- Email: [email protected]
On quasilinear boundary value problems in domains with corners
Nonlinear Analysis,
Theory, FVintcd in Great Britain.
Methods & Applicstiom,
ON QUASILINEAR
Vol. 5,No.7,
0362.546W81M70721-15 SU2.WO 0 1981 Pergamon Press Ltd.
1981.
pp. 721-735,
BOUNDARY VALUE PROBLEMS WITH CORNERS
IN DOMAINS
PETER TOLKSDORF* Inst. f. Angew. Anal., Univetsitlt
Bonn, Betingstt. 6, 53 Bonn 1, Fed. Rep. of Germany
(Received 27 October 1980) Key wora’s: Ditichlet problem for a class of quasilinear gradient neat cornets.
equations, behaviout of the solution and its
0. INTRODUCTION BEHAVIOUR near singular boundary points of elliptic boundary value problems is of particular interest for physical applications (cf. [l]) an d numerical computations (cf. [2]). For linear equations this subject is almost completely treated by Grisvard [3] and Kondratjev [4]. They proved that the solution can be split near a plane corner or conical point into a smooth part and a linear combination of certain singular functions. For the model problem THE
-Au =
f,
u =g,
inQ, on aQ,
posed on a plane domain Q, the solution behaves like (x -x~lniBo, where 0, is the interior angle of a corner at xn. The knowledge of these singular functions can be used to improve numerical algorithms (cf. [ 11). Only recently some papers appeared treating the behaviour near corners or conical points of the solutions of certain semi&ear equations or systems. Sperner [5] and Dziuk [6] gave estimates of the type lu(x) - U(Q) 1G c . Ix - x#,
(0.1)
[Vu(x)\ c c . Ix -x&y
for vector valued solutions of the problem -Au(x)
= f(x, u(x), Vu(x)),
u(x) = 0,
in Q,
(0.2)
on asl,
near a singular boundary point XO,where the function f satisfies the growth condition
if@,4x1, VW) ( s a . IV+4 I2+ b, for some constants a, b. In this work, we consider quasilinear boundary value problems of the form -div{(k + IVU(P-~) *Vu} =
f,
u = g, * This research was supported by the Sondetfotschungsbeteich 721
in &, on 8s2,
72 of the Deutsche Fotschungsgemeinschaft.
(0.3)
PETER TOLKSDORF
722
(p > 1, k 3 0). We give estimates
of the type (0.1) and U(X) - U(Xc,)2 c . /x - X”iA,
(0.4)
(c > 0) in the neighbourhood of a corner with vertex at x0 and interior angle Q0 E (x, 2~). In the case p = 2, i.e., the Laplace equation, we can choose A and h arbitrarily close to JC/&. Forp > 2, (0.1) and (0.4) are valid with A and A greater than X/C&, and forp < 2, with Iz and A smaller than n/0,. Generally, the solution becomes smoother near the corner with increasing p and decreasing interior angle @,. The exponents A and A in (0.1) and (0.4) can be chosen independently of the constant k in (0.3). We also treat three-dimensional corners and give inequalities corresponding to (0.1). The proofs of the estimates for the solution itself are based on the observation that the functions c * r’ * sin (~0) are usable as barriers for problems of the type (0.3), with suitably chosen constants c > 0 and A, ,U E (0, 1) ( r and 8 are the plane polar coordinates). The gradient bounds are obtained similarly as by Dziuk [6] for the problem (0.2). His analysis rests on an inequality of the type ,sig, IOU(X)] d c, + c2 . r-l . sup {I+)
- u(Y) I IX>Y E
B217
(0.5)
where B, and B2 are balls with radius r and 2r, respectively. Such an estimate was proved by Nagumo [7, p. 2121, for linear equations, and by Heinz [8, p. 2281, for the semilinear problem (0.2). Our Proposition 1 generalizes Heinz’s result to the quasilinear equations considered here. As a byproduct of our proofs, we also obtain estimates of the type (0.1) near singularities of the boundary function similar to 1.x - XO/* (0 < A < 1). 1. ASSUMPTIONS
We consider
the boundary
AND
RESULTS
value problem -div
in Q,
{~(/VU(~) . VU} =f,
(1.1)
on &?.
u =g,
Here, Q is an open bounded subset of R”. The “right side” f belongs to L”(Q), and g to #P(Q) fl L”(Q), for somep E (1, 00). The function a( . ) satisfies the following conditions, for some positive constants cl, ~2, ~3, cd and all t > 0: 4t *)
3 cl . min {tp-2. (t + l)p -“},
a(t 2,
< c2. max{tP-2,
* t2J S c3. max {tp -2, (t + 1)pm2},
Ia’ a’(t) . t
a’(t).t !iEZ
(t + l)“-*},
a(t)
3 (cd - 112) . a(t),
(1.2) (1.3) (1.4) (1.5)
p-2 = ___2
.
(1.6)
In the following, u always stands for the weak H’,P(Q)-solution of (1.1). It exists, because, on account of Lemma 1, the usual assumptions for “monotone operators” (cf. Theorem 1.1 of [9]) are satisfied by the differential operator considered here. The uniqueness of u can also easily be derived from Lemma 1.
On quasilinear boundary value problems in domains with corners
123
From now on these assumptions are used without being mentioned explicitly. They allow us to apply the results of this work to the following equations, for arbitrary c 3 0: -div {(c + IVU\P-~) . Vu} = f, -div {(c + \VU[*)@-~)‘*.VU}= f. Finally, we introduce two functions A(8) and A(0) depending on 8 E (n, 27r) and p. 0(p - 2) + d4??@ - 1) + e*(p - 2)2 2*8*(JJ-1) 7
A(B) =
- 1)(2p - 3) + e*@ - 2)2 2 .8.(2~ - 3) ) p f 1.5,
e(p - 2) + q4,;(,
A(e) = i
n2/e2,
p = 1.5
Plane corners
Let XA have a plane corner with vertex precisely: There is a ball Br centered C2-curves I’i, I2 which meet at x0 including boundary function g to each Ii belongs to radius smaller than that of B1.
at x0 E &? and interior angle t9,,E (n, 279, more at x0 so that aQ n B, consists of two regular the angle f30.We assume that the restriction of the C*(IJ. By B2, we denote a ball with center x0 and
THEOREM 1.
The additional assumptions cited above imply that, for every E > 0, there is a constant c so that the solution u of (1.1) satisfies: \u(x) - u(xlJ) / c c . (x - Xol*(@y (Vu(x)( s C. lx - ~~l*(~~)-~-‘,
vx E 22, VX E
B2 n Q.
Remarks.
The influence of the various parameters upon the constant c is very difficult to describe. For example, c depends strongly on how well (p - 2)/2 is approximated by (a’(t) - f)la(t), for t large. In particular, for the gradient bound near the corner, an arbitrarily small variation of E in the exponent has more influence than any variation of the constant c. The next theorem shows how sharp the estimates of Theorem 1 are.
THEOREM 2. Let Q be the plane wedge {(r . cos 8, r .
sin @IO < r -C R and 0 < 8 < eo} with opening angle 0, E (,7~,27r), and let g = 0 in 52. Then, for every E > 0, there is a constant c such that the solution u of (1.1) satisfies
ne
1I
u(r, 19)2 den)+& . sin e. for
,
all r E (0, R/2) and 8 E (0, eo), provided that f 2 c in S2.
Because of the trivial equality div {\V(c * u)(Pd2 . V(c * u)} = cP_l . div {IVU[P-~ . VU} (c > 0), we can formulate Theorem 2 in a simpler way, for this special case.
PETERTOLKSDORF
724
THEOREM2’. Let D be as in Theorem 2. Assume that f satisfies the inequality f 2 6 in Q, for some positive 6. Then, for every E > 0, there is a positive constant c such that the solution u of the problem -div{(VuJP-2 * VU}=f, u = 0,
in 52, on an,
satisfies the following inequality, for all r E (0, R/2) and 6 E (0, 0,):
Three-dimensional
corners
We suppose that S2 has a corner with interior angle 19,E (Ed,2~)~ more precisely: There is a ball B1 centered at 0 satisfying
We assume that g vanishes in B1 fl Q. By Bz, we denote a ball with center 0 and radius smaller than that of B1. THEOREM 3. The additional assumptions cited above imply that, for every E > 0, there is a constant c such that the following inequalities hold, for all x =(x1, x2, xx) E Bz n Q: lug
s c ’ I(XI,X2))*@?
pu(x)I
=z c. J(c~,X2)/*(oo)-~--.
Singular boundary data
Let 0 be a plane domain, and p boundary function g has a singularity The function g belongs to C’(c) rl following inequalities hold, for all x
be greater than 1.5. We suppose that, at x0 E aQ, the of the type Ix - x0(*, for some /1 E (0, l), more precisely: C2(S2). M oreover, there is a constant go such that the E BI n 52 and all i, j E (1, 2):
[g(x) - g(xo)l c go * lx - &dAj
Further, we assume that there is a ball B1 centered at x0 such that a&! fl BI is a regular C2curve. By Bz, we denote a ball with center x0 and radius smaller than that of B,. THEOREM4. The assumptions cited above imply that the following inequalities hold, for some constant c: lu(x) - U(X”>l=Sc . Ix - xoia, p4(x)(
6 c * Ix - x01*-~,
vx E !a. Vx E Bz n Q.
725
On quasilinear boundary value problems in domains with corners 2. SOME
SIMPLE
INEQUALITIES
We use the abbreviations a,(V) = a(M*) * vi,
The assumptions (1.2), (1.3), (1.4) (1.5) imply that the following ellipticity and growth conditions hold, for some positive constants p and p, and all g, n E R”, q # 0:
(2.1)
(2.2) From (2.1) and (2.2), we obtain positive constants y and r such that, for every E, q E R” satisfying (q( 2 1, (2.3)
il Iail
=sr. (1 + lnl)p-?
(2.4)
LEMMA 1. There are positive constants y’ and l? so that, for every q, 9’ E R”, the following inequalities are true:
p E (L21, P E 1% cc)),
(2.5) (2.6)
Proof. For 117’1 > 1~1and t E [0, l/4], we have
~~4~l~-~‘I~Irl’+~~~r7-rl’~l~lIrll+l11’I, and
X
(qj - qj)(qi - vi)
drn
These inequalities and (2.1) imply (2.5). The estimate (2.6) is almost trivial. From the assumptions (1.2) and (1.4), we get lima(t)*IA f--trn
= 00,
lim a’(t)la(t) = 0. f--t_
(2.7) (2.8)
726 The functions
PETERTOLKSDORF n(13) and A(0) defined
in Section
q(A, l9) = -n$J
- 1) + h(p - 2) + Jt*/e*,
Q(& 0) = 472~ We need some properties
1 are the zeros of the functions
- 3) - il(p - 2) + n2(p - 1)/02.
of A, A, q, Q which we formulate
in the following
lemma.
LEMMA 2. 0
< A(e) < J-r/%, = A(0) = n(e),
p = 2,
(2.9)
Jc/nlB < A(0) < A(0) < 1, q{A( 0) + E, 13) < 0, Q(A(0)
(2.10)
- c, 0} > 0,
Proof. For p = 2, these statements are obvious. In the other from the following properties of q and Q with 0 being fixed.
cases,
they can be derived
(i) p < 2: The f unction q and, for p > 1.5, also Q has one negative zero. For p S 1.5, Q has at most one zero smaller than ~18. Moreover, q(d0, 0) =Q(d/e, 0) < 0, q(0, 0) > 0, Q(0, 0) > 0, q(A, 0) >Q(A, 0), for 0 < il < ~10. (ii) p > 2: The f unctions q and Q have one negative zero each. Moreover, q(d0, 0) = Q(d0, 0) > 0, q(1, 0) -=c0, Q(1, 0) < 0, q(A, 0) > Q(A, f3), for A > n/8.
3. STANDARD
BARRIERS
In this section, we use standard barriers to obtain two inequalities. In the first one, we estimate the solution u of (1.1) near a smooth part of the boundary. In the second one, we establish a bound for (u l,_=(o). The dependence of the constants stated there is very important for us. Therefore, we give proofs though similar estimates are known (cf. [9, p. 2951).
LEMMA 3. Let bi, b2 E H1,p(SL) be such that bi - & s 0 on IXJ, and
(3.1) for every nonnegative
Q, EHA-P(Q). Then we have 61 d 62 almost
everywhere
in Q.
Proof. In [lo] (p. 266) it is shown, for p = 2, that max (bl - b2, 0) belongs to HA,p(Q). from Therefore, we may set Q, = max (bi - b2, 0) in (3.1), and we obtain the proposition (2.5). For the proof I4C-VV
of the next lemma,
we have to define
explicitly
the quantities
Iu/c~(~J and
121
On quasilinear boundary value problems in domains with corners
Here, Q’ is an open subset of R”, u E C’(Q’),
w E C’(Q’).
4. Let B1 be a ball centered at x” with radius r E (0, l), and B1 fl a = {x’}. Suppose that the boundary function g can be extended to a function g E C’(&), where B2 is the ball with center x” and radius 2r. Let go and uo be constants such that LEMMA
IMCI(&)s r-l . go, sup iI+>
lMC+%)c r-* ’ go,
- 4.y) I Ix, Y E B21 s uo.
Then, there is a constant c depending
only on y, r, lfl~-(o),~ such that, for all x E B2 n
Q2, ILL(x)- u(x’)l s c ’ go + y + r . (x - x’ I. Proof. We set a =r . y-l . r-*, p = min (2,p), bo = em3”Y= e-3arz,fo = IflLm(Q), and
b(x) = k . (1 - exp (u(r* - lx - x”I’))}, where k is the maximum of the following four constants:
2uo+ go l-b0
r . y *(go *r-l + 1) _ go . r-l
’
2.bo+I-
2arbo ' * a-‘.
By definition of the barrier b, we get in BZ lsIV(b+g)/G12.k.a.r-1. Now, using (2.3) and (2.4) it is easy to show that, in B2, 2kabo(2ayr2 c IflL”(R)
r) - rgd.*
12(ka)* -P
The definition of k implies that u - (b + g) s 0 on a(B, fl Q). Therefore, Lemma 3 to (b + g) and u to obtain
we may apply
U(X) - u(x’) s 12 * k . a . r * lx - x’ 1, for x E Bz fl Sz. The corresponding completes the proof.
inequality for -(u(x)
- u(x’)) is shown similarly. This
728 LEMMA
PETERTOLKSDORF 5. There
is a constant c depending only on y, r, p, (fl~l(a), Igj~-cn~, R such that lul L”(Q)=Sc.
Proof. We choose a point x” E R” satisfying dist (xl’, Q) = 1, and we set
a = I- . y-i, D = sup {IX- yl lx, y E 52) + 1,
b@) = k . (1 _ e-+x”/2), where k is chosen sufficiently large such that the following inequalities hold: k
*(1- ema> 3 IgL-(~1, 2ka . eeaD2P 1,
2kaT . eMaD 3 max {IflL-p,,
!~IL-(QJ*(1 + 2kaD)‘-?.
Now, in the same way as in the proof of Lemma 4, one can show that Ju( Gb,
inn.
4,SPECIALBARRIERS
In this section, we use the cylindrical coordinates (r . cos 8, r . sin 0, z) and the plane polar coordinates (r . cos 8, r . sin 0). As indicated in the introduction, we construct barriers with the aid of the functions r’ . sin (~0). The equalities (4.1) and (4.2) of Lemma 6 are obtained by elementary calculations. In the Corollaries 1-3, we give the barriers which can be used to obtain the estimates of the Theorems 1-4, for the solution itself. The proofs of the corollaries consist mainly of exploiting the equalities (4.1) and (4.2). LEMMA
6. Let
c be a positive constant, A E (0, l), cc E (0, l), b2 E Cm(R). We set bl(r, 0)
= c . rh. sin (PO),
b(r, 8, z) = b,(r, 13)f by. Then, IVb/*= c2. r2(‘-l) . {A2 . sin’(@) + ,u~. cos*(@)} + (b$)‘,
-div {a(lVb(‘) . Vb} =
[i
{
-b;
. a(lVb12) . rAe2}
-AZ@ - 1) + A@ - 2) + p*(p - 1) -
X
+
{c. sin (~0)
2.
a’Wl*) . WI2 41Vb12)
.{a(lVbl*)
(4.1)
(p - 2) - (b;)*
+ 2. (bi)*.a’(lVb)?}.
(p - 2) .Ik2.J + p2 . cos2(pf3) I
A2 .sin2(@)
. ;;,;;,!t)}] (4.2)
129
On quasilinear boundary value problems in domains with corners
Proof. The equality (4.1) is verified directly. By elementary -div{a(lVb/‘)
‘Vb} = - $!{a(iV6/‘)
calculations,
~c~A~r”~sin(@)}
- &$j{a(lV61Z) . c . p. ?J * cos (/A@}
=
{c. sin (~0) + {c. ti-“. X
. a(lVbl*) * rid2. (p* - A”)}
a’(lV612)}
-A. sin (~0) . r-i
(IV6j’)
- p . cos (~0) -$ (lV612)) - 5 {a(lV612) * 65). This, and the equalities r *-$-(jV6[2) = 2(A-- 1) . (Vbr(*,
p * cos (pe) & (lV6l’) = 2 * sin (@I) . b’ . A* . c* ’ ?(h-l) - p*)Vbrj*}, imply (4.2). COROLLARY
1. Let go, gl, f. be positive constants, and let Sz be {(r.cose,r.sin8,z)IO
for some e, E (n, 27~). We set ~0 = Ace,), for p c 2, and Q = (A(eo) - x/0,), forp > 2. Then, for every E E (0, ~g), there are positive constants c and 6 so that the function b(r, 8, z) = c . r*(@-E . sin{$oe:z]
+ g2.,r2
satisfies, in &, -div {a(lV6(’ . Vb} >:fo,
63
gl
.
r +
g2
.
z*.
Proof. On account of (1.6), (2.8), (2.9), (2.10), Lemma 6, for every E E (0, ~g), we can find constants COand S such that, for every c 3 co, -div {a(lV612) *Vb} 3 l/2 . c * sin {zoe++i)} in 52. Corollary 1 is now easily seen to be true.
.a((V6(2) .rA(B”)-E-2.Q{A(eo)
- E, eo},
PETERTOLKSDORF
730
COROLLARY 2. Let Q be the plane wedge {(r cos 0, Y . sin 0) (0
satisfies -div
{a(/Vb(‘)
Proof. Let E E (0, GJ) be arbitrary. a r-0> 0 such that
Lemma
-div in 52, for 0 < r < rO. From
&= n/4 * (lllh
in 9,
bs0,
onaQ.
6, (1.6), (2.8),
(2.9), (2.10) imply that there
is
{a(lVbj’) . Vb} G 0,
this, the corollary
COROLLARY 3. Let R,fo, go be positive
Vb) G c,
follows.
constants,
A E (0, l), p 3 1.5. We set
- l), 80 = min {~-r/2. (lid
+ l), 3~c/2}.
We suppose that Q is contained in {(r. cos 0, r . sin 0)/O < 0 < @aand 0 < r < R}. Then, there is a positive constant c such that the function b(r, 0) = c . r’ * sin {vi satisfies,
. (8 + E)}
in Q. - div {a(/Vb/*) . Vb} zfo, b(r, 0) 2 go ’ r’.
ProoJ
For ,u =fi,
we have
-AZ@ - 1) + il(p - 2) + &JJ - 1) Therefore,
the corollary
can be derived
5. AN
0, - 2) . AZ. p2 A2sin’(@)
from Lemma
L”-BOUND
FOR
THE
> o
+ p2 cos (~0)
’
6 using (1.6) and (2.7).
GRADIENT
In the introduction, we announced the La-bound (0.5) for the gradient which depends essentially on the oscillation of the solution. This section is devoted to the proof of that estimate. Our proof requires some regularity of the solution u of (1.1) which is not known to hold for equations satisfying only the ellipticity and growth conditions (2.1) and (2.2). In order to avoid this difficulty, we approximate u by solutions of equations satisfying stronger ellipticity and growth conditions.
On quasilinear boundary value problems in domains with corners
731
PROPOSITION 1.Let B3 be a ball with center x0 and radius 3r contained in Q, and let B1 be a ball centered at x0 with radius r. Further, suppose that there is a constant uo such that
Then, there is a constant c independent
For the proof of Proposition LEMMA 7.
of x0, r and the solution u of (1.1) such that
1, we need the following lemmas.
Let E > 0, cx > 0, u E L’(Q) satisfy IAk
(u - k) dx s LY.{meas (Ak)}l+E,
for every k 2 0, where Ak is {x E Qju(x) > k}. Then l+& esjEstp n(x) 2 _ E . (ylw + E)’ (1 u
IL1(A,,))E’(l
+ E).
Proof. This result can be found in [ll, p. 711 or in [9, p. 2811. It is easily seen that one may choose the constants in the estimate in this way.
For E > 0, we set a’(t) = a(t + E), and we define a; and u$ correspondingly and aij. Let uE be the solution of (1.1) with a replaced by a’. Then, we have:
LEMMA 8.
to Uj
(9 The
inequalities (1.2), (1.3), (1.4), (1.5), (2.1), (2.2), (2.3), (2.4), (2.5), (2.6) are valid for a” with constants independent of F. (ii) For every E > 0, there are positive constants yEand JYEsuch that the following ellipticity and growth conditions hold, for every 5, r] E R”: iil
#rll5&j
i& lG(rl)
I
a YE ’ C1 +
c rE *(1 +
ld)p-2*15‘1*, I~I)@.
(5.1)
(5.2)
(iii) liiO Iu - UE(#G(Q)= 0. Proof. The statements (i) and (ii) are easily checked. For (iii), we get from the definition of U&using Holder’s inequality
732
PETER TOLKSDORF
This, together with (i), (2.5) and (2.6), G 2, we use additionally the inequality
implies
(iii), in the case p 2 2. In this other
case, p
i i, (Vu - Vzf,~ dx j2u 4 / I, (1 + (VUJ + ,Vu”,)P dx ((2-p)‘p
x * (1 + /vu/ + IVU”l)P~2. pu - Vu’)%, I
to obtain
(iii).
Proof of Proposition 1. Let us suppose that Proposition 1 holds, for ZY (E > 0), with a constant c independent of E. Moreover, suppose that B3 is contained in Q. These assumptions imply that z/ is Lipschitz-continuous uniformly with respect to E, since Lemma 5 and Lemma 8 yield a uniform bound for sup {/U”(X) -~~(y)j (x, y E B3}. Hence by Lemma 8, up + u uniformly on B3. Consequently, Proposition 1 will hold for the function II itself once the above assumptions about uE and B3 have been established. Moreover, it is easily seen that the condition B3 C Q can be dropped. Now it remains to prove Proposition 1, for up. According to Lemma 8, we may assume that (5.1) and (5.2) hold. Hence by the regularity theory (cf. [ll]), we know that U&belongs to C’(Q) fl H?$(Q). Consequently,
(5.3) holds, for every s E (1, 2, . . , n} and v E H’,2(Q) having compact support. During this proof, c stands for a generic constant independent of x0, r, ~8, E. From now on, we write u instead of uE, ai instead of a:, etc. The proof consists mainly of three parts. (i) We estimate J . (1 + (Vu/)” d_x in terms of (UO + r)/r and r. (ii) By means of a version of the iteration technique used in [ll, pp. 259-2631, we bound _r& (I +lV4)q d.x in terms of (~0 + r)lr and r, for every q E [l, cc) (B2 is the ball with center x0 and radius 2r). (iii) We show the proposition with the aid of (5.3), (ii), Lemma 7. During the whole procedure, we have to be careful not to lose powers of (uO + r)lr and r. We take a function p E C”(R) with values in [0, 11, equal to 1, for t c 1, and equal to 0, forta1.5.Fork=1,2,3 ,... ands=1,2 ,..., n,weset Q?k(x) =
u,(x) 0,
u,(x) = i (i) Testing
(1.1) with (U -Z&O)).
~(2~+’ . (Ix -x&r - 1,
4$(X)+ 19
q4 we get
u,(x)
- 2)) 2 I,
733
On quasilinear boundary value problems in domains with corners
Using (2.5) and (2.6) we obtain
Finally, by applying Holder’s inequality,
6 (ii) We shall o b tain the announced estimate ). . . . Consequently, we suppose, for k 3 1,
by induction,
for q =p + 2,p f 4,p +
(5.5)
sR
In (5.3), we set t/ =IR
$+l
* &+I,
and we use (2.3), (2.4) to obtain
(1 + jVu))P-2 * IUp
)vu,p * &+I dx
+ c * r-l . * (1 + IVUl)P+2k~lpk+l dx. I
With the aid of (5.5) and Cauchy’s inequality, this is transformed IQ
(1 + IVul)p-2.lz@
* pu,(*.
Q$+, &s
c. t-2.
into p+2k.
7 1
I
(5.6)
Only by integrating by parts one gets IQ
ux, . 4 . IU,IP+*k. cp;+, dx
s c. zig. Q (Vu,l * (1 + I +c ‘?*I,(1
(VuJ)P_ . IU,l2k+l.
+ jV~()P+2k+1’P)k+,‘k
q;+,
dx
(5.7)
Finally, using the trivial inequality
I
R
(1+IVUI)P+2(k+l).Q?2k+l~~C.r”+C.
I ~u,~u,.Iu~,p+2k.(Pi+ldX as=1
the inequalities (5.5), (5.6), (5.7), and Cauchy’s inequality, we obtain (5.5) with k replaced by k + 1. For k = 1, (5.5) can be shown in the same way using (5.4) instead of (5.5), for k
134
= 0. From
PETER TOLKSDORF
(_5.5), we obtain
with the aid of Holder’s
inequality u”+r
4
1 1
I
(1 + /VU~)~~XGCC~~. B2
-
r
(5.8)
’
for all q E [l, co). (iii) We set T(X) =p(lx - x0//r), (J = max(2, p), Ak = {x E Q / u,(x) . q(x) > k}, for k 2 0. In the following, the generic constant c will also be independent of k. We use (5.3) with i$ = +-1. to obtain max (u, . cp -k, 0), (2.3), (2.4), and Cauchy’s inequality
IAk
(1 + ]VU/)~-~ * /Vu,)‘. @d.z G c .r-‘.
Now we have to distinguish
The integral
inequality
and the Sobolev
Ak (u, . q - k) dx 4 c . {meas (Ak)}l’n
on the right side can be estimated
Ii
C’
(1 +
I
(1 + pq
imbedding
theorem,
we get
L, lV(u, . q)I dx.
by
puj)p-2 * (Vu,12. p12 dx
At
+ c.rml.
(5.9)
two cases
p c 2: With the aid of Holder’s
i
(1 + )VuI)” &. IAk
11’2. Ii,/
+ ~VUl)‘~“&~“~
dx.
Ak
Combining
this with (5.8) and (5.9) we obtain
by Holder’s
Ak(U, . q _ k) & < c . r-l/O . !f!$
inequality
that
. {meas (,+)}l+[l’h’)l.
(5.10)
I p a 2: In this case, we get with the aid of Holder’s theorem
I The integral
Ak (us
inequality
imbedding
(Ak)}l-[(“-2)‘@‘d . /i,, IV(4. dPi212 d.x1Ih. *Q,- k) b < c. {meas
on the right side can be estimated c.
I
by
A,(1 + JV~~)p~2~~V~.~Jz-~pdx+c~r~2~~
Now we can conclude in the same way as above In both cases, (5.10) and Lemma 7 imply
inequality
for -u,
(1 + IVul)pdX. Ak
that (5.10) holds.
u0 + r ess sup U,(X) G c . r XEB, The corresponding
and the Sobolev
can be shown similarly.
On quasilinear boundary value problems in domains with corners 6. PROOF
135
OF THE THEOREMS
Proof of Theorem 1. The estimate for the solution itself follows from Lemma 3, Lemma 5 and Corollary 1. In some extreme cases it may happen that the barrier of Corollary 1 does not belong to H’lP(Q). However, one can approximate this barrier in a suitable manner, for example by drawing the singularity away from Q. The smoothness of the curves Ii guarantees the existence of an & E (0, l/4) so that, for every x’ E as2flBt, there is a ball B’ with radius a . (x’ - xg( satisfying 3 fl B’ = {x’}. Employing Proposition 1, Lemma 4, and the estimate for the solution itself we get the gradient bound, for all x E B2 fl Q whose distance to the boundary is smaller than a/2 . Ix - x01. For the remaining x E BZ II S2, we obtain the gradient bound directly from Proposition 1 and the estimate for the solution. Theorem 2 follows directly from Lemma 3 and Corollary 2. Theorem 3 and 4 can be proven similarly as Theorem 1. For Theorem 4 one has to use Corollary 3 instead of Corollary 1. REFERENCES 1. JACKSONJ. D., Clussical Electrodynamics, John Wiley & Sons, New York (1975). 2. STRANGG. & FIX G. J., An Analysis ofrhe Finite Element Mefhod, Prentice-Hall, Englewood Cliffs (1973). 3. GRISVARDp., Behaviour of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Num. Sol. part. diff. Eqns 3, 207-274 (1976). 4. KONDRATJEVV. A., Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Sot. 1967, 227-313 (1968).
5. SPERNERE., Nichtlineare elliptische Differentia[gleichungen in singuliiren Gebieten, Habilitationsschrift, Universitat Bayreuth (1979). 6. DZUK G., Das Verhalten von Losungen semilinearer elliptischer Systeme an Ecken eines Gebietes, Math, Z. 159, 88-100 (1978).
7. NAGUMOM., On principally linear elliptic differential equations of the second order, Osaka math. J. 6, 207-229 (1954).
8. HEINZ E., On certain nonlinear elliptic differential
equations and univalent mappings, J. d’Ana1. Math. 5,
197-272 (1956157).
9. HARTMANP. & STAMPACCHIA G., On some non-linear elliptic differential-functional equations, Acfa Math. 115, 271-310 (1966). 10. TREVESF., Basic Linear Partial Differential Equations, Academic Press, New York (1975). 11. LADYZHENSKAYA 0. A. & URALTSEVAN. N., Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968).
Theory, FVintcd in Great Britain.
Methods & Applicstiom,
ON QUASILINEAR
Vol. 5,No.7,
0362.546W81M70721-15 SU2.WO 0 1981 Pergamon Press Ltd.
1981.
pp. 721-735,
BOUNDARY VALUE PROBLEMS WITH CORNERS
IN DOMAINS
PETER TOLKSDORF* Inst. f. Angew. Anal., Univetsitlt
Bonn, Betingstt. 6, 53 Bonn 1, Fed. Rep. of Germany
(Received 27 October 1980) Key wora’s: Ditichlet problem for a class of quasilinear gradient neat cornets.
equations, behaviout of the solution and its
0. INTRODUCTION BEHAVIOUR near singular boundary points of elliptic boundary value problems is of particular interest for physical applications (cf. [l]) an d numerical computations (cf. [2]). For linear equations this subject is almost completely treated by Grisvard [3] and Kondratjev [4]. They proved that the solution can be split near a plane corner or conical point into a smooth part and a linear combination of certain singular functions. For the model problem THE
-Au =
f,
u =g,
inQ, on aQ,
posed on a plane domain Q, the solution behaves like (x -x~lniBo, where 0, is the interior angle of a corner at xn. The knowledge of these singular functions can be used to improve numerical algorithms (cf. [ 11). Only recently some papers appeared treating the behaviour near corners or conical points of the solutions of certain semi&ear equations or systems. Sperner [5] and Dziuk [6] gave estimates of the type lu(x) - U(Q) 1G c . Ix - x#,
(0.1)
[Vu(x)\ c c . Ix -x&y
for vector valued solutions of the problem -Au(x)
= f(x, u(x), Vu(x)),
u(x) = 0,
in Q,
(0.2)
on asl,
near a singular boundary point XO,where the function f satisfies the growth condition
if@,4x1, VW) ( s a . IV+4 I2+ b, for some constants a, b. In this work, we consider quasilinear boundary value problems of the form -div{(k + IVU(P-~) *Vu} =
f,
u = g, * This research was supported by the Sondetfotschungsbeteich 721
in &, on 8s2,
72 of the Deutsche Fotschungsgemeinschaft.
(0.3)
PETER TOLKSDORF
722
(p > 1, k 3 0). We give estimates
of the type (0.1) and U(X) - U(Xc,)2 c . /x - X”iA,
(0.4)
(c > 0) in the neighbourhood of a corner with vertex at x0 and interior angle Q0 E (x, 2~). In the case p = 2, i.e., the Laplace equation, we can choose A and h arbitrarily close to JC/&. Forp > 2, (0.1) and (0.4) are valid with A and A greater than X/C&, and forp < 2, with Iz and A smaller than n/0,. Generally, the solution becomes smoother near the corner with increasing p and decreasing interior angle @,. The exponents A and A in (0.1) and (0.4) can be chosen independently of the constant k in (0.3). We also treat three-dimensional corners and give inequalities corresponding to (0.1). The proofs of the estimates for the solution itself are based on the observation that the functions c * r’ * sin (~0) are usable as barriers for problems of the type (0.3), with suitably chosen constants c > 0 and A, ,U E (0, 1) ( r and 8 are the plane polar coordinates). The gradient bounds are obtained similarly as by Dziuk [6] for the problem (0.2). His analysis rests on an inequality of the type ,sig, IOU(X)] d c, + c2 . r-l . sup {I+)
- u(Y) I IX>Y E
B217
(0.5)
where B, and B2 are balls with radius r and 2r, respectively. Such an estimate was proved by Nagumo [7, p. 2121, for linear equations, and by Heinz [8, p. 2281, for the semilinear problem (0.2). Our Proposition 1 generalizes Heinz’s result to the quasilinear equations considered here. As a byproduct of our proofs, we also obtain estimates of the type (0.1) near singularities of the boundary function similar to 1.x - XO/* (0 < A < 1). 1. ASSUMPTIONS
We consider
the boundary
AND
RESULTS
value problem -div
in Q,
{~(/VU(~) . VU} =f,
(1.1)
on &?.
u =g,
Here, Q is an open bounded subset of R”. The “right side” f belongs to L”(Q), and g to #P(Q) fl L”(Q), for somep E (1, 00). The function a( . ) satisfies the following conditions, for some positive constants cl, ~2, ~3, cd and all t > 0: 4t *)
3 cl . min {tp-2. (t + l)p -“},
a(t 2,
< c2. max{tP-2,
* t2J S c3. max {tp -2, (t + 1)pm2},
Ia’ a’(t) . t
a’(t).t !iEZ
(t + l)“-*},
a(t)
3 (cd - 112) . a(t),
(1.2) (1.3) (1.4) (1.5)
p-2 = ___2
.
(1.6)
In the following, u always stands for the weak H’,P(Q)-solution of (1.1). It exists, because, on account of Lemma 1, the usual assumptions for “monotone operators” (cf. Theorem 1.1 of [9]) are satisfied by the differential operator considered here. The uniqueness of u can also easily be derived from Lemma 1.
On quasilinear boundary value problems in domains with corners
123
From now on these assumptions are used without being mentioned explicitly. They allow us to apply the results of this work to the following equations, for arbitrary c 3 0: -div {(c + IVU\P-~) . Vu} = f, -div {(c + \VU[*)@-~)‘*.VU}= f. Finally, we introduce two functions A(8) and A(0) depending on 8 E (n, 27r) and p. 0(p - 2) + d4??@ - 1) + e*(p - 2)2 2*8*(JJ-1) 7
A(B) =
- 1)(2p - 3) + e*@ - 2)2 2 .8.(2~ - 3) ) p f 1.5,
e(p - 2) + q4,;(,
A(e) = i
n2/e2,
p = 1.5
Plane corners
Let XA have a plane corner with vertex precisely: There is a ball Br centered C2-curves I’i, I2 which meet at x0 including boundary function g to each Ii belongs to radius smaller than that of B1.
at x0 E &? and interior angle t9,,E (n, 279, more at x0 so that aQ n B, consists of two regular the angle f30.We assume that the restriction of the C*(IJ. By B2, we denote a ball with center x0 and
THEOREM 1.
The additional assumptions cited above imply that, for every E > 0, there is a constant c so that the solution u of (1.1) satisfies: \u(x) - u(xlJ) / c c . (x - Xol*(@y (Vu(x)( s C. lx - ~~l*(~~)-~-‘,
vx E 22, VX E
B2 n Q.
Remarks.
The influence of the various parameters upon the constant c is very difficult to describe. For example, c depends strongly on how well (p - 2)/2 is approximated by (a’(t) - f)la(t), for t large. In particular, for the gradient bound near the corner, an arbitrarily small variation of E in the exponent has more influence than any variation of the constant c. The next theorem shows how sharp the estimates of Theorem 1 are.
THEOREM 2. Let Q be the plane wedge {(r . cos 8, r .
sin @IO < r -C R and 0 < 8 < eo} with opening angle 0, E (,7~,27r), and let g = 0 in 52. Then, for every E > 0, there is a constant c such that the solution u of (1.1) satisfies
ne
1I
u(r, 19)2 den)+& . sin e. for
,
all r E (0, R/2) and 8 E (0, eo), provided that f 2 c in S2.
Because of the trivial equality div {\V(c * u)(Pd2 . V(c * u)} = cP_l . div {IVU[P-~ . VU} (c > 0), we can formulate Theorem 2 in a simpler way, for this special case.
PETERTOLKSDORF
724
THEOREM2’. Let D be as in Theorem 2. Assume that f satisfies the inequality f 2 6 in Q, for some positive 6. Then, for every E > 0, there is a positive constant c such that the solution u of the problem -div{(VuJP-2 * VU}=f, u = 0,
in 52, on an,
satisfies the following inequality, for all r E (0, R/2) and 6 E (0, 0,):
Three-dimensional
corners
We suppose that S2 has a corner with interior angle 19,E (Ed,2~)~ more precisely: There is a ball B1 centered at 0 satisfying
We assume that g vanishes in B1 fl Q. By Bz, we denote a ball with center 0 and radius smaller than that of B1. THEOREM 3. The additional assumptions cited above imply that, for every E > 0, there is a constant c such that the following inequalities hold, for all x =(x1, x2, xx) E Bz n Q: lug
s c ’ I(XI,X2))*@?
pu(x)I
=z c. J(c~,X2)/*(oo)-~--.
Singular boundary data
Let 0 be a plane domain, and p boundary function g has a singularity The function g belongs to C’(c) rl following inequalities hold, for all x
be greater than 1.5. We suppose that, at x0 E aQ, the of the type Ix - x0(*, for some /1 E (0, l), more precisely: C2(S2). M oreover, there is a constant go such that the E BI n 52 and all i, j E (1, 2):
[g(x) - g(xo)l c go * lx - &dAj
Further, we assume that there is a ball B1 centered at x0 such that a&! fl BI is a regular C2curve. By Bz, we denote a ball with center x0 and radius smaller than that of B,. THEOREM4. The assumptions cited above imply that the following inequalities hold, for some constant c: lu(x) - U(X”>l=Sc . Ix - xoia, p4(x)(
6 c * Ix - x01*-~,
vx E !a. Vx E Bz n Q.
725
On quasilinear boundary value problems in domains with corners 2. SOME
SIMPLE
INEQUALITIES
We use the abbreviations a,(V) = a(M*) * vi,
The assumptions (1.2), (1.3), (1.4) (1.5) imply that the following ellipticity and growth conditions hold, for some positive constants p and p, and all g, n E R”, q # 0:
(2.1)
(2.2) From (2.1) and (2.2), we obtain positive constants y and r such that, for every E, q E R” satisfying (q( 2 1, (2.3)
il Iail
=sr. (1 + lnl)p-?
(2.4)
LEMMA 1. There are positive constants y’ and l? so that, for every q, 9’ E R”, the following inequalities are true:
p E (L21, P E 1% cc)),
(2.5) (2.6)
Proof. For 117’1 > 1~1and t E [0, l/4], we have
~~4~l~-~‘I~Irl’+~~~r7-rl’~l~lIrll+l11’I, and
X
(qj - qj)(qi - vi)
drn
These inequalities and (2.1) imply (2.5). The estimate (2.6) is almost trivial. From the assumptions (1.2) and (1.4), we get lima(t)*IA f--trn
= 00,
lim a’(t)la(t) = 0. f--t_
(2.7) (2.8)
726 The functions
PETERTOLKSDORF n(13) and A(0) defined
in Section
q(A, l9) = -n$J
- 1) + h(p - 2) + Jt*/e*,
Q(& 0) = 472~ We need some properties
1 are the zeros of the functions
- 3) - il(p - 2) + n2(p - 1)/02.
of A, A, q, Q which we formulate
in the following
lemma.
LEMMA 2. 0
< A(e) < J-r/%, = A(0) = n(e),
p = 2,
(2.9)
Jc/nlB < A(0) < A(0) < 1, q{A( 0) + E, 13) < 0, Q(A(0)
(2.10)
- c, 0} > 0,
Proof. For p = 2, these statements are obvious. In the other from the following properties of q and Q with 0 being fixed.
cases,
they can be derived
(i) p < 2: The f unction q and, for p > 1.5, also Q has one negative zero. For p S 1.5, Q has at most one zero smaller than ~18. Moreover, q(d0, 0) =Q(d/e, 0) < 0, q(0, 0) > 0, Q(0, 0) > 0, q(A, 0) >Q(A, 0), for 0 < il < ~10. (ii) p > 2: The f unctions q and Q have one negative zero each. Moreover, q(d0, 0) = Q(d0, 0) > 0, q(1, 0) -=c0, Q(1, 0) < 0, q(A, 0) > Q(A, f3), for A > n/8.
3. STANDARD
BARRIERS
In this section, we use standard barriers to obtain two inequalities. In the first one, we estimate the solution u of (1.1) near a smooth part of the boundary. In the second one, we establish a bound for (u l,_=(o). The dependence of the constants stated there is very important for us. Therefore, we give proofs though similar estimates are known (cf. [9, p. 2951).
LEMMA 3. Let bi, b2 E H1,p(SL) be such that bi - & s 0 on IXJ, and
(3.1) for every nonnegative
Q, EHA-P(Q). Then we have 61 d 62 almost
everywhere
in Q.
Proof. In [lo] (p. 266) it is shown, for p = 2, that max (bl - b2, 0) belongs to HA,p(Q). from Therefore, we may set Q, = max (bi - b2, 0) in (3.1), and we obtain the proposition (2.5). For the proof I4C-VV
of the next lemma,
we have to define
explicitly
the quantities
Iu/c~(~J and
121
On quasilinear boundary value problems in domains with corners
Here, Q’ is an open subset of R”, u E C’(Q’),
w E C’(Q’).
4. Let B1 be a ball centered at x” with radius r E (0, l), and B1 fl a = {x’}. Suppose that the boundary function g can be extended to a function g E C’(&), where B2 is the ball with center x” and radius 2r. Let go and uo be constants such that LEMMA
IMCI(&)s r-l . go, sup iI+>
lMC+%)c r-* ’ go,
- 4.y) I Ix, Y E B21 s uo.
Then, there is a constant c depending
only on y, r, lfl~-(o),~ such that, for all x E B2 n
Q2, ILL(x)- u(x’)l s c ’ go + y + r . (x - x’ I. Proof. We set a =r . y-l . r-*, p = min (2,p), bo = em3”Y= e-3arz,fo = IflLm(Q), and
b(x) = k . (1 - exp (u(r* - lx - x”I’))}, where k is the maximum of the following four constants:
2uo+ go l-b0
r . y *(go *r-l + 1) _ go . r-l
’
2.bo+I-
2arbo ' * a-‘.
By definition of the barrier b, we get in BZ lsIV(b+g)/G12.k.a.r-1. Now, using (2.3) and (2.4) it is easy to show that, in B2, 2kabo(2ayr2 c IflL”(R)
r) - rgd.*
12(ka)* -P
The definition of k implies that u - (b + g) s 0 on a(B, fl Q). Therefore, Lemma 3 to (b + g) and u to obtain
we may apply
U(X) - u(x’) s 12 * k . a . r * lx - x’ 1, for x E Bz fl Sz. The corresponding completes the proof.
inequality for -(u(x)
- u(x’)) is shown similarly. This
728 LEMMA
PETERTOLKSDORF 5. There
is a constant c depending only on y, r, p, (fl~l(a), Igj~-cn~, R such that lul L”(Q)=Sc.
Proof. We choose a point x” E R” satisfying dist (xl’, Q) = 1, and we set
a = I- . y-i, D = sup {IX- yl lx, y E 52) + 1,
b@) = k . (1 _ e-+x”/2), where k is chosen sufficiently large such that the following inequalities hold: k
*(1- ema> 3 IgL-(~1, 2ka . eeaD2P 1,
2kaT . eMaD 3 max {IflL-p,,
!~IL-(QJ*(1 + 2kaD)‘-?.
Now, in the same way as in the proof of Lemma 4, one can show that Ju( Gb,
inn.
4,SPECIALBARRIERS
In this section, we use the cylindrical coordinates (r . cos 8, r . sin 0, z) and the plane polar coordinates (r . cos 8, r . sin 0). As indicated in the introduction, we construct barriers with the aid of the functions r’ . sin (~0). The equalities (4.1) and (4.2) of Lemma 6 are obtained by elementary calculations. In the Corollaries 1-3, we give the barriers which can be used to obtain the estimates of the Theorems 1-4, for the solution itself. The proofs of the corollaries consist mainly of exploiting the equalities (4.1) and (4.2). LEMMA
6. Let
c be a positive constant, A E (0, l), cc E (0, l), b2 E Cm(R). We set bl(r, 0)
= c . rh. sin (PO),
b(r, 8, z) = b,(r, 13)f by. Then, IVb/*= c2. r2(‘-l) . {A2 . sin’(@) + ,u~. cos*(@)} + (b$)‘,
-div {a(lVb(‘) . Vb} =
[i
{
-b;
. a(lVb12) . rAe2}
-AZ@ - 1) + A@ - 2) + p*(p - 1) -
X
+
{c. sin (~0)
2.
a’Wl*) . WI2 41Vb12)
.{a(lVbl*)
(4.1)
(p - 2) - (b;)*
+ 2. (bi)*.a’(lVb)?}.
(p - 2) .Ik2.J + p2 . cos2(pf3) I
A2 .sin2(@)
. ;;,;;,!t)}] (4.2)
129
On quasilinear boundary value problems in domains with corners
Proof. The equality (4.1) is verified directly. By elementary -div{a(lVb/‘)
‘Vb} = - $!{a(iV6/‘)
calculations,
~c~A~r”~sin(@)}
- &$j{a(lV61Z) . c . p. ?J * cos (/A@}
=
{c. sin (~0) + {c. ti-“. X
. a(lVbl*) * rid2. (p* - A”)}
a’(lV612)}
-A. sin (~0) . r-i
(IV6j’)
- p . cos (~0) -$ (lV612)) - 5 {a(lV612) * 65). This, and the equalities r *-$-(jV6[2) = 2(A-- 1) . (Vbr(*,
p * cos (pe) & (lV6l’) = 2 * sin (@I) . b’ . A* . c* ’ ?(h-l) - p*)Vbrj*}, imply (4.2). COROLLARY
1. Let go, gl, f. be positive constants, and let Sz be {(r.cose,r.sin8,z)IO
for some e, E (n, 27~). We set ~0 = Ace,), for p c 2, and Q = (A(eo) - x/0,), forp > 2. Then, for every E E (0, ~g), there are positive constants c and 6 so that the function b(r, 8, z) = c . r*(@-E . sin{$oe:z]
+ g2.,r2
satisfies, in &, -div {a(lV6(’ . Vb} >:fo,
63
gl
.
r +
g2
.
z*.
Proof. On account of (1.6), (2.8), (2.9), (2.10), Lemma 6, for every E E (0, ~g), we can find constants COand S such that, for every c 3 co, -div {a(lV612) *Vb} 3 l/2 . c * sin {zoe++i)} in 52. Corollary 1 is now easily seen to be true.
.a((V6(2) .rA(B”)-E-2.Q{A(eo)
- E, eo},
PETERTOLKSDORF
730
COROLLARY 2. Let Q be the plane wedge {(r cos 0, Y . sin 0) (0
satisfies -div
{a(/Vb(‘)
Proof. Let E E (0, GJ) be arbitrary. a r-0> 0 such that
Lemma
-div in 52, for 0 < r < rO. From
&= n/4 * (lllh
in 9,
bs0,
onaQ.
6, (1.6), (2.8),
(2.9), (2.10) imply that there
is
{a(lVbj’) . Vb} G 0,
this, the corollary
COROLLARY 3. Let R,fo, go be positive
Vb) G c,
follows.
constants,
A E (0, l), p 3 1.5. We set
- l), 80 = min {~-r/2. (lid
+ l), 3~c/2}.
We suppose that Q is contained in {(r. cos 0, r . sin 0)/O < 0 < @aand 0 < r < R}. Then, there is a positive constant c such that the function b(r, 0) = c . r’ * sin {vi satisfies,
. (8 + E)}
in Q. - div {a(/Vb/*) . Vb} zfo, b(r, 0) 2 go ’ r’.
ProoJ
For ,u =fi,
we have
-AZ@ - 1) + il(p - 2) + &JJ - 1) Therefore,
the corollary
can be derived
5. AN
0, - 2) . AZ. p2 A2sin’(@)
from Lemma
L”-BOUND
FOR
THE
> o
+ p2 cos (~0)
’
6 using (1.6) and (2.7).
GRADIENT
In the introduction, we announced the La-bound (0.5) for the gradient which depends essentially on the oscillation of the solution. This section is devoted to the proof of that estimate. Our proof requires some regularity of the solution u of (1.1) which is not known to hold for equations satisfying only the ellipticity and growth conditions (2.1) and (2.2). In order to avoid this difficulty, we approximate u by solutions of equations satisfying stronger ellipticity and growth conditions.
On quasilinear boundary value problems in domains with corners
731
PROPOSITION 1.Let B3 be a ball with center x0 and radius 3r contained in Q, and let B1 be a ball centered at x0 with radius r. Further, suppose that there is a constant uo such that
Then, there is a constant c independent
For the proof of Proposition LEMMA 7.
of x0, r and the solution u of (1.1) such that
1, we need the following lemmas.
Let E > 0, cx > 0, u E L’(Q) satisfy IAk
(u - k) dx s LY.{meas (Ak)}l+E,
for every k 2 0, where Ak is {x E Qju(x) > k}. Then l+& esjEstp n(x) 2 _ E . (ylw + E)’ (1 u
IL1(A,,))E’(l
+ E).
Proof. This result can be found in [ll, p. 711 or in [9, p. 2811. It is easily seen that one may choose the constants in the estimate in this way.
For E > 0, we set a’(t) = a(t + E), and we define a; and u$ correspondingly and aij. Let uE be the solution of (1.1) with a replaced by a’. Then, we have:
LEMMA 8.
to Uj
(9 The
inequalities (1.2), (1.3), (1.4), (1.5), (2.1), (2.2), (2.3), (2.4), (2.5), (2.6) are valid for a” with constants independent of F. (ii) For every E > 0, there are positive constants yEand JYEsuch that the following ellipticity and growth conditions hold, for every 5, r] E R”: iil
#rll5&j
i& lG(rl)
I
a YE ’ C1 +
c rE *(1 +
ld)p-2*15‘1*, I~I)@.
(5.1)
(5.2)
(iii) liiO Iu - UE(#G(Q)= 0. Proof. The statements (i) and (ii) are easily checked. For (iii), we get from the definition of U&using Holder’s inequality
732
PETER TOLKSDORF
This, together with (i), (2.5) and (2.6), G 2, we use additionally the inequality
implies
(iii), in the case p 2 2. In this other
case, p
i i, (Vu - Vzf,~ dx j2u 4 / I, (1 + (VUJ + ,Vu”,)P dx ((2-p)‘p
x * (1 + /vu/ + IVU”l)P~2. pu - Vu’)%, I
to obtain
(iii).
Proof of Proposition 1. Let us suppose that Proposition 1 holds, for ZY (E > 0), with a constant c independent of E. Moreover, suppose that B3 is contained in Q. These assumptions imply that z/ is Lipschitz-continuous uniformly with respect to E, since Lemma 5 and Lemma 8 yield a uniform bound for sup {/U”(X) -~~(y)j (x, y E B3}. Hence by Lemma 8, up + u uniformly on B3. Consequently, Proposition 1 will hold for the function II itself once the above assumptions about uE and B3 have been established. Moreover, it is easily seen that the condition B3 C Q can be dropped. Now it remains to prove Proposition 1, for up. According to Lemma 8, we may assume that (5.1) and (5.2) hold. Hence by the regularity theory (cf. [ll]), we know that U&belongs to C’(Q) fl H?$(Q). Consequently,
(5.3) holds, for every s E (1, 2, . . , n} and v E H’,2(Q) having compact support. During this proof, c stands for a generic constant independent of x0, r, ~8, E. From now on, we write u instead of uE, ai instead of a:, etc. The proof consists mainly of three parts. (i) We estimate J . (1 + (Vu/)” d_x in terms of (UO + r)/r and r. (ii) By means of a version of the iteration technique used in [ll, pp. 259-2631, we bound _r& (I +lV4)q d.x in terms of (~0 + r)lr and r, for every q E [l, cc) (B2 is the ball with center x0 and radius 2r). (iii) We show the proposition with the aid of (5.3), (ii), Lemma 7. During the whole procedure, we have to be careful not to lose powers of (uO + r)lr and r. We take a function p E C”(R) with values in [0, 11, equal to 1, for t c 1, and equal to 0, forta1.5.Fork=1,2,3 ,... ands=1,2 ,..., n,weset Q?k(x) =
u,(x) 0,
u,(x) = i (i) Testing
(1.1) with (U -Z&O)).
~(2~+’ . (Ix -x&r - 1,
4$(X)+ 19
q4 we get
u,(x)
- 2)) 2 I,
733
On quasilinear boundary value problems in domains with corners
Using (2.5) and (2.6) we obtain
Finally, by applying Holder’s inequality,
6 (ii) We shall o b tain the announced estimate ). . . . Consequently, we suppose, for k 3 1,
by induction,
for q =p + 2,p f 4,p +
(5.5)
sR
In (5.3), we set t/ =IR
$+l
* &+I,
and we use (2.3), (2.4) to obtain
(1 + jVu))P-2 * IUp
)vu,p * &+I dx
+ c * r-l . * (1 + IVUl)P+2k~lpk+l dx. I
With the aid of (5.5) and Cauchy’s inequality, this is transformed IQ
(1 + IVul)p-2.lz@
* pu,(*.
Q$+, &s
c. t-2.
into p+2k.
7 1
I
(5.6)
Only by integrating by parts one gets IQ
ux, . 4 . IU,IP+*k. cp;+, dx
s c. zig. Q (Vu,l * (1 + I +c ‘?*I,(1
(VuJ)P_ . IU,l2k+l.
+ jV~()P+2k+1’P)k+,‘k
q;+,
dx
(5.7)
Finally, using the trivial inequality
I
R
(1+IVUI)P+2(k+l).Q?2k+l~~C.r”+C.
I ~u,~u,.Iu~,p+2k.(Pi+ldX as=1
the inequalities (5.5), (5.6), (5.7), and Cauchy’s inequality, we obtain (5.5) with k replaced by k + 1. For k = 1, (5.5) can be shown in the same way using (5.4) instead of (5.5), for k
134
= 0. From
PETER TOLKSDORF
(_5.5), we obtain
with the aid of Holder’s
inequality u”+r
4
1 1
I
(1 + /VU~)~~XGCC~~. B2
-
r
(5.8)
’
for all q E [l, co). (iii) We set T(X) =p(lx - x0//r), (J = max(2, p), Ak = {x E Q / u,(x) . q(x) > k}, for k 2 0. In the following, the generic constant c will also be independent of k. We use (5.3) with i$ = +-1. to obtain max (u, . cp -k, 0), (2.3), (2.4), and Cauchy’s inequality
IAk
(1 + ]VU/)~-~ * /Vu,)‘. @d.z G c .r-‘.
Now we have to distinguish
The integral
inequality
and the Sobolev
Ak (u, . q - k) dx 4 c . {meas (Ak)}l’n
on the right side can be estimated
Ii
C’
(1 +
I
(1 + pq
imbedding
theorem,
we get
L, lV(u, . q)I dx.
by
puj)p-2 * (Vu,12. p12 dx
At
+ c.rml.
(5.9)
two cases
p c 2: With the aid of Holder’s
i
(1 + )VuI)” &. IAk
11’2. Ii,/
+ ~VUl)‘~“&~“~
dx.
Ak
Combining
this with (5.8) and (5.9) we obtain
by Holder’s
Ak(U, . q _ k) & < c . r-l/O . !f!$
inequality
that
. {meas (,+)}l+[l’h’)l.
(5.10)
I p a 2: In this case, we get with the aid of Holder’s theorem
I The integral
Ak (us
inequality
imbedding
(Ak)}l-[(“-2)‘@‘d . /i,, IV(4. dPi212 d.x1Ih. *Q,- k) b < c. {meas
on the right side can be estimated c.
I
by
A,(1 + JV~~)p~2~~V~.~Jz-~pdx+c~r~2~~
Now we can conclude in the same way as above In both cases, (5.10) and Lemma 7 imply
inequality
for -u,
(1 + IVul)pdX. Ak
that (5.10) holds.
u0 + r ess sup U,(X) G c . r XEB, The corresponding
and the Sobolev
can be shown similarly.
On quasilinear boundary value problems in domains with corners 6. PROOF
135
OF THE THEOREMS
Proof of Theorem 1. The estimate for the solution itself follows from Lemma 3, Lemma 5 and Corollary 1. In some extreme cases it may happen that the barrier of Corollary 1 does not belong to H’lP(Q). However, one can approximate this barrier in a suitable manner, for example by drawing the singularity away from Q. The smoothness of the curves Ii guarantees the existence of an & E (0, l/4) so that, for every x’ E as2flBt, there is a ball B’ with radius a . (x’ - xg( satisfying 3 fl B’ = {x’}. Employing Proposition 1, Lemma 4, and the estimate for the solution itself we get the gradient bound, for all x E B2 fl Q whose distance to the boundary is smaller than a/2 . Ix - x01. For the remaining x E BZ II S2, we obtain the gradient bound directly from Proposition 1 and the estimate for the solution. Theorem 2 follows directly from Lemma 3 and Corollary 2. Theorem 3 and 4 can be proven similarly as Theorem 1. For Theorem 4 one has to use Corollary 3 instead of Corollary 1. REFERENCES 1. JACKSONJ. D., Clussical Electrodynamics, John Wiley & Sons, New York (1975). 2. STRANGG. & FIX G. J., An Analysis ofrhe Finite Element Mefhod, Prentice-Hall, Englewood Cliffs (1973). 3. GRISVARDp., Behaviour of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Num. Sol. part. diff. Eqns 3, 207-274 (1976). 4. KONDRATJEVV. A., Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Sot. 1967, 227-313 (1968).
5. SPERNERE., Nichtlineare elliptische Differentia[gleichungen in singuliiren Gebieten, Habilitationsschrift, Universitat Bayreuth (1979). 6. DZUK G., Das Verhalten von Losungen semilinearer elliptischer Systeme an Ecken eines Gebietes, Math, Z. 159, 88-100 (1978).
7. NAGUMOM., On principally linear elliptic differential equations of the second order, Osaka math. J. 6, 207-229 (1954).
8. HEINZ E., On certain nonlinear elliptic differential
equations and univalent mappings, J. d’Ana1. Math. 5,
197-272 (1956157).
9. HARTMANP. & STAMPACCHIA G., On some non-linear elliptic differential-functional equations, Acfa Math. 115, 271-310 (1966). 10. TREVESF., Basic Linear Partial Differential Equations, Academic Press, New York (1975). 11. LADYZHENSKAYA 0. A. & URALTSEVAN. N., Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968).