CC BY
11UDC 517.918
EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR VARIATIONAL DIRICHLET PROBLEMS OF A NONLINEAR DEGENERATE DIFFERENTIAL EQUATION S, A. Iskhokov
1. Embedding theorems of different metrics for weighted function spaces
Let 0 be a bounded n-dimensional domain in the space Rn of points x = (x\,x2,... ,xn) with (n — l)-dimensional boundary dfi. Denote by p(x) the regularized distance from a point x e fi to dfi (the definition see, for example, in [1]) and by Uk the Sobolev generalized derivative d H kn °f the function u(x), where k = (ki,... , kn) is a multi-index and |k| = ki + ... + kn.
Suppose that r is a nonnegative integer, a and p are real numbers, and p ^ 1. The symbol Lpa(Q) denotes the class of measurable real-valued functions u(x) defined on fi which have all the generalized Sobolev derivatives Uk (x) of order r(|k| = r) with the finite seminorm
\u;Lla(Q )\\ = { E / P?a(x) U k (x) |p dx} ^
When r = 0, the class Lrp, a(fi) coincides with the weighted space Lp,a(fi), which has the norm
i/p
\u,LPMn )\\ = I pP^ x Hx) lp dx'
© 2008 Iskhokov S. A.
We denote by Wpa(Q) and Vpa(Q) the spaces with the respective norms IbWyfi)\\ = {\\u;Lrpa(n)f + \\u;Lp(fiF}1/p , (1.1)
||u;VpУfi)|| = {||u;Lp,a(fi)||p + \u,Lp,a-^mP) 'P ■ (1.2)
The space Wp,a(fi) is a Banach space with the norm (1.1) for any p > 1, a G R and a natural number r. For a negative integer r and p > 1, the space Vp o(ii) is defined by the equality Vpa(Q) = (V—-a(fi))*, where q = p/{p - 1).
Let C^(fi) denote the set of infinitely diflterentiable functions in fi
o
with compact support. The symbol Wp,a(0) denotes the closure of C^(fi) with respect to the norm of the space Wpa(Q).
Henceforth the symbol Bm(OQ) stands for Besov space of functions defined on the boundary 8Q (for its definition, see, for example, [2,3]).
Basic properties of spaces Wp,a(Q) and Vpa(ii) were examined by S. M. Nikol'skii, P. I. Lizorkin and N. V. Miroshin in [2,4-6]. In particular, the following theorems were proven.
Theorem 1.1. Let OQ be a dosed differentiable manifold. Then:
1) for all a G (—<x>,+ <x), p G [1, ro), and a nonnegative integer r, Vpa(Q) is a Banach space with the norm (1.2) and the set C^(fi) is dense inV;,a(n);
2) ifa+ 1/p G{ ,2,... ,r}, then Vpa(Q) = W p,a(fl) (up to the norm equivalence).
Theorem 1.2. Let r G N,p > 1,
— — < a < r — (1.3)
pp
and so be a natural number for which
r — a--<so < r — a---hi. (1-4)
pp
Let Oil G CSo+1+E, where £ G (0,1). Then any function u G Wp,a(il) has the s0 boundary functions
dns
= M X e Brp-a-s-1 /p( dQ), s = 0,l,...,so - 1, (1.5)
an
d su
and
||VS; Brp-a-s-1 m )|| < C\\u,wr a(Q )||, s = 0,1,... ,s0 - 1,
where the positive constant C is independent of u(x). Here n is the normal vector to OQ.
Conversely, for given functions
Mx) e Bp-a-s-/p{3U) (s = 0,1,... - 1) there exists a function u e Wp, aa(Q) satisfying (1.5) and
so-i
\u,wra(tt)|| < C £ ||Vs;B/p(on)||,
s=0
where the constant C > 0 is independent of functions ^s (x) (s = 0,1,... , s0 -1 ).
Theorem 1.3. 1. If dQ is a closed differentiable manifold and a < -1 /p or a > r - I/p, then Wrp, a(Q) = WrPia(Q).
2. If Oil e Cwhere the integer so is the same as in Theorem 1.2 and £ e (0,1), then under the condition (1.3) the equality
° ( d su
wrp,a(n) = G w^(O) : —
holds.
= 0, s = 0,1,... , s - 1
on
Further in this section we prove embedding theorems of different metrics for weighted spaces aa (fi), V^ (fi), that will be applied in the next sections to investigating unique solvability of variational Dirichlet problems.
For normed spaces Bi, B2 with norms B ||, B || an embedding Bi —► B2 means that all elements of the space Bi can be regarded as elements of the space B2 and, moreover, Hf; B21| < C||f; Bi || for any f G B where the constant C > 0 does not depend on f.
Theorem 1.4. Let dQ be a dosed differentiable manifold, r be a natural number, and s be a integer satisfying 0 < s < r. Then for real numbers p, q, a, as such that
1 < p ^ q < oo, a.s >--, a — s ^ as H--------(1-6)
q q P
we have the embedding Wpa(Q) —► W^^il).
Proof. It is enough to prove Theorem 1.4 for the case n = 1 and Q = (0, a), where a > 0. Then the assertion of the theorem can be extended to the case n > 1 and a bounded domain iic Rn by the approach developed in the book [2].
Proposition 1.5 [7,8]. Let w0(t) and wi(t) be positive measurable functions defined in (0, œ), 1 < p < q < œ. Then the following Hardy inequality
o / o \1 \ 1 /q / o W/P
J ¡J v(t) dT I w0(t) dt I < C ¡J t)u!(t) dt
holds for nonnegative functions p(t) if and only if the weight functions w0(t), wi(t) satisfy the condition
1 /q / o w/p'
W (t) dT | = C < œ
sup I / wo (t) dt 1 l w P t dt 0<t<o \ J I \ J
where p = p/(p — 1).
It follows from proposition 1.5 that the inequality
1/q /a W /p
' X'
■M
f (y) dy
dxI « (/dx) (!,,
holds if/?</x+l + i — i, —| < /x. Henceforth the shorthand notation A(u) < B(u) stands for the inequality A(u) < cB(u), where c > 0 is a constant independent of u(x).
First we consider the case s = 1. So a\ > —| and + l + | — K
Let u(x) be a function defined on (0, a) which has all generalized derivatives of order < r. Since (see [2])
| ur- >(a) I << \\u;Lp( Ja) \\ + \\xYur (x)-,Lp( Ja) \\, Ja = (0,a),
for arbitrary p > 1 and arbitrary 7 e R, and
u{r-V {x)= u{r-1) (a)+ / u
(y)
by applying (1.7) we have
||xai(x); Lq( Ja) y « \\u;Lp( Ja) \ + \\xau{r) (x)-,Lp( Ja) y q \ 1/q
dx) « \\u; Lp J W + Wx^r) (x);Lp( J J
raiq
u r y dy
Thus
Wra J — W^-] J . (1.8)
Let s ^ 1 and a, as satisfy (1.6). Denote a.j = as - (s - j) for j = 1, 2,... ,s - 1; applying embedding (1.8), we obtain
w— J.
W— Ja) — W— (Ja) — ••• The theorem is proven.
rs 0 < s < r. Then under the conditions
nn n n
1 ^ P ^ 1 < 00, s---1--JiO, a-sH----^ as,
p q p q
we have the embedding
Vp^m — Vqr-Ss((î) .
x
Proof. The embedding Vpra(fl) —► Vr-S (0), in the case p = q, was proven in [5]. We consider the case p < q. Let £ e 0 be an arbitrary-point, and J(£) be the ball around it of radius p(£)/2. Then by the Troisi's theorem (see [4,9]) for any real number t and qi >1 the relation
J(pt-n/qi (0\\u-,Lp( J(m\)qi d£ ~ j(/(x)\u(x)\)qi dx (1.9) n n
holds, where the sign ~ means that there exists a two-sided estimate of the left-hand side of the relation in terms of the right-hand side with positive constants.
It follows from the results of [1] that
£ fc);Lq(mn < £ y^'>.lp(myLp(m)y (i.io)
for 1 < p < q < +oo, 0 < to < r and r — to — ^ + ^ > 0, where I (0) = {z e Rn : | z | < 1/2}■
Let £ be a fixed point in fi. For v (z) = u (£+ \zp (£)'), by applying (1.10) we have
k\=m I{0)
q \ i/q dz
<
I
p \ i/p dz
l\=r Vo) V J
p \ 1 /p dz\ . (1.11)
By performing the change of variables z = (y — £) in integrals of inequality (1.11) and multiplying both sides of obtained inequality by pa-r+(n/p)-(n/q) £ we get af^er some transformations and integrating with respect to £ eii
E f Pqa-rq+iqn/p)-n{£)pqm{off x(v,0uk(y)|qdy) d£
\k\=mQ Q
< E J Pqa-n( £)(/ y Ip dyj q'P d£
f x(y,0 Iu(y) Ip dy^J q'P d£, (1.12)
n
fl(a-r)-v,i
l
k
m
r
where x(y, 0 the characteristic function for the set J(£). By the Troisi's inequality (1.9) the left-hand side of the inequality (1.12) can be estimated from below by-
const Y, Jpaq+qm(y)p-rq+{qn/p)(y)\Uk(y) \q dy.
\k\= mQ
To estimate from above the integrals on the right-hand side of the inequality (1.12) we first apply the following generalized Minkowski inequality
r /r \q/p {fir \p/q i q/p
J U \F(x,y)\p dyj dx J \F(x,y)\q dxj dy\
X Y Y X
and then use the inequality (1.9). We get as a result
| {pa-r+m+(n/p)-n/q x\uk x\)q dA11 / < \\uVPra(Q)\\. \k\= ™fi
(1.13)
Now, if we choose m = r — s, then under the condition as ^ a — s + ^ — ^ the inequality (1.13) implies the embedding Vp a(ii) —► V^-^^}). The theorem is proven.
2. Homogeneous variational Dirichlet problem
Let r be a natural number and pk > 2 for all k : k < r. In a bounded domain fi c Rn with (n — l)-dimensional boundary dfi of class CC we consider the nonlinear differential equation
Lu = J2 —!)\k\ (ak J) \Uk (x)\pk - Uk (x))k = F, x G Q, (2.1)
\k\^r
with real-valued coefficients ak(x).
A function U(x) is said to be a generalized solution to the equation (2.1) if it satisfies the integral identity
]T ak(x)\Uk (x) \p— Uk {x)vik) (x) dx = (F, v) (2.2)
Q \k\^r
for any v e C^ (fi). Henceforth (F, v) stands for the value of the functional F on the function vX- If F is an ordinary function, then (F, v) is the inner product of FX and v(x) in L2(0).
Suppose that the coefficients o,k(x) (|k| ^ r) satisfy the following inequalities
ClpPkia-r+\k\+(n/p))-n X < ok (x) < cippk*a-r+\k\+{n/p))-n X , (2.3)
where ci and c2 are positive constants independent of x e 0. The numbers pk (|k| < r) have to satisfy the conditions
pk = p for |k| = 0, |k| = r; n n
r-\k\----> 0, 2 < p < pk for k : 0 < |A;| < r. (2.4)
p p k
The equation (2.1) was studied in [10] in the case when its right-hand side F is an ordinary function and 2 < pk < p (Vk : |k| < r). Here we F
metrics proven in section 1, study unique solvability of variational Dirichlet problems for equation (2.1) when 2 < p < pk (Vk : |k| < r), that is, in our
pk
Dirichlet problem for class of degenerate nonlinear differential equations was also studied in [11].
In this section we study solvability of the following homogeneous variational Dirichlet problem
Problem Do- For a given functional F e Vq--a (fi), q = p/ (p — 1), find the generalized solution to the differential equation (2.1) belonging the space Vpr^iil).
Consider the functional associated with the equation (2.1)
E(u)= V — /ak X h{k) X)\Pk dx. pk
\kKr fi
Under the conditions (2.4) with the aid of the Theorem 1.6 it is easily-established that
Jp-n(x)(pa-r+'k\+(n/p(XukX|)Pk dx << \\u-,v;a{ii)\\P n
for any multi-index k : |k| < r and any function u G Vp a(Q). So it follows from (2.3) that functional E(u) is finite for u G Vp a(ii). Let F be a given functional from V""a(fi) and let $ (u) = E (u) — (F, u).
Theorem 2.1. Under the conditions (2.3) and (2.4) for any given functional F G V""a (fi) there exists a unique function U G Vpa (fi) such
that
inf$ (u) = $(U) = E(U) — {F,U), where the in£mum is taken over all u(x) G Vpa (fi). Proof. By Theorem 1.6 it follows from (2.3) that
)ir << E{u). (2.5)
Let e be an sufficiently small arbitrary positive number. Applying the
Young inequality
em a iw e - \h\M2
\ab\ < -+-U_ (£ > o, 1/Ml + 2/M2 = 1), (2.6)
pi
we have
HF,u)l < \\F;V-ra(nt)\\ \\u;Vpria(n)\\
ep „ e "q ,, „ « - N VP,a (°) + — h ^,7-a (°) • (2-7)
p ii f, ii q ii h, ii
From (2.5) and (2.7) we obtain
> E(u) -\(F,U)\2 -—11^^(0)11. u
A0 = inf $t(u), where the infimum is taken over all u(x) G Vpa(ii). Applying the Clarkson inequality it is easy to show that
F
^ /u — v\ T / u + v\
Consider a sequence {um}^i=1 C Vpa(Q) such that $ (um) = A0 + £m, £m > 0, and £m ^<x> as m Applying inequality (2.8) we find
E < A0 + \ {£m+s + em) - $ (Mm+S2+Mm) (2.9)
for any natural number s. By virtue of (2.5) it follows that $ ium+s + um\ ^ Aq + 1 ^^ + ^
So
Um $ ( Um+S + Um ] = An.
Hence, by letting m ^<x> in the inequality (2.9) we obtain
Um E (= o. m—V 2 J
It implies that (see (2.5)) the sequence {um}^=1 is a Cauchy sequence in Vpra(Q). Since Vpra (fi) is a complete space, there exists unique function U e Vp 0 (fi) such that \\um — U : Vpa(Q)\\ ^ 0 as m
By virtue of Theorem 1.6 and condition (2.3) we have
|E(um) — E(U)J PPHa — r+ W + (n/p)) — n(x)|
\k\^rQ
x ^ X^" — |U(k xr1 dx
« E \\pa-r+{n/p)-{n/Pk)x(u(m)X — U(kx);Lpk(Q)\\
\k\^r
x {\\pa-r+{n/p)-{n/Pk)Xum X; Lpjn)\ \
+ \\p«-r+\k\+ (n/p)- n/pk) xU k X; Lpk(n) \ \ }Pk/P
« \\um — U;Vpria(il)\\ £ {\\um-, Vpam\\ + \\U-,Vpom\\}Pk/P.
>,aV"/|| / , XW^m, <■ p,a\
\k\^r
Thus lim E (um) = E (U) and, consequently,
m—
um U .
Let us prove uniqueness of the function U(x)
another function U G Vpa (fi) such that A0 = $ (U 'U - U
Suppose there exists Then by virtue of
(2.8) we get
E
< A0 - $
U + Ui
Since (see (2.5))
and
it implies that
Hence
E fc^'.iO
A0 < $
UU
E №'.=0.
U - UvXaW)|l =°
Ux
Theorem 2.2. Let conditions (2.3), (2.4) be satisfied. Then the function U(x) from Theorem 2.1 is the unique solution to the problem D0 and the inequality
||U;V;ia(0)f < M llF;V-la(n)llq is valid, where constant M > 0 does not depend on F.
(2.10)
Ux
$ (u) on the space V^ ^ (fi). So (see, for example, [12]) the Gateaux derivative d$ (u, v) of the functional $ (u, v) at the point u in the direction v exists for any u,v G Vra (fi) and (U,v) = 0 (Vv G Vpa(ii)), that implies identity (2.2). Thus U(x) is a solution to the problem D0.
Conversely, if U(x) is a solution to the problem Do, then U(x) satisfies the identity (2.2) and, consequently, d$(U,v) = 0 (Vv G Vp a(ii)). By-virtue of results of [12, Chapter IV] the second Gateaux derivative d2$ (u, vi,v2) of the functional $ (u) exists and d2$ (U,v,v) > 0 for all
v e Vpa (fi). Therefore (see [12, Chapter IV]), the functional $ (u) has a U
Since U X is a solution to the problem D0 and belongs to Vra (fi), by substituting v(x) = U(x) in (2.2) we get E(U) = (F,U); that implies inequality (2.10) by virtue of (2.5). The theorem is proven.
Under some additional requirements on the smoothness of the boundary dfi and on the power of the degeneracy a we can weaken the condition (2.3) for lower coefficients ak(x) (|&| < r) of the equation (2.1).
Theorem 2.3. Let
a+- {1,2,... ,r-l}, —— < a. < r — — (2.11)
p p p
and let the boundary dQ of the domain fi C Rn belong to class Cs°+1) s
the following conditions is satisfied:
(A) 2s0 > r.
(B) dfi is not the zero set of a polynomial of degree at most r — 1. Then assertions of Theorems 2.1 and 2.2 are valid if the conditions
(2.3) for |fc| < r are replaced by
0 < a,k(xx) < C2pPfc({n/p))-n X , (2.12)
where c2 > 0 does not depend on x e 0.
Proof. We note that according to Theorem 2 from [13] under conditions of Theorem 2.3 for u e Vpa(Q) the following Poincare type inequality
E \\pa-r+\k\u{k); Lp(fi)\\ < c\\u-,Lrpta(n)\ (2.13)
\k\<r
holds, where the constant c > 0 does not depend on u(x). Using the inequality (2.13), it is proven that the inequality (2.5) holds when the leading coefficients ak(X (N = r) satisfy condition (2.3) and the lower coefficients ak(X (|^| < r) satisfy the condition (2.12). The remaining part of the proof IS clS those of Theorems 2.1 and 2.2.
3. Nonhomogeneous boundary conditions
If conditions (2.11) are satisfied and dfi e where s0 is the
natural number from (1.4) and e e (0,1), then by virtue of Theorems 1.1 and 1.3
Vpr,am = wp^m
(up to the norm equivalence) and every function u(x,t) e V^^ii) satisfies boundary conditions
dsu dns
= 0, s = 0,1,... ,so — 1-
dQ
D
Problem Do- For given functional F e V--a (fi) find a generalized
r
p,a
conditions
d sU
-r -a '
solution U(x) e WIa(fl) to the equation (2.1) satisfying the boundary
= 0, s = 0,l,...,so - 1. (3.1)
dfi
dn
In this section we study the solvability of a variational problem with nonhomogeneous boundary conditions of the form (3.1).
Problem D. For a given functional F e W--a(Q) =f (W^^^l))* and a given set of boundary functions
(x) e Bi-a-s-1 /p (dQ), s = 0,1,... ,s0 - 1,
find a generalized solution U(x) e Wpa (Q) to differential equation (2.1) mnd
dsU
TT
' P,a
that satisfies the boundary conditions
dns
= ^s, s = 0,1,..., so - 1. (3.2)
dfi
Suppose that coefficients o,k(x) of the differential equation (2.1) satisfy the condition
clflP*(a+(1/p/p^ (x) < ok (x) < c2pPk{-a+{1/p)/pk)) (x), (3.3)
= ^s, s = 0,1,... , So - 1.
an
where ci, c2 are positive constants independent of x.
Let 1 be a given set of boundary functions G BTp a s 1/p (dd).
By virtue of Theorem 1.2 there exists a function ^ (x) G a (fi) such that
dns
We denote by W^(fi) the set of functions u(x) G Wpa(fi) such that
u(x) — V(x) G WPaM-
By virtue of Theorem 1.4 applying the same technique as in proof of Theorem 2.1, one can prove the following theorem.
Theorem 3.1. Oil G where e G (0,1). Let pk = p for
|k| = 0, |k| = r, and 2 < p < pk for 0 < |k| < r. Then under the conditions (2.10) and (3.3) for every given functional F G W""a (fi) and every given set {^KLq of boundary functions ^s G
BP"a"s"1/p (dtt), there exists a
unique function U G Wy, (fi) such that
inf $ (u) = $ (U) = E (U) — {F, U), (3.4)
u G W
Ux D
and it satisfies the inequality
\\U;WPa(0)\\p << ^ \\U; WpyO)\\Pfc"^^ W^JO)\\
\k\<r
+ \\F;Wrjn)f + \\*; W^il)\\p. (3.5)
Moreover, if
Pk <p+l (Vk: k <r), (3.6)
Ux
\\U;WPa(0)\\p << \\*; WprJQ)f0 + \\F-Wl,_a(il)\\q , (3.7) where = max {p, p/(p + 1 — pk)} if > Wpa(fi)\\ > 1 and = p if
\\*; W,jn)\\ <1. '
Proof. The first part of the proof is as that of Theorem 2.2. We prove the estimates (3.5), (3.7).
Ux
(2.2). Since UX X e Wp,a (fi), by substituting vX = UX) X in (2.2) we obtain
E(U) — B[U= {F,U — *), (3.8)
where
B[U-,^} = ]T akX U k X \P— U(k (x)$ (k)( X dx.
\ k \ <rQ
By the embedding Theorem 1.4 from (3.3), it follows that
\\u; wpa(n)\\P << E(u) (Vu e W^l)). (3.9)
Applying condition (3.3) and the Holder inequality with exponents Ai = pk/(pk — 1 ),A2=pk we obtain
\ k \ <rQ
{Pk-1 )/Pk
< E { /(/+^XI^(fc)XI)Pfc dx
I k | ^
X {/dx
Q
«J2 )\P||Ф; wpjn)\\p. (3.10)
I k I <r
The last inequality is valid by virtue of the embedding Theorem 1.4. Now, using inequality
\(F-U -Ф>\ < \\F;W-La\\ {\\U;WPaa(n)\\ + \\Ф; Wpjfi)\\}
and (3.9) and (3.10) from (3.8), we find that
\\U;wpa(fi)\\p << E \\U;wp,a(n)\ГЦФ;wpa(fi)\\
I k I <r
+ \\F-,Wq-r-a\\ \\U;Wpjn)\\ + \\ \\Ф;wpjtt)\\.
Furthermore, by applying the Young inequality (2.6) we obtain inequality
Uwpyo)ir 'll*; wrAn)ll
< £ ||u w;ia(n)||p + K(e) ||*; w;ia(n)f2 .
e
obtain (3.7) from (3.5). Theorem 3.2 is proven.
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(3.5).
In the case when condition (3.6) is satisfied
^=p^pk — l) > ^ and it follows from (2.6) that
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г. Душанбе
25 декабря 2006 г.
