The trace theorem for anisotropic Sobolev - Slobodetskii spaces with applications to nonhomogeneous Elliptic BVPs Текст научной статьи по специальности «Математика»

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УДК 514.157.2

The Trace Theorem for Anisotropic Sobolev — Slobodetskii Spaces with Applications to Nonhomogeneous Elliptic BVPs

S.A. Sazhenkov1-2-3 and E.V. Sazhenkova4

1 Lavrentiev Institute of Hydrodynamics of the Siberian Branch of the RAS (Novosibirsk, Russia)

2 Novosibirsk State University (Novosibirsk, Russia)

3 Heilongjiang University, Sino-Russian Institute (Harbin, PR China)

4 Novosibirsk State University of Economics and Management (Novosibirsk, Russia)

Теорема о следах в анизотропных пространствах Соболева — Слободецкого и ее приложения к неоднородным краевым эллиптическим задачам

С.А. Саженков 1-2-3, Е.В. Саженкова4

1 Институт гидродинамики им. М.А. Лаврентьева СО РАН (Новосибирск, Россия)

2 Новосибирский государственный университет (Новосибирск, Россия)

3 Хэйлунцзянский университет, КРИ (Харбин, Китай)

4 Новосибирский государственный университет экономики и управления (Новосибирск, Россия)

In this paper, anisotropic Sobolev — Slobodetskii spaces in poly-cylindrical domains of any dimension N are considered. In the first part of the paper we revisit the well-known Lions — Magenes Trace Theorem (1961) and, naturally, extend regularity results for the trace and lift operators onto the anisotropic case. As a byproduct, we build a generalization of the Kruzhkov — Korolev Trace Theorem for the first-order Sobolev Spaces (1985). In the second part of the paper we observe the nonhomogeneous Dirichlet, Neumann, and Robin problems for p-elliptic equations. The well-posedness theory for these problems can be successfully constructed using isotropic theory, and the corresponding results are outlined in the paper. Clearly, in such a unilateral approach, the anisotropic features are ignored and the results are far beyond the critical regularity. In the paper, the refinement of the trace theorem is done by the constructed extension. Namely, we formulate proper weakly regular anisotropic classes for boundary conditions, so that the boundary value problems appear to be well-posed. Finally, the analogous results are formulated for the p-parabolic problems. Key words: anisotropic fractional Sobolev spaces, boundary trace of function, p-Laplace operator.

DOI 10.14258/izvasu(2018)4-19

Рассматриваются анизотропные пространства Соболева — Слободецкого в ограниченных полицилиндрических областях произвольной размерности N. В первой части статьи конструируется естественное продолжение результатов теоремы Лионса — Мадженеса о граничных следах функций, принадлежащих изотропным пространствам (1961), на случай анизотропных пространств. Как следствие, также устанавливается некоторое обобщение теоремы Кружкова — Королёва о граничных следах функций (1985) для соболевских пространств первого порядка. Во второй части статьи рассматриваются основные неоднородные краевые задачи для вырождающихся эллиптических уравнений с анизотропным ¿»-лапласианом. Результаты о корректности этих задач, получаемые с помощью результатов и техники Лионса — Мадженеса, разработанной для изотропных пространств, корректны, но не учитывают особенностей, связанных с анизотропией. В связи с этим проводится уточнение: с помощью построенного в первой части статьи продолжения теории следов формулируются надлежащие слабо регулярные анизотропные классы для граничных условий, для которых рассматриваемые задачи оказываются корректно поставленными.

Ключевые слова: анизотропное пространство Соболева дробного порядка, р-лапласиан, граничный след функции.

The first author's work was supported by the Ministry of Higher Education and Science of the Russian Federation (project no. III.22.4.2) and by the Russian Foundation For Basic Research (grant code 18-01-00649).

Introduction This article is devoted to a study of a class of anisotropic Sobolev — Slobodetskii spaces Ws,p{0) and their dual spaces W~s'p (0) Here domain O is a polycylinder in M^f, i.e., it meets the following requirements.

Condition C. • O = B' x B2 x ... x Bk,

k

• Bi c RN; (i =1,2,...,k), ^Ni = N,

i=i

• Bi are nonempty bounded open sets with Lipschitz boundaries dBi.

Multi-indices s = (s',s2,... ,sk) and p = (pi,p2,... ,pk) are such that si e (0,1] and pi e (1, +to). Thus Ws,p(O) are fractional Sobolev spaces, in general. Multi-index p' = (p',p'2,... ,p'k) is such that p-i + (pi)-1 = 1 (i = 1,2,...,k), thereby pi e (1, +ro). We focus on the question about regularity properties of traces of functions from Ws,p(O) on subsets of dO. Our interest to this question is motivated by applications to p-elliptic equations (see Eq. (7) in Section 2.) supplemented by either Dirichlet or Neumann nonhomogeneous conditions. Such equations arise in modelling of heat transfer, gas diffusion, etc [1-6]. Anisotropy pi = pj means that diffusion rates differ in different directions xi and Xj.

The general theory of isotropic Sobolev -Slobodetskii spaces was built in sixtieth of 20th century [7-9]. Within its framework, regularity properties of traces of functions RN n- R on (N — 1) -dimensional manifolds were investigated in detail (see, for example, [7, Theorem 5.1], [10, Theorems 2.21 and 2.22]). Some applications of trace theorems to the Dirichlet, Neumann, and Robin problems for the isotropic p-Laplacian equation can be found in [11, Chapter 2]. Embedding in anisotropic Sobolev spaces of the first order were studied in [12], and the following result was established [12, Inequality (12)] (see also [1, Lemma 3.6 and Remark 3.6]):

Proposition 1. Let Q c RN be a bounded domain with the smooth boundary r, u e Lr(Q), dXiu e LP;(Q), 1 <pi < <x, i = 1,...,N.

Then

In Section 1, we formulate the trace theorem for anisotropic Sobolev — Slobodetskii spaces (Theorem 1). Then we outline a brief scheme of its proof. This theorem somewhat extends the result of Proposition 1, in particular.

In Section 2, we give some examples of application of this theorem to the basic nonhomogeneous boundary value problems.

1. The Trace Theorem in Anisotropic Sobolev — Slobodetskii Spaces

1.1. Some useful notation. In Introduction we have entered a polycylindrical domain O c RN satisfying Condition C. Now let us introduce some relevant convenient notation in order to study traces of functions from Ws,p(O) on (N — 1)-dimensional manifolds M c dO. For x e O write

X = (xi, X2, ...,Xk) eO, where xi e Bi (i = 1,..., k), and denote Xi := (x', X3, ...,Xk) eO-£l,

O*

B2 x B3 x ...xBk c RN—Nl,

Xj . ... 7 Xj — 1, Xj + 1, . . . , Xk ) G OXj ,

o*

B1 x ... x Bj—1 x B

C

r>N — Nj

j+i x ...xBk c (j = 2, 3,... ,k - 1),

Xk := (Xi,.. .,Xk—2, Xk—i) G Oxk,

Oxk := Bi x ... x Bk—2 x Bk—i C Rn—Nk.

For the sake of conciseness, for O n- R we write

xi; Ci ) := 4>(Z i, X2, X3, ...,Xk) (C i G Bi),

$(Xj;Cj) := HX1,...,xj—l,Cj,Xj+i,...,Xk)

(Zj G Bj, j = 2, 3,.. . ,k - 1),

\\u\\Lq(r) <

< c (n) ^ ^\dXi u\

LPi(a) + \MLr(n) )

ii-8

r(a) \\u\\Lr (a),

where

N

N < q, 1 <r < q, p* = V —,

T>* < J r>-

Pi

1 > 0,

N(1 - p*) r

e ={1 - N-. )/( I+

r Nq J / \N Np* r

Also, worth to notice the results on traces in anisotropic spaces that were achieved in [13,14]. The present article is organized as follows.

HXh; Z k) := &(xi,..., Xk-i, Z k) (Zk e Bk),

when it is suitable.

Quite analogously, we also introduce further the notation Xi and pi.

Remark 1. In line with the above introduced

k

notation, remark that dO = dBi x O^.

i=i

Of course, in the right-hand side here we mean that the order of Cartesian product is proper, that is, by dBi x O-£i we mean

B' x .. .x B—i x dBi x B+ x ... x Bk.

The above introduced notation will be systematically used throughout the rest of the article.

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1.2. Notion of anisotropic Sobolev — Slobodetskii spaces For p = (p1,p2,...,pk) introduce the anisotropic Lebesgue space Lp(O) equipped with the standard norm

W^WlP(O) = where, as usually,

\LP1 (Bi)ll LP2 (B2)...

Lpk (Bk)

\\Lpi (Bi)

^ \*(xi;Zi)\PidZ^

for sufficiently regular

Further, for s^ £ (0,1) and sufficiently smooth 04 R introduce the seminorm

m

ws

i (O)

O a

Bi Bi

\m( x i; z i) - m(xi; m)\Pi \z i - m\ni+pisi

dZidVi

dxi.

Iw s,P(O)

\lP(O) + (m)wl i=l

i (O)

WmWW (M) : = WmWLr (M) (4>)w2i,'

i (MndBi),

(1)

where

WmWLr (M)

k

= T

i=1

1/ri

Xi)WLri (Os,i d Xi

\ MndBi

(m)w2i,ri (MndBi)

J J J A(Xi, Vi, Zi)daCidaVidXi

A(xi, Vi, Zi) =

\m(xi;Zi) - m(xi; Vi)

\Zi - Vi\Ni-1+ri^i

For si = 1 we canonically define

(m>)Wl'pi (0) = \Wxi mWLPi (O).

Now we are in a position to define anisotropic Sobolev — Slobodetskii spaces Ws'p(O) in a rigorous way.

Definition 1. The space of functions O 4 R equipped with the Sobolev — Slobodetskii norm

Yi £ (0,1), ri £ (1, and dax., da^i, and dani

are elements of Mn dBi.

Definition 2. The space of functions M 4 R equipped with the norm (1) is called the Sobolev -Slobodetskii space W1,r(M), M C dO.

Remark 2. The important case is M = dO. Obviously, restriction of $ £ W 1'r (dO) to M C dO (M Lipschitz) belongs to W 1'r (M). If MndBi = 0 then y and ri are dumb indices and therefore can be taken arbitrarily.

Remark 3. If p1 = ... = pk = p £ (1, +to) and s1 = ... = sk = s £ (0,1] then Wsp(O) := Wsp(O) is the classical isotropic Soboles (s = 1) or Sobolev — Slobodetskii (s £ (0,1)) space. If s1 = ... = sk = 1 and pi = pj for i = j, in general, then W 1p(O) is the anisotropic first-order Sobolev space, as in Proposition 1.

We finish this subsection by introducing the notion of the negative spaces.

Definition 3. For £ (0,1) and ri £ (1, +to) (i = 1,... , k), for an arbitrarily fixed (N — 1)-dimensional Lipschitz manifold M C dO, the space W-1,r' (M) is defined as the dual space of W1,r (M). Hereby we have 1/ri + 1/ri = 1, and the associated norm is

W-1,r' (M)

is called the Sobolev — Slobodetskii space Ws'p(O).

In order to study properties of traces of $ £ Ws,p(O) on (N-1)-dimensional manifolds M C dO, we also introduce the notion of W1,r(M). To this end, we naturally induce the norm of Wsp(O) onto any (N — 1)-dimensional Lipschitz manifold M C dO via its atlas, and set

sup

4 e w1,r(M)

4 = o

\(Km)M\

4>\\w 1,r (M)

Analogously, W-1'r'(O) = (W1r(O))*.

By (•, )m we denote the duality bracket between Wir(M) and W-1'r'(M). Definition 3 generalizes the notion of negative Sobolev — Slobodetskii spaces from isotropic case [7, Section 4.3], [10, Section 2.2] onto anisotropic case.

1.3. Notion of interior trace. Formulation of the trace theorem in anisotropic spaces.

Definition 4. For $ £ V(O) the mapping j0nt defined by the formula

font^(x) :=

m(x) for x £ dO (2)

Oa- MndBi MndBi

lim

X 4 x X eO X e dO

is called the interior boundary trace operator.

Here D(O) is the space of Cfunctions with compact support contained in O.

The following theorem is the first main result of the article.

r

p

11 1

Y I 1 , / , ' ' ' , 1 Vl Pk

Theorem 1. (The Trace Theorem in Anisotropic Spaces.) Let O c be a polycylinder that meets the requirements of Condition C. Then the following assertions hold true.

(i) For any p = (pi,p2, • • .,Pk), Pi € (1, the interior boundary trace operator Yont, defined by (2) for $ € D(O), admits a continuous extension

Yn € L(W 1p(O), WYp(dO)), (3)

- + A = 1, (4)

Pi Pi

hereby there is a constant cT > 0 such that

Wlo^^WwYP^O) < cT \\$\\w 1,P(O),

V$ € W 1p(O).

(Constant cT is independent of $.)

(ii) For any p = (pi,p2, • • • ,Pk) Pi € (1,

the interior boundary trace operator Yont has a continuous right inverse operator (called the lift operator)

E€L(wy 'p(dO),W1'p(O)) (5)

satisfying YontE^ = ^ for all ^ € WY,p(dO) as well as

WE^ww 1,P(O) < citW^\\wY,P(8O),

V^ € WY'p(dO).

(Constant cit is independent of ^. Notation j is the same as in (4) .)

Remark 4. Let M c dO be a (N — 1)-dimensional Lipschitz manifold. On the strength of Remark 2, the both assertions of Theorem 1 hold true with M on the place of dO.

1.4. Brief scheme of proof of Theorem 1.

The idea of the proof is very simple. Firstly, we directly apply the Lions — Magenes Trace Theorem [7, Theorem 5.1] (in the isotropic case) to the space of functions (xi n- $(xi; xi)) € W1,pi (Bi) for a.e. xi € O-£i and construct the trace operators

Y0ntB : Bi ^ dBi,

Yi0nt'Bi € L(W1p(Bi),W1/pip(dBi))

(i = 1, 2, • • • ,k). These operators depend on xi € OXi parametrically.

Secondly, we notice that, since O is polycylinder, then variables x1, x2, • • • ,xk are separated, so that the operator Yont defined by the rule

Y0nt4>(x) = %

int.Bi

Xi; Xi) for x e dBi (i = 1, 2, . . ,k)

satisfies (3) and is the trace operator by construction.

Thirdly, thus constructed Yont is surjective since all ji"t'Bi are surjective and the sets dBi x O^ do not overlap each other. Hence the right inverse operator E is also defined and satisfies (5). □

2. On Boundary Value Problems for the Anisotropic p-Laplacian Equations. In this section we revisit the theory of weak generalized solutions to the Dirichlet, Neumann, and Robin problems for the isotropic p-Laplacian equation. This theory was mainly built in [11, Chapter 2]. With the help of Theorem 1, we somewhat extend it onto the anisotropic case.

2.1. Formulations of the basic problems for the anisotropic p-Laplacian equation. Let O c R^ be a polycylinder that meets Condition C. Let p = (p1,p2, • • • ,Pk), Pi € (1, (i = 1, 2, • • • ,k), p* = min _ Pi. Let f = f(x) be given

such that

i=1,...,k

f e w-1p (O) (p-1 + (p'*)-1 = 1). (6)

We consider the anisotropic p-Laplacian equation

k

- Y, divxiVu\Pi-2VXzu) + \u\P'*-2u = f in O

i=1

(7)

supplemented with either the Dirichlet boundary condition

u = g on dO, (8)

where g = g(x) is given, or the Neumann boundary condition

k

u\pi-2VX%u) ■ vi = h on dO, (9)

i=1

where h = h(x) is given, or the Robin boundary condition

k

^QV^u\pi-2VXzu) ■ vi + k \u\q*-2u = H on dO,

i=1

(10)

where —— 1 — < —, H = H(x) and k = p* N — 1 q*

const > 0 are given.

In (9), (10), and further, vi is the unit outward normal to dBi. (If x e dO but x e dBi then in (9) and (10) we simply take zero vector for vi(x).)

In the isotropic case, the following existence and uniqueness results were established in [11, Chapter 2, Sections 1.6 and 2.3].

Proposition 2. Let (p* =)pi = r e (1, (i = 1,2,...,k). Let f satisfy (6). The following assertions hold true

(i) Whenever g e W1/r',r(dO), the Dirichlet problem (7), (8) has the unique weak generalized solution u e W1,r(O).

(ii) Whenever h e W-1/r ,r (dO), the Neumann problem (7), (9) has the unique weak generalized solution u e W1,r(O).

(iii) Whenever H e W-1/r',r' (dO), the Robin problem (7), (10) has the unique weak generalized solution u e W1,r(O).

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Remark 5. The straightforward application of the monotonicity method, as in [11, Chapter 2], of the Lions — Magenes Trace Theorem, and of the fact that W 1'a (O) 4 W1>P(O) for a > 3, in the general anisotropic case leads to the assertions of Proposition 2 with g e W 1/p'*p* (dO), h e W-1/p*,p* (dO), H e W-1/p'*p'* (dO), and u e W1p* (O) on the places of g e W 1/r',r(dO), h e W-1/r',r'(dO), H e W-1/r'r(dO), and u e W1'r(O). (Here

p* = min pi.) Clearly, the anisotropic peculiarity

i=1,...,k

is ignored in this approach.

2.2. Refinement of regularity via Theorem 1. Applying Theorem 1 we arrive at a more precise conclusion than in Remark 5. This conclusion is the second main result of the article.

Theorem 2. Let p = (p1,p2,.. .,Pk), Pi e (1,

(i = 1, 2,... ,k), p* = min pi. Let j be defined by i=1,...,k

(4) and f satisfy (6). The following assertions hold true

(i) Whenever g e WY,p(dO), the Dirichlet problem (7), (8) has the unique weak generalized solution u e W 1'P(O).

(ii) Whenever h e W Y'p (dO), the Neumann problem (7), (9) has the unique weak generalized solution u e W 1p(O).

(iii) Whenever H e W f'p (dO), the Robin problem (7), (10) has the unique weak generalized solution u e W 1p(O).

Remark 6. Note that W1p(O) 4 W1p* (O). Hence Theorem 2 extends the result in Remark 5 indeed.

Remark 7. The approach mentioned above can be also applied to the p-parabolic problems. For example, in Ox (0,T), T = const > 0 consider the non-stationary equation

k

ut - Y, div*, u\Pi-2Vxiu) + \u\p'*-2u = f (11) i=1

supplemented with the Cauchy data

u\t=0 = uo on O, (12)

and either the Dirichlet condition (8), or the Neumann condition (9), or the Robin condition (10). Let the given functions in (11), (12), (8)-(10) depend, in general, on t (except for u0) and have the following regularity:

f e Lp* (0,T; W-1p'* (O)), uo e L (O), g e Lp* (0,T; WYp(dO)), h e Lp* (0,T; W-Yp' (dO)), H e Lp* (0,T; W-Yp'(dO)) + Lq* (O x {t = 0}).

Then each of the problems

• (11), (12), (8) (Cauchy — Dirichlet),

• (11), (12), (9) (Cauchy — Neumann),

• (11), (12), (10) (Cauchy — Robin)

has the unique solution u e L^(0,T; L2(O)), u e Lp,(O x (0,T)).

Acknowledgement. The authors are grateful to Dr. Ivan Kuznetsov (Novosibirsk State University) for many fruitful discussions.

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