by

~ (x, ~)[x(1 - x)] 2-~/> 8 > 0,

for some positive constants ~ e (0, 1) and 8, and for all (x, ~) e (0, 1) • R.

(iii) The semigroup solution given by the Crandall-Liggett theorem is a strong solution in Xp whenever 1 < p < cr This follows from the remark at the end of Section 2. Then the solutions are classical solutions, provided that the coefficients are suf- ficiently smooth [12].

4 . - R e s u l t s i n n s p a c e d i m e n s i o n s .

Of concern is

n

au /a t = i=lZ ~-~i(~i(x,. Vu)) +f(x , u).

Here u = u(x, t), x e ~, a smooth bounded domain in R n , n ~ 2, and t I> 0. We associate the initial condition u(x, O)= uo(x), x E t), and various boundary conditions to this equation.

Let Xp = LP (t)) for 1 ~< p < ~ and X~ = C(~). We take these to be Banach spaces of real functions on ~9. Define A formally by

(6)

216 J. A. GOLDSTEIN - C.-Y. LIN: A n L ~ semigroup approach etc.

Let

D1 = {u e C 2 (~): A u e C(~)},

D2 = {u e C 2 (~) n C 1 (~): A u e C(~)},

D3 = {u e C2(D) • C1 (~): A u e C(~)}.

(UE) There is an ~ > 0 such that for 1 <~ i <~ n, 9i e C 2 + ~ (~ x R ~ ), .~i (x, O) = 0 for all x e ~, and

09 i i,j =1 ~ j (x, e) ~ j >/el~l 2 (1 + le] 2) ' ,

holds for all (x, ~, ~) e t? x R ~ x R ~ and some 8> 0 and some ~ > - 1.

(DE) There is an ~ > 0 such that fo r 1 <~ i <. n, 9i e C 2 + ~ (~ x R ~ ), 9i (x, 0) = O for all x e ~ and

~ ( x , ~ ) ~ i V j ~ o ( X ) [ ~ [ 2 ( 1 + 1~[2); , i , j = 1

holds for all (x, ~, ~) e ~ x R ~ x R ~ and some 8> 0 and some ~ > - 1, where ~oe C(~),

9o > 0 on t?, and 1/po e L q (t~) for some q > n/2.

(DE*) There is an a > 0 such that for 1 <~ i <. n, 9i e Ce + ~ (~ x R ~ ) , 9i ( x , O) = O for all x e ~, and

~-~-/(x, :)V~ Vj" ~> 90 (x)]~[~ ( 1 + I:1')', ~,j=l ~j

holds for all (x, ~, ~) e ~ x R " x R " and some 8> 0 and some ~ > - 1, where ~o e C(~),

9o > 0 on ~, and

90 (x) >I 8o [dist (x, at))] 2 .z ,

holds for all x e t? with constants ~ > 0 and 0 < fl < 1.

(H I) f e C ~ (~ x R ) for some ~ e (0, 1), and for all (x, ~) e ~ x R , f (x , O) = 0 and f (x , ~ is monotone nonincreasing in ~ for each x e ~.

$ (H~ ~ ) There are constants C > 0, ~ > - 1 such that for all (x, ~, u) e ~ x R ~ x R ,

and all i , j e {1, ..., n},

Here 9i = 9i( x, ~), f = f ( x , u).

J. A. G O L D S T E I N - C.-Y. LIN: A n L p semigroup approach etc. 217

This last condition is a standard one for parabolic problems in divergence form; cf. [12].

THEOREM 5 (Dirichlet condition). - A s s u m e (DE), (Hf) and ( H ~ ) . Then A with domain

D3 c~ {u: u = 0 on at~} ,

is essentially m-dissipative on Xp , 1 <. p ~ ~. The same conclusion holds with D 1 re- placing D3, provided (UE) holds. Final ly , the same conclusion holds provided (DE) is replaced by (DE*).

THEOREM 6 (Nonlinear Robin condition). - Assume (DE) with q > n, (Hf), and (H~9 with la.~/axj(x, ~)] <~ C(1 + I~t) 2 holding as well. Let

(~(A) ~- D 2 ~ Iu: - i=1 ~ ~i(x, Vu),~ i -- ~(u) on 1 , where ,J = ('~1, ..., ~ ) is the unit outer normal vector to x ~ ate, and 8: R---> R is a C 2 strictly increasing funct ion with 8(0) = O. Then A is essentially m-dissipative on Xp, 1 <. p <. ~. I f (UE) holds, then the same conclusion holds when D1 replaces D~ in the

definition of (9(A).

THEOREM 7 (Neumann condition). - A s s u m e (UE), (Hf) and (H~). Let

where ~ = (vl, ..., '~) is the uni t outer normal vector to x e Ot~. Then A is essentially

m-dissipative on Xp , 1 <~ p <. ~.

5. - The dissipativity proof.

Recall that A (on a suitable domain) is described by (5) or (6). Let 1 < p < 2 and for j = 1, 2 let hj EXp. If uj ~ d)(A) satisfies

u s - = h i

for ), > 0 and j = 1, 2, we must show that

I lu l - u rlp < Ifhl - II .

Once we establish this we may let p -~ 1 and p -* ~ and deduce that A is dissipative on all the Xp, 1 ~ p ~< 2.

218 J. A. GOLDSTEIN - C.-Y. LIN: An L p semigroup approach etc.

So let, for j = 1, 2,

uj-)~ ~(x, Vu j )+f (x , uj) =hj.

Let a0(s)= sign(s)lsl p-1 (with l < p < ~) and let, for m = 1, 2, ...,

[sign(s)lsl p-~ for Isl >I 1/m , ~m (S)

[sign(s)mlslP for Isl <~ 1/m.

Then am is increasing, odd, locally Lipschitz continuous, and am (s)-~ a0 (s) as m ~ for all real s. Furthermore, Isl p-1 I> lain(s) I holds for all s eR , which is equivalent to sa(s) >I Iam(s)l q where q-1 +p-1 = 1. Then

faro (Ul -- u2)(hl - h2) dx = f am (ul - u2)(ul - u2) dx + ;~J t) t~

where

J = - f am (Ul -- U2)(f(Xl, Ul ) -- f(x, U2)) dx t~

fn + ~ (v~(x, V u l ) - v~(x, V u ~ ) ) ~ (u~ - u~) (u~ - u2) dx i = 1

s

f n - am (U~ -- U2) ~ (~i (X, VUl ) -- ~ (X, VU2 ))~i d S ~ ,

i = 1

3

by the divergence theorem. Rewriting this expression as J = ~ Ji, we next show that J~ I> 0 for each i. First, ~= 1

am(u1 - ue)( f (x , Ul) - f (x, u2)) <~ O,

for all x e t) since am is odd and increasing while f(x, u) is nonincreasing in u. Thus J 1 ~ 0. Next, J8 = 0 if the Dirichlet boundary condition holds since am (ul - u2) -= 0 on at), while if the nonlinear Robin boundary condition holds, then

J8 = f (ul - u2 )(/~(Ul ) -~(u2 ))dS~ >i O ,

a~

since again the integrand is nonnegative because of the monotonicity properties of am and ~. (Strict monotonicity of~ is not needed for this argument.) Also it is easy to see

J. A. GOLDSTEIN - C.-Y. LIN: A n L p semigroup approach etc. 219

that Js = 0 if ei ther Neumann or periodic boundary conditions hold. Next, for i = 1, ..., n,

(~i(x, Vu 1 ) - v i ( x , v u 2 ) ) a m(u 1 - u 2) ~ / ( u 1 - u 2) ~ 0 ,

since ~i (x, ~, ..., In) is nondecreasing in ~i for each i. Consequently J2 ~> 0 and so J >/O. Thus

faro (U l - - U2 ) ( h i - h2 ) dx ~ f ~m ( u l - u 2 ) ( U l - u 2 ) dx.

Let t ing m ~ ~ we conclude that

IlUI -- U 2 II p ---- f ~(U 1 -- 72 2 )(U 1 -- U 2 ) dx < t)

lid0 ( U l - u2)ilqllh I - h211p = HUl - u21~p-111hl- h211p , (HSlder' s inequality),

and the dissipativity of A on each Xp follows. This holds for the operator A in each of the seven theorems.

6. - T h e r a n g e c o n d i t i o n .

I t remains to show that the range of I - )~4 is, for some ;~ > 0, dense in each Xp. In various one dimensional cases A will be a closed operator on X , , and the range of I - L4 is all of X . in those cases.

For h e C[0, 1] (n = 1) or else h e C~(~) (n/> 2), we consider the elliptic problem (with ;( = 1 for convenience)

(7) u - -~x ~(X, x ' ) - f (x, u) = h,

or

n (8) u - E �9 Vu) - f ( x , u) = h , z=l

together with appropriate boundary conditions, which can be summarized by writing u e 0~(A). This is an ordinary or partial differential equation, according as, n = 1 or n 1> 2. The lat ter case requires more regulari ty on h.

Writ ing ~ for (~1,..-, ~n) equation (8) can be rewri t ten as

u - V. ~(x, Vu) - f ( x , u) = h,

and thus we have a unified notation for (7) and (8). We t rea t the case of n = 1 and consider (7). Assume the uniform ellipticity condi-

220 J. A. GOLDSTEIN - C.-Y. LIN: A n L p semigroup approach etc.

tion (UE). Rewrite (7) as

u - a l ~ ( x , u ' ) u " - f ( x , u ) = h , u ~ ( ~ ( A ) .

Let ~e [0, 1], v ~ C1[0, 1] and let u = T(v, ~) be the unique solution of

(9) u - ~ a l ~ ( x , v ' ) - ~ 2 ~ ( x , v ' ) u " - ~ f ( x , v ) = ~ h , u e ( ~ ( A ) .

This is a linear equation with linear or nonlinear boundary conditions. The existence of u = T(v, ~) follows from[10]. Furthermore, the estimates of[10] imply that T is continuous as a map from C 1 [0, 1] • [0, 1] to C 2 [0, 1] and compact from C 1 [0, 1] • [0, 1] to C 1 [0, 1].

The solution of (7) will be a fixed point of the map v ---> T(v, 1). To that end we shall apply the Leray-Schauder fixed point theorem (cf. [8, Thm. 11.6, p. 286]). We check the hypotheses of that theorem. Clearly T(v, O) = 0 for all v e C 1 [0, 1]. It therefore on- ly remains to prove the a priori bound: There exists a constant K such that

IlulLcl < K,

whenever u = T(u, ~) for some = e [0, 1]. But if this equation holds, i.e.

(10) u - ~ ~d ~ ( x , u ' ) - ~ f ( x , u ) = ~ h , u e ( ~ ( A ) ,

choose Xo �9 [0, 1] such that lU(Xo)l = Ilull~. If x0 e (0, 1) then by the second derivative test, u' (Xo) = O, U(Xo)U"(Xo)<-O. Under most of the boundary conditions under con- sideration, Xo cannot be an endpoint (unless u - constant, in which case u' - 0). Un- der the periodic boundary condition, we are effectively looking at a circle with no boundary points, so that the second derivative criteria applies.

So let u solve (9) with v = u. We deduce

llulloo < (11hII + lla,

a uniform bound on u. Integrating (10) yields

0

~[~(x, u ' (x)) - ~(Xo, u ' (xo))] = ~ ( u ( y ) - ~h(y) - ~f((y, u(y ) ) ) d y .

~o

The right hand side is uniformly bounded. Since T(v, O) = 0 and T is continuous on C 1 [0, 1] • [0, 1], we may assume ~ I> 80 > 0. Then

u' (x)) - (Xo, u' (xo))

is uniformly bounded. Since u ' ( x o ) = O, it follows that ~(x, u ' (x ) ) is uniformly bound- ed. By the uniform eUipticity (a2 ~(x, ~)~> ~ > 0) it follows that

Ilu'll~ < constant.

Thus I[Ullc~ <~ K and there is a solution u of u = T(u, 1), as desired. When the operator A is not uniformly elliptic we use the ,mpproximation from

J. A. GOLDSTEIN - C.-Y. LIN: A n L p s e m i g r o u p approach etc. 221

above, technique of [10]. For m = 1, 2, ... let Am be the operator A with ~ replaced by ~,~ where p~ is defined as follows. Let

I a2 ~(x, 02 ~,n (x, ~) = [1 /m

and let

~) if a2 ~(x, ~) >I 1 / m ,

if 02 ~(x, ~) < 1 / m ,

~m (x, ~) = ~(x, 0) + f a2 ~,~ (x, v) dr . o

Then for each h e C[0, 1] there is a (unique) um �9 0~(A) such that

u m - A,~ u,~ -- h, u,~ �9 (~(A);

this follows from the above proof applied to Am which satisfies (UE). Since the degeneracy of a2 ~ occurs only near the endpoint x = 0, 1 (see hypothesis

(DEq) on ~0) we can rewrite (11) as

d u,~ - -~X ~m ( X , U , J -- f ( x , um ) = h

and establish the compactness of {urn } in 2 Cloc(0, 1) using the argumentation of[10]. The control of {u,~ (x)} at and near x = 0, I is based on the techniques of [10] also. The hardest case to consider involves the nonlinear Robin conditions, and for that we use Serizawa's Green's function construction as in [10].

This completes the proof of Theorems 1-4. The proof of Remark (li) (following the statement of Theorem 4) follows from the techniques of Lin [14].

7. - T h e r a n g e c o n d i t i o n w h e n n ~> 2.

We shall solve (8) for h �9 C" (~). Define the operators L and L0 by

L u = V . ~(x, Vu) + f ( x , u) - u, Lo u = A u - u ,

with ~ ( L ) = 0~(L0)= (~(A). For Lt = tL + ( 1 - t)Lo we want to solve L t u = - h with t = 1. For this we use the continuity method (cf. [8]). Let

S = (t e [0, 1]: L t u = - h has a solution u in (~(A)}.

First, L o u = - h is solvable by LIN [14], under all of the boundary conditions un- der consideration. Thus 0 �9 S. In order to show that 1 �9 S, which is the desired result, it is enough to show that S is both open and closed.

Note that L reduces to Lo when f - 0 and ~(x, ~) = $ for all x and ~. In particular, when L is uniformly elliptic, so is Lt, uniformly so for 0 < t ~< 1.

We begin by assuming uniform ellipticity, i.e. let (UE) hold.

222 J. A. GOLDSTEIN - C.-Y. LIN: An L p semigroup approach etc.

Let Y1 = C2+~(~), ]12 = C~(~) • Cl+~(a t)) and define G: Y1 • [0, 1]-~ ]12 by

G(u, t) = (Ltut + h, Nu).

The operator Lt was defined above, and N incorporates the boundary condition. In the context of Theorem 5, 6, 7, the definitions are, respectively,

Nu = O,

n

Nu = ~ ~ (x, Vu) v~ + fl(u), i = 1

Nu = ~ ~(x, Vu) ~ . i = 1

To show that S is open we use the implicit function theorem (cf. [8, Thin. 17.6, p. 447]).

Let z0 e S. Then there is a Uo e 0~(A) such that G(uo, ~o) = O. Since ~i e C 2+~ , a~ is smooth, and/~ e C e, it follows that G is continuously differentiable from Y1 • [0, 1] to Y2 with the partial derivative Du G(u, t)[(~0,:0) being an invertible bounded linear oper- ator from Y1 to Y2 by standard results from linear elliptic theory (cf. [8, Thm. 6.31, p. 128]). The implicit function theorem applies and we conclude that S is open.

To show that S is closed, let tk e S and suppose tk-~ t. We must show that t e S. For each k choose uk e 0~(A) such that Lt~uk = -h , i.e. G(uk, tk)= 0,.

As in the proof of dissipativity (see Section 5) we have, using ~i(x, 0) = 0,

faro (Uk)(-h) dx = f ~ (uk) uk dx + J

where

J = - am(Uk)tkf(x, uk)dx+ tk~i(x, Vuk)+(1--tk)~X i a~(Uk)~ixiukdx

- a~(Uk) tk~i(x, Vuk)+(1-- k)~x~JvidS~.

Using J I> 0 and letting m-- . ~ gives

Ilukll, < LIhll �9

Letting p ---) ~ yields

llukll < NhlL �9

Next apply a result in Ladyzhenskaya's and Ural'tseva's book [13, Thm. 2.1,

J. A. GOLDSTEIN - C.-Y. LIN: A n L p semigroup approach etc. 223

pp. 476-477] to deduce the existence of a constant K, such that

sup llukllc~ < K 1 . k

Finally, the global Schauder estimate (see [8, Thin. 6.30, p. 127]) gives a constant K2 such that

sup tf- < k

The Arzela-Ascoli theorem implies the existence of a subsequence {tk } (which we again denote by {tk}) such that uk-~ u and aruk--~ aru uniformly on ~ for I]'1 <2 , where u e C 2' ~ (~). Thus

- h = lim Ltk u k = Lu in t~, k--> ~

0 = lim Nuk = Nu on &9, k o 0

and so t e S. Thus I e S and A is essentially m-dissipative on Xp. Theorems 5-7 follows when

the assumption (UE) holds. When the operator A is not uniformly elliptic we use the ,~approximation from

above, technique of [10] as in the one dimensional case (cf. Section 6). First, for each positive integer m, let An be the operator A modified by replacing ~ by ~m where, for i , j = l , . . . , n ,

I O~i if - - > ~ l / m

( x , =

8~. [1/m if O--~j.j<l/m,

and

~m (x, ~) = ~i (x, 0) + f V~ ~ (x, O" d r , C

where C is the directed line segment from 0 to ~ in R n. Then for each h e C~(~) there is a (unique) solution um e O(A) of

u m - Am Um --- h;

this follows from the above proof applied to Am which satisfies (UE). Thus um satisfies (8) (with ~ replaced by ~m) and the boundary condition Nu = O,

where N was previously defined in this section. For definiteness we consider the non-

224 J. A. GOLDSTEIN - C.-Y. LIN: A n L p semigroup approach etc.

linear Robin condition, so that we have

I u ~ - V.~ '~(x , V u ~ ) - f ( x , u ~ ) = h in t~, (11) [ - ~ ( x , V u ~ ) . v = f l ( u ~ ) ~ on aQ.

To implement the compactness arguments we need estimates on u,~. As before, the dissipativity proof implies

Next, dividing the first equation in (11) by ~o(X)(1 + IVu~[Y we get

s~-k (x, Vu~)

f = F~ (x), x e D, a~j a 2

Um

i,j=l ~0(x)(l+lVUml} ~ axiax~

where

u,~ - h - f (x, u,~) - (x, Vu,~) i=1 r~=

~0(x)(1 + IVu~ I) ~

It follows from our assumptions that

IFn(x)l <<. constant ~ol(X).

Thus, by (DE), {Fn } is uniformly bounded in L q (f~) for some q > n/2. By standard in- terior L q estimates for elliptic problems [8, Thm. 9.11, pp. 235-235], {urn } is uniform- ly bounded in W2,q(~ o) for Q0 r162 (9. Let fh , fh , .-o for a sequence of smooth domains such that

and

~e~ 1 C ~1 C ~'~2 C . . . C ~ n C ~ n C .. . C~(~,

co

U Qk = f2, k = l

s~r ~0(x)l~l ~ ~> �88

for all x e ~k and all ~ e R ~. The uniform bound for u,~ in Lq (t~k) (with q > n/2) implies a uniform bound for u~

in C 1' ~ (~k). (For this, ~ > 0 may have to be decreased.) This follows from the Sobolev embedding theorem. Next, the interior Schauder estimate [8, Cor. 6.3, p. 93] implies that

sup llUmllC~+o(~)< oo, m

for each k. Consequently {u~ } has a subsequence converging in C~oc (Q) to a function

J. A. G O L D S T E I N - C.-Y. LIN: An L p semigroup approach etc. 225

u e C 2 (~9). Clearly u satisfies

u - V. ~(x, Vu) - f ( x , u) = h in ~ .

It remains to show that u satisfies the boundary condition. Of the three boundary conditions under consideration, the nonlinear Robin condition (see (11)) is the most complicated; this is the one we shall deal with. Easy modifications of our proof give the other two cases.

Since Um satisfies the desired boundary condition it suffices to find a uniform bound for u~ in C: (~). Write (11) as

w ( X , VUn) a 2urn : U m - h - f ( x , u m ) - i ~=1 ~Xi (x 'v~m)

i,j=l (1 "J- [VUn j) z axiaxj (1+ IVuml):

and, renaming the terms we have

(12) ~ a S (x) a2u------~m --- H m (x) (x G ~). :,~ = : axi aj

The boundary condition, using the mean value theorem together with fl(0)= = ~n (x, 0) = 0, becomes

( / ) i--~1{ / ) a ,~ ~Um

(13) fl'(Oum)dOum+ -~i ? (x, VO~tm).vdO ~ =0 on at). 0 0

Next let Xo e ~ be fnxed but otherwise arbitrary, and let No be a small neighbor- hood of xo. By flattening the boundary in No n at) we can view No n t) as

{(Xl , ... , X n ): 0 < X n • R(X:,..., xn- 1 ), (Xl , "", Xn- 1 ) E Q}

and No n at) as

{ ( x : , . . . , x ~ ) : ( x : , . . . , x ~ - l ) , ~ Q and Xn=O},

for some open set Q c R ~-1. And we can view (12) and (13) as being of the form

I j~__ a2Um . : a[~ (x) axi--Oxj " ~ H,~ (x), in No n t),

(14) ~bm(x)u,~ + n a E = o , on No a t ) . [ i = 1 ~Xi

The various norms for urn, etc. in these two (i.e. flat and curved) coordinates systems

226 J. A. GOLDSTEIN - C.-Y. LIN: An L p semigroup approach etc.

are equivalent (cf. [8, p. 96]), and we may conclude that

sup {Ib~(x)l + Ic~(x)t + IH~(x)i} < ~ .

This enables us to find an integral representation for the known solution u,~ (in No n t~) of (14) in the domain No n ~. This representation takes the form

(15) um (x) = f Gm (x, y) H,~ (y) dy, N 0 ~

for x ~ No n t~, where

IO(~o(y)-l lx-yle-~) i f n ~ > 3 ,

Gm(x,y)=[O(~o(y)_ l l o g l x _ y l ) if n = 2 ,

(by [7, p. 152 if] and [8, p. 121]). Here we are using the fact that if (~r (y)) is the n x n inverse matrix of (a~ (y)) then

5~(y)~5.j ~ ko~0 (y)l~l 2 i , j = 1

for some positive constant k0 and all ~ e R ~ all m = 1, 2, ..., and all y e No n t Z And the constant in the big ~(O, term is independent of m.

We apply the Lebesgue dominated convergence theorem to (15) and conclude that, in the limit as m--* ~, we have

u(x) = f G(x, y) H(y) dy. No n a

This requires the above estimate plus ~1 e Lq(~) for q > n and [8, p. 159]. Then by an L q potential theory estimate [15, p. 31] we conclude that u e W2'q(No n ~). It follows that u eW2'q(D). Since q>n we deduce that u e cx+~(~) for some ~ > 0 by the Sobolev embedding theorem. This completes the proof. �9

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J~ A. GOLDSTEIN - C.-Y. LIN: A n L p semigroup approach etc. 227

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