On phragme´ n - lindelo¨ f principle for non-divergence type elliptic equations and mixed boundary conditions Текст научной статьи по специальности «Математика»

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DOI: https://doi.Org/10.15688/mpcm.jvolsu.2017.3.5

UDC 517 LBC 22.161

ON PHRAGMÉN - LINDELOF PRINCIPLE FOR NON-DIVERGENCE TYPE ELLIPTIC EQUATIONS AND MIXED BOUNDARY CONDITIONS1

Akif Ibraguimov

Doctor of Physical and Mathematical Sciences, Professor,

Department of Mathematics and Statistics,

Texas Tech University

[email protected]

Box 41042, Lubbock, TX 79409-1042, USA

Alexander I. Nazarov

Doctor of Physical and Mathematical Sciences, Professor, Department of Mathematical Physics, Saint Petersburg State University,

Prosp. Universitetsky, 28, 198504 Saint Petersburg, Peterhof, Russian Federation; [email protected]

Senior Researcher, St. Petersburg Department of Steklov Institute of Mathematics of RAS,

Fontanka St. 27, 191023 St. Petersburg, Russian Federation

^ Abstract. The paper is dedicated to qualitative study of the solution of the

§ Zaremba-type problem in Lipschitz domain with respect to the elliptic equation in

~ non-divergent form. Main result is Landis type Growth Lemma in spherical layer

> for Mixed Boundary Value Problem in the class of "admissible domain". Based on

1 the Growth Lemma Phragmen — Lindelof theorem is proved at junction point of

^ Dirichlet boundary and boundary over which derivative in non-tangential direction is defined.

<

>

о

E

D M га

©

Key words: elliptic equation in non-divergent form, Mixed Boundary Value Problem, Growth Lemma, Phragmen — Lindelof theorem, Zaremba-type problem.

1. Introduction

We consider non-divergence elliptic operator

n

Cu := — ^ aij(x)DiDju in Q. (1.1)

i,j=l

Such operators arise in theory of stochastic processes and various applications.

In (1.1) Q is a domain in Rra, n > 3, and Di stands for the differentiation with respect to Xi. We suppose that the boundary dQ is split dQ = r U {Z} U r2. Here ri is support of the Dirichlet condition, and r2 is support of the oblique derivative condition:

. du. . v u(x) — u(x — 61)

u(x) = $(x) on ri; -r^ix) := lim --- = ^(x) on r2,

di 6^+0 6

where I = l(x) is a measurable, and uniformly non-tangential outward vector field on r2. Without loss of generality we can suppose |l| = 1. We call ri Dirichlet boundary, and r2 Neumann boundary.

At point Z e ri n r2 function u is not defined, and we investigate asymptotic properies of the solution at this point.

For divergence type equation in case of Dirichlet Data this type of theorem first was proved in very general case by Mazya in [9]. Criteria for regularity for Zaremba problem first was obtained by Mazya in [3].

Here we consider the case of non-divergence equation in bounded domain Q where Neumann r2 is Lipschitz in a neighborhood of the point Z.

In the case r2 = 0 the similar question was discussed by E.M. Landis (see [5; 6]) and sharpened by Yu.A. Alkhutov [2].

We always assume that the matrix of leading coefficients (aij) is bounded, measurable and symmetric, and satisfies the uniform ellipticity condition:

max sup e(x, 4) =: ei < ro, ItM

where e is the ellipticity function (see [2; 6])

5Zi=ian(x)

e(x, 4)

For simplicity we consider the operators without lower-order terms, a more general case can be easily managed.

The paper is organized as follows.

In Sec. 2 we formulate some known results about non-divergence equations: lemma on non-tangential derivatives at point of maximum (minimum) on the boundary in the form of Nadirashvili [10], the Landis Growth Lemma in case r2 = 0, and Growth Lemma in Krylov's form.

The Growth Lemma for elliptic and parabolic equations first was introduced by Landis in [4; 7]. Growth Lemma is a fundamental tool to study qualitative properties and regularity of solutions in bounded and unbounded domain. Recent review on Growth Lemma and its applications was published in [12] (see also [1]).

In Sec. 3 we prove strict Growth Lemma near Neumann boundary.

Sec. 4 glues two Growth Lemmas. This result was obtained under some admissibility constraint on the boundary r2, which is an analog of isoperimetric condition.

In the last Sec. 5, dichotomy theorem is proved for solutions of mixed boundary value problem to non-divergence elliptic equation.

We use the following notation. x = (x',xn) = (x1}... ,xn-1,xn is a point in Rn. B(x,R) is the ball centered in x with radius R.

2. Preliminary Results

Here we recall some known results and prove auxiliary lemmas for the sub- and supersolution of the equation Lu = 0. We call function u sub-elliptic (super-elliptic) if u E W2(Q)f)C 1(Q U r2), and Lu < 0 (respectively, Lu > 0).

We say that r2 satisfies inner cone condition (see, e.g., [10]) if there are 0 < p < n/2 and h > 0 such that for any y E r2 there exists a right cone K(y) c Q with the apex at y, apex angle p and of the height h.

cone/ axis/

Fig. 1. Inner cone condition

In [10] N. Nadirashvili obtained fundamental generalization of Oleinik — Hopf lemma2, the so-called "lemma on non-tangential derivative":

Lemma 2.1. Let r2 satisfy inner cone condition. Let a non-constant function u be super-elliptic (sub-elliptic) Lu > 0 (Lu < 0) in Q. Suppose that y E r2 and u(y) < < u(x) (u(y) > u(x)) for all x E r2. Then for any neighborhood S of y on r2 and for any e < p there exists a point x E S s.t.

dufdu\ «(x) <0 («(x) >0)

for any outward direction 1 s.t. the angle y between 1 and the axis of K(x) is not greater then p — e.

From standard maximum principle and Lemma 2.1 follows comparison theorem for mixed boundary value problem.

Lemma 2.2. Let Q be a bounded domain, dQ = r U r2. Let r2 satisfy inner cone condition. Suppose that vector field 1 satisfies the same condition as in Lemma 2.1. Let functions u and v belong to W;2(Q) f|c:(Q U r2) n C(Q). _

Then, if Lu < Lv in Q, u < v on ri, and || < || on r2 then u > v in Q. Definition 2.1. Let Q be a domain, dQ = r1 U r2. Define "small ball" B(0,R) and "big ball" B(0,aR), a > 1 (see Fig. 2).

We call the function w barrier with respect to mixed boundary value problem in these two balls if it posses properties:

w is sub-elliptic (Cw < 0) in the intersection Q n B(0,aR); (2.1)

w(x) < 1 on ri n B(0,aR); (2.2) dw

— < 0 on r2 n B(0, aR); (2.3)

w < 0 on Q n 3B(0,aR); (2.4)

w(x) > n0 in the intesection B(0, R) n Q (2.5)

for some constant no-

Fig. 2. Domain G and two balls 5(0, R) and 5(0, aR) (a > 1)

Now we are in the position to prove the following strict growth property for subsolutions of the mixed boundary value problem.

Lemma 2.3. Let Q be a domain, dQ = r U r2. Suppose that a function u be sub-elliptic in Q n B(0, aR), u> 0 in Q, u = 0 on r n B(0, aR) and % < 0 on r2 n B(0, aR). Let r2 satisfy inner cone condition.

Assume that there is a barrier w in balls B(0,R) and B(0, aR). Then

sup « > SUPlnB(°>R) U. (2.6)

QnB(°,aR) 1 — n°

Proof. Let M = supnnB(° aR) u, and let the barrier w(x) be as in Definition 2.1. Define

v(x) = M(1 - w(x)).

Obviously Cv > Cu in Q, v > u on ri n B(0,aR), % > ^ on r2, and v > M > u on dB(0,aR) n Q. Applying comparison Lemma 2.2 to functions v and u in the domain Q n B(0, aR) we get that v > u. In the intersection Q n B(0, R) this gives with regard of (2.5)

M(1 - n°) > m(1 - inf w) > sup u.

1nB(°,R) QnB(°,R)

The latter is equivalent to statement in (2.6).

We recall the well-known notion of s-capacity, see, e.g., [6, Sec. I.2].

Definition 2.2. Let H be a Borel set. Let a measure h be defined on Borel subsets of H. We call h admissible and write h e M(H) if

i ^L < 1, for x E Rn\H. Jh \x — y\s

Then the quantity

Cs( H)= sup h( H) h^m(h)

is called s-capacity of H.

We also recall the following simple statement. Proposition 2.1. If s > e1 — 2 then L\x\-S < 0.

Now we formulate a variant of the Landis Growth Lemma, see [6, Sec. I.4]. Lemma 2.4. Let function u be sub-elliptic in Q n B(0,aR), u > 0 in Q, u = 0 on r1 = dQ n B(0,aR). Let s > e1 — 2. Then there exists 0 < n1 < 1 depending only on s s.t.

supQnB(0,R) u

sup u > ---.

QnB(0,aR) 1 — mC4 H )R S

Here H = r nB(0,R).

Consequently if B(0, R) \ Q contains a ball with radius bR then

^ supnns(o,_R) u sup u > -----,

QnB(0,aR) 1 — T|1

where the constant n depend on s and b.

3. Growth Lemma near Neumann boundary

Here we prove the Growth Lemmas in the domain adjunct to r2 under some assumption on r1.

We recall that r2 is uniformly Lipschitz in a neighborhood of x0. This means that there is b > 0 s.t. the set r2 n B(x0, b) is the graph xn = f(x') in a local Cartesian coordinate system, and the function f is Lipschitz. Moreover, we suppose that its Lipschitz constant does not exceed L. Without loss of generality we assume that Q n B(x0, b) C {xn < f(x')} (see Fig. 3). This implies the inner cone condition if we direct the axis of the cone K along —xn and set p = cot-1(L).

Lemma 3.1. Let r2 n B(0, R) = and x0 E r2 n dB(0, R), for some R < §. Assume that Q n B(0, aR) = 0 for some 0 < a < 1 (see Fig. 3).

Suppose that the vector field 1 satisfies conditions in Lemma 2.1 uniformly on r2 (that is, e does not depend on x E r2).

Let function u be sub-elliptic (Lu < 0 in Q), u > 0 in Q, u = 0 on r and g < 0 on r2.

Then there exists a > 1 depending on the Lipschitz constant L, e and ellipticity constant e1 s.t.

sup u > SUp°nB(0,R) u. (3.1)

QnB(0,aR) 1 — n2

Here n2 E (0,1) is defined by a and a.

Fig. 3. Domain Q, boundary r2 and balls B(0, R), B(0, aR) and B(0, aR)

Proof. We take s > e1 — 2 and set

w(x) =

asRs

l^F"

a"

We claim that for a sufficiently close to 1 this function satisfies all conditions in Definition 2.1. Indeed:

1. From Proposition 2.1 function w is sub-elliptic, condition (2.1) holds.

2. Evidently w = 0 on dB(0,aR), condition (2.4) holds,

3. while Q n B(0, aR) = 0 implies w < 1 in Q n B(0,aR) (and therefore on ri) condition (2.2) holds.

Now we check condition (2.3). We introduce the Cartesian coordinate system with axes collinear with those of local coordinate system at x°. We observe that the assumption r2 n B(0, R) = 0 and Lipschitz condition imply that for x G r2 n B(0, aR)

\x'| <

R

VT+L2

(L + Va2 - 1); xn >

R

VTTl2

(1 - lV^—1).

Moreover, our assumption on the vector field I means that

|l'| < sin(cot-i(L) - e) <

£n > cos(cot-i(L) - e) >

where 1 depends only on L and e.

Therefore, the direct calculation gives

1

v/TTX2

L

VTTX2

- e;

+ e

dw

rn(x)

s a rs

И

5+2

(Xnta + £ • X )

<

sasRs

R

N*+2 V/TTX2

(V^2—г • (VTTÏ2 + 1(L - 1)) - ê(L + 1)).

It is easy to see that, given e > 0, there is a > 1 depending only on e and L s.t. ff(x) < 0, and (2.3) holds.

Finally, for x e Q n B(0, R), w(x) > as(1 - a-s) =: n2, and (2.5) holds. Thus, the claim follows, and w is the barrier in the balls B(0,R), B(0,aR). From Lemma 2.3 we get (3.1).

s

a

4. Growth Lemma in the Spherical Layer

In this section we prove Growth Lemma in spherical layer near junction point of interest Z = r1 n r2. Without loss of generality we put Z = 0.

First we will introduce admissible class of domains in the spherical layer. Definition 4.1. Fix five constants 0 < q1 < q2 < q* < q3 < q4. Define two spherical layers

Ur C Ur :

UR = B(0, q4R) \ B(0, qR, UR = B(0, q3R) \ B(0, ^R).

We call Q admissible in the layer Ur if for some 0 > 0 there is finite set of the balls (see Fig. 4)

B = {Bk = B(lk, 0R)}£=0; Bk c UR

s.t. the following holds:

1. Cs(B0 n r1) > kCs(r1 n UR), for some constant k > 0.

2. Bk n r2 = 0, k = 1,..,N, and B fa, a0R) n r2 = 0, where a > 1 is defined in Lemma 3.1. 3

3. There is b E (0,1/2) s.t. every ball in B can be connected with B0 by a subsequence of balls Bj s.t. any intersection Bj n Bj+1 n Q contains the ball B(E,j+1, bR).

4. The set SR = dB(0, q*R) n Q is covered by balls in B. Fig. 4 schematically illustrate Definition 4.1.

Fig. 4. On the left: domain Q admissible in Spherical Layer Ur. On the right: domain and layer

zoomed near boundary r2 (bold line)

Lemma 4.1. Let function u be sub-elliptic, u > 0 in Q. Suppose that u < 0 on r1 and

du dl

jH < 0 on r2. Let domain Q be admissible in the layer UR. Then

supSrи

supu > _ . . . np - 1 — nCs (H )R-S

Here H = r1 nUR while n depends on s, the ellipticity constant e1, the Lipschitz constant L, the vector field 1, constants 0, k, b in Definition 4.1 and the number N of balls in the set .

Proof. Without loss of generality we set 0 = 1. Let sups u =: m = u(y), here y e SR. By assumption 4 in Definition 4.1, y e Bk for some k. By assumption 3, we can choose a subsequence Bj connecting B0 and Bk.

Consider the ball B0 and the ball B(L00,aR), a > 1, concentric to it. Due to assumptions 1 and 2 in Definition 4.1, we can apply Lemma 2.4 to get:

us \ \ sup_B0nn u

M :=sup u > sup u >--^ / mp.

n nnB(lo,aR) 1 - KniCs(n )K A

Suppose that

sup u > m(1 - 6o), where 60 = p ^. (4.1)

Bonn 2(1 - xniCs(H)R-S)

Then after some calculation we get

m

M >

1 - mCs(H )r-

for some n3 depending on and the statement follows. If (4.1) does not hold, we consider the function

u1(x) = u(x) - m(1 - 60), (4.2)

then ui(x) < 0 in B0 n Q.

By assumption 3, B0 n B1 n Q contains a ball of radius 6R. Let Q1 := [x : u1(x) > 0}. Assume that B1 n Q1 = 0, otherwise we consider the first ball in the subsequence Bj for which this property holds. Suppose that

sup u1 > m60(1 - t), (4.3)

B1 nn

here the constant t will be chosen later.

Consider any simply connected component of the domain B(^1,aR) n Q1 in which the supremum in (4.3) is realised. There are two possibilities:

a) B(h.aR) n r = 0;

b) B(h,aR) n r2 = 0

(recall that a = a(L,i, e1) > 1 is defined in Lemma 3.1).

Let us start with case (a). Due to assumption 3, Lemma 2.4 and (4.3) it follows that

sup «1 > sUMn^i > m6°(1 - t) . (4.4)

B{l1,aR)nn 1 - Hi 1 - Hi

Using (4.2) and (4.4) we deduce

. h . 60(11 - t) x

sup u > m[ 1+---—).

B{l1,aR)nn 1 - "Hi

Letting t = f we get

M > sup u > m(1 + 60T ), (4.5)

B(l1,aR)nn 1 - 2t

and the statement follows.

In case of (b) we proceed with the same arguments but instead of Lemma 2.4 we apply Lemma 3.1 and put t = nf. Thus, if (4.3) holds with t = 2 min{ti1,n2} then (4.5) is satisfied in any case, and Lemma is proved.

If (4.3) does not hold then function u satisfies

sup u < m(1 — b0T).

sinn

As in previous step we consider the function

u2(x) = u(x) — m(1 — b0T),

u2(x) < 0 in B1 n Q.

Repeating previous argument we deduce that if

sup u2 > mb0t(1 — t) (4.6)

B2nn

then

b0T2

M >m{i 1 + ,

and Lemma is proved.

If (4.6) does not hold, then

sup и < m(l — 60t2).

В2П П

Repeating this process we either prove Lemma or arrive at the inequality

sup u < m(1 — b0Tk)

Bknn

that is impossible since y E Bk and u(y) = m.

5. Dichotomy of solutions

In this section we will apply obtained Growth Lemma in spherical layer to prove dichotomy of solutions near point Z of the junction of Dirichlet and Neumann boundaries. As in previous section we put Z = 0.

Let Q C {x : xn < f(x')} and r2 is a graph of the function xn = f(x'), /(0) = 0. Set Rm = Q-m for some Q> 1, Sm = dB(0, q*Rm), and

Um = B(0, q4Rm) \ B(0, q1Rm), Um = B(0, ^Rm) \ B(0, ^R™).

We fix N0 E N and q1 < q2 < q* < q3 < q4 s.t. q* < q1Q. Suppose that for all m > N0 the domain Q with boundaries r1 and r2 is admissible in the layer Um in the sense of Definition 4.1 with R = Rm, and all constants in Definition 4.1 do not depend on m.

Lemma 5.1. Let function u be sub-elliptic, u > 0 in Q. Suppose that u < 0 on r n nB(0, qiRNo) and g < 0 on r2 nB(0, q4RNo). Let domain Q be admissible in the layers Um, m > Nq.

Let Mm = sup^nQ u. Then one of two statements holds: either MNl+1 > MNl for some Ni, and for all m > Ni

Mm+1 > 1 - nCMMHm)Q«n, (5-1)

or for all m > N0

M- —1 - Jz^. <5-2>

Here Hm = ri n Um, and n is the constant from Lemma 4.1.

Proof. Due to Lemma 2.2, there are two possibilities:

(a) if MNl+i — MNlfor some Ni > N0 then M(p) = supaB(0 p)nn u > Mm, m > Ni for any p < q*Rm;

(b) otherwise Mm > Mm+i for all m > N0.

Now Lemma 4.1 gives (5.1) in the case (a) and (5.2) in the case (b).

Remark 5.1. Let function u be sub-elliptic, u > 0 in Q. Suppose that u < 0 on rinB(0, p0) and || < 0 on r2 n B(0, p0). Then the maximum principle implies the following dichotomy (we recall that M(p) = sup^^nn u):

either there is p* < p0 s.t. for p2 < pi < p* we have M(p2) > M(pi); or M(p2) < M(pi) for all p2 < pi < p0.

Applying recursively alternative in Lemma 5.1 and using Remark 5.1 we get asymptotic dichotomy.

Theorem 5.2. Let the assumptions of Lemma 5.1 be satisfied. Suppose that Em=0 C( Hm)Qsm = <, where Hm = ri n Um. Then one of two statements holds:

either M(p) ^ < as p ^ 0, and

[c ln p]

liminfM(p)exp( - n V Cs(Hm)Qsm) > 0,

p—^^o V ^—' /

m=0

or M(p) ^ 0 as p ^ 0, and

[c ln p]

limsupM(p)exp (n ^ C(Hm)Qsm) =0,

' m=0

Here if and c depend on the same quantities as n in Lemma 4.1.

REMARKS

i Akif Ibraguimov partially supported by DMS NSF grant № 1412796 and Alexander I. Nazarov supported by RFBR grant № 15-01-07650.

2 In [10] classical solutions u G C2(Q) nC 1(Q) are used but due to the Aleksandrov — Bakel'man maximum principle it is transferred to u G W2(Q)Ç]C1(Q U r2).

3 Note that boundaries of some balls Bk may touch r2.

REFERENCES

1. Aimar H., Forzani L., Toledano R. Holder regularity of solutions of PDE's: a geometrical view. Comm. PDE, 2001, vol. 26, no. 7-8, pp. 1145-1173.

2. Alkhutov Yu.A. On the regularity of boundary points with respect to the Dirichlet problem for second-order elliptic equations. Math. Notes, 1981, vol. 30, no. 3, pp. 655660.

3. Kerimov T.M., Maz'ya V.G., Novruzov A.A. An analogue of the Wiener criterion for the Zaremba problem in a cylindrical domain. Funct. Analysis and Its Applic., 1982, vol. 16, no. 4, pp. 301-303.

4. Landis E.M. On some properties of the solutions of elliptic equations. Dokl. Akad. Nauk SSSR, 1956, vol. 107, no. 4, pp. 640-643.

5. Landis E.M. s-capacity and its application to the study of solutions of a second order elliptic equation with discontinuous coefficients. Math. USSR-Sb., 1968, vol. 5, no. 2, pp. 177204.

6. Landis E.M. Second Order Equations of Elliptic and Parabolic Type. Providence, Rhode Island, AMS, 1998. 278 p.

7. Landis E.M. Some problems of the qualitative theory of elliptic and parabolic equations. UMN, 1959, vol. 14, no. 1 (85), pp. 21-85. (in Russian)

8. Landis E.M. Some problems of the qualitative theory of second order elliptic equations (case of several independent variables). Russian Math. Surveys, 1963, vol. 18, no. 1, pp. 162.

9. Maz'ya V.G. The behavior near the boundary of the solution of the Dirichlet problem for an elliptic equation of the second order in divergence form. Math. Notes, 1967, vol. 2, no. 2, pp. 610-617.

10. Nadirashvili N.S. Lemma on the interior derivative and uniqueness of the solution of the second boundary value problem for second-order elliptic equations. Dokl. Akad. Nauk SSSR, 1981, vol. 261, no. 4, pp. 804-808. (in Russian)

11. Nadirashvili N.S. On the question of the uniqueness of the solution of the second boundary value problem for second-order elliptic equations. Math. USSR-Sb., 1985, vol. 50, no. 2, pp. 325-341.

12. Safonov M.V. Non-divergence Elliptic Equations of Second Order with Unbounded Drift. Amer. Math. Soc. Transl. Ser. 2. Providence, RI, AMS, 2010, vol. 229, pp. 211-232.

ПРИНЦИП ФРАГМЕНА - ЛИНДЕЛЕФА ДЛЯ ЭЛЛИПТИЧЕСКИХ УРАВНЕНИЙ НЕДИВЕРГЕНТНОГО ТИПА И СМЕШАННЫХ ГРАНИЧНЫХ УСЛОВИЙ

Акиф Ибрагимов

Доктор физико-математических наук,

профессор факультета математики и статистики,

Texas Tech University

[email protected]

Box 41042, Lubbock, TX 79409-1042, USA

Александр Ильич Назаров

Доктор физико-математических наук, профессор кафедры Математической физики, Санкт-Петербургский государственный университет,

просп. Университетский, 28, 198504 г. Санкт-Петербург, Петергоф, Российская Федерация;

[email protected] Ведущий научный сотрудник,

Санкт-Петербургское отделение Математического института им. В. А. Стеклова РАН,

ул. Фонтанка, 27, 191023 г. Санкт-Петербург, Российская Федерация

Аннотация. Статья посвящена качественному исследованию решения задачи типа Зарембы в липшицевой области, поставленной для эллиптического уравнения в недивергентной форме. Основной результат — лемма о росте типа Ландиса в сферическом слое для смешанной краевой задачи в классе «допустимой области». На основе леммы о росте доказана теорема Фрагмена — Линделефа в точке соединения границы Дирихле и границы, над которой определена производная в некасательном направлении.

Ключевые слова: эллиптическое уравнение в недивергентной форме, смешанная краевая задача, лемма о росте, теорема Фрагмена — Линделефа, задача типа Зарембы.