A behavior of generalized solutions of the Dirichlet problem for quasilienar elliptic divergence equations of second order near a conical point

A BEHAVIOR OF GENERALIZED SOLUTIONS OF THE DIRICHLET PROBLEM FOR

QUASILIENARELLIPTICDIVERGENCE

EQUATIONS OF SECOND ORDER NEAR A

CONICAL POINT M. V. Borsuk

UDC 517.956.25

The behavior of solutions of boundary-value problems for linear elliptic equations near corner, conical, and other irregular boundary points, is well known. The modern status of the theory of linear boundary-value problems in nonsmooth domains has been presented in the survey [i]. This can be appended by a recently published paper [2]. Also, during the last ten years, there has been developed the theory of regularity of solutions of elliptic boundary-value problems in nonsmooth domains for equations of second order [3-12]~ In [3-5] there has been discussed a generalization of solutions of the elliptic equation diva(Vu) = f(x), where a(p) is a strongly monotone coercion field of class C I. In papers [6-9] there was studied an equation for the p-harmonic (p > i) Laplace operator. There was considered positive generalizations of the solution of the Dirichlet problem and obtained almost exact estimates of the decrease rate, and also estimates of the gradient of the modulus of the solution in a neighborhood of a conical boundary point. As Tolksdorf [9, po 310, (iv)] has indicated, it would be desirable to establish such estimates regardless of the sign of a solution. In [i0] there was proved the solubility of the Dirichlet problem for the equation div(~.(Vu) V ~ ) ~ / ~ > 0 , ]~:C i,~,0 < ~ < I with i n space ~!~p(~) n w2~P(~), where 2 < p < 2 / ( 2 ~/~), D is a convex polygon in the plane, ~ is the largest angular value on the polygon's boundary. Finally~ recently in [ii], there has been established a Lipschitz estimate for divergence elliptic equations of second order in an arbitrary convex domain. The goal of the present paper is an extension of the result of [121 into the multidimensional case. Namely, we will investigate the behavior of a bounded generalized solution of the Dirichlet problem

7~i a~ (x, u, u~) = a (x, u, u~),

(])

x ~ G, ~ (z) = O, z ~ oG

(by reiterating indices, one means the summation from 1 to n) near a conical point ~ e ~G of a domain G c R ~ (n ~ 2) that is assumed to be smooth surface everywhere except for the origin ~. Our notation is widely accepted~ We assume that Eq. (i) is elliptic, and its coefficients satisfy the minimal smoothness condition and some growth restriction with respect to IvuI. We will prove that in some neighborhood of point

u(x)= O(Ix1~), Vu(x)= O ( I z t ~ ) with an exact value of % > 0 and that u(x) has second order generalized derivatives that are square sum~ab!e with some exact weight. In addition, we do not make any assumptions concerning the sign of u(x). i.

Notations,

Definitions,

Auxiliary Inegu_alities

We admit the following notation: Bd(0) - the ball in R ~ of radius d > 0 and center at the origin

~;

Go d = G N Bd(0) - the cone in R ~ with the vertex at ~ for some sufficiently small d > 0, i.e., Go d = {(r, ~)I0 < r < d; ~ = (~i, ~2 ..... ~n-1) e ~}; (r, m) - spherical coordinates of a point x e lRn; - a domain in the unit sphere S n-l with infinitely differentiable

boundary 3~;

L'vov. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, NO. 6:~ ppo 25-38, November-December, 1990. Original article submitted June 14, 1988.

0037-4466/90/3106-0891512.50

9 1991 Plenum Publishing Corporation

891

F0 d = {(r, ~)I0 < r < d; m e 3fi} c a G

- the side surface of the cone

G~;

~p = G0dln {Ix I = p}, 0 < p < d. Recall some elementary formulas [13]:

(2)

dx = r~-Idr do~, df~ = p~-IdcD,

where dm is the area element of the surface of the unit sphere;

(3)

iv=l= (e?

where IVu I is the modulus of the gradient u(x), and IVmul is the projection of a vector Vu onto the plane that is tangent to the sphere S n-~ at the point w; Au = - -O~u + n - - t Ou + ~ A~u, Or2 r Or r~

(4)

where h is the Laplace operator in R '~, and h~ is the Beltrami-Laplace operator on the unit sphere. We introduce the following symbols for functional spaces: Lq(G) is the Banach space of measurable functions on G, which are summable to the power q ~ I, and which is endowed with the norm llUllq,G. Wk(G) is the Sobolev space of function from L2(G) possessing generalized o

derivatives up to the order k that are square summable on G. Wk(G) is the subspace of the space Wk(G), whose dense subset is the totality of all infinitely differentiable functions with supports in G. W01(G) is the subspace of the space WI(G), whose dense subset is the collection of all functions that are continuously differentiable on G and vanish on 8G. o

w a k ( G ) is the space of functions possessing generalized derivatives up to the order k in the interior of G and endowed with the norm k

I [14, 15], in particular

II

= 5

+

iv

I +

G n 2

U~xx

E

2

{,j=l

We will need certain inequalities. Ha~dy Inequality [16, p. 296, Theorem 330; 17, p. 238, Theorem V.IO].

For

every

u(r) e WI(]0; d[) d

d

y r=-~+=u 2 dr ~ ] 4 _ n _4 a I ~ j?r ~-a+=uTdr, 0

a 0 .

Gao

Turning into the last integral in (24), by a Holder-continuity of u(x), by virtue of the Cauchy inequality which is satisfied for any o > 0, and by (24), we have

l Gd

l(Ivul

+/(x))

+

9

lw

+

d

d

Go

dx

W>0.

(30)

GO

Returning to inequality (24), by virtue of inequalities (26), (29), and (30), we infer that

d %

d GO

(31)

P'2(M~ f . I rc~/2(x)dx -I- 3~ -- c~j" J~ rsr162 , ~ , / ~dx +

where, by inequality (27) and condition (12),

C(~,n,a)= i - - ( 2 - - ~ ) ( 2 - - - ~ ) H ( ~ , n , a ) > O .

896

(32)

Since h(x) e W~_2~

f(x) e We~ 93

e.~JrO

then there exists

~ ~ r~

(x)dx = C~ r~-~h~ (x)dx,

llmoS!r~]2(x)dx=~!ref(x)dx"

(33)

The numbers 6 and o are chosen as follows: (%'2) ~2 (Mo) H (~, n, ~) = 6 [(5 - - ~)t2 + (2 - - ~)//(~,, ~, ~)] = C (~, n, ~)/0,

(34)

while the number d is selected in such a way that inequality (28) with 6, taken from (34), and the inequality

~(Mo)~od T~ C(%, n, ~)i6, hold.

(35)

Then inequality (31) takes the form

S .I r2-:l W l ~ * d Go

~ ~o.f J [,-2L(_Q),

OO.

In inequalities (43)-(45) we will assume 0 < s 5 7, so that 07 ! p~. Returning to inequality (39), by virtue of estimates (42)-(45), and taking into account the notation in (40), (41), we have

..> 2% t -- ~ (p) Y(p) v' (P) ~ 7 i + o (,o)

2s

(n, Mo) ~-~-e, t + ~ (p)

(46)

where o(p) = c(n, X, M0) [6(0) + pe], 0 < e _< y, 0 < p < d. It is known that the solution of the differential equation (46) is majorized by a solution of the Cauchy problem for the ordinary differential equation

w'(p) -- 2~ i -- o (p) w(p) p i+o(p)

2~kcl -- -i-V~(p) pS 1 ~,

O n. o

Then u ( x )

e Wa=(G) and t h e r e

holds

the estimate

II ull~a(c ) < c,(! + Ilglb,G + [1] llq/=,~q- []% II,~/=,G-A-]1% ]~a/2,G+ IIqolb.~ + (51)

::/~(x) + :(x)]dx}

"~4/q

,

u 0 depends on the same parameters as in (13), and c a > 0 depends only on M0, v, p, v0, Pl, P2, Pd, Ps. Proof.

Consider a sequence of domains Gk, p = Go d n {p/2 k+1 < r < p/2k}, k = 0, i, .... so

0 < p < d.

Obviously,

G~.pcG~,

O Gh.p=G~ ,

In the domain Gk, p we consider Eq. (I) and per-

h=o

form a transformation of coordinates x i = xi'/2k-i ~ i = 1 . . . . . n. Under such transformation, the domain Gk, p is mapped into the domain Gz, p of the space of points x' elR h. The function v(x') = u(x'/2 k-l) is a bounded generalized d

':h),

solution of the equation

,

-77ai (x , v, Ux,) + a (k)(x',~v,vx,)=0, ax i

~i-(k~"(x', v, v~,) ------21-ka~ (21-kx ', v (x'), 2~-1t,'~,),

a(h)(x ', u, V,,)==2~(1--k)a(2*-~X', U(X'), 2h--'V~,)

x'~Gl,p, i -- 1, . .., n,

(Ik')

(k = O, l, 2 . . . . . ).

To equation (ik') we apply the well known results [18, 21] on smoothness of a generalized solution inside the domain and near a smooth piece of the boundary. Namely, from the proof of Theorem 1 [21] the validity of the following estimate is easily seen:

Gl,p

P

P

P

GoUG1UG2

P

,P, P

(Go UGI [JG2 ,~4lq

q- iq/2(x ') ~, q:~ (x') -.}- gq(x')]/z' } ,

q > n,

) where ci, c2 depend only on M0, v, p, Pl, P2, Pd, Ps. Reverting in this inequality to the variable x, by virtue of the definition of the sets Gk, p we obtain the inequality

Gh,p

Gk--l,pUGk,pUOh+ l,p

(52) [ h--l,pUCh,pUah+l,p

Let us sum up all equations (52) over k = I, 2 . . . . . The inequality obtained this way, combined with Theorem i and a result from paper [21], means that u(x) e W~2(G) and estimate (51) is valid. This completes the proof of Theorem 3.

901

(12).

COROLLARY. Let assumptions of Theorem 3 be fulfilled with the exceptions of condition The generalized solution of problem (i) u(x) ~ W2(G), if (i) n ~ 4, (ii) n = 2 and 0 < m 0 < ~ (cf. the remark on page 898),

(iii) n ~ 3 , Q c ~ 0 = { ~ = ( 0 ; ~)10 0 is defined with the help of quantities v, v0, ~, ~z, D2, ~4, ~s, and also of vraimax Iz(x')l, which can be estimated using inequality (50). Returning to the function

Q,

u(x), according to (48), we will obtain

I v u (~)1