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УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА Том 148, кн. 2 Физико-математические пауки 2006
UDK 517.5
ISOPERIMETRIC INEQUALITIES FOR Lp-NORMS OF THE DISTANCE FUNCTION TO THE BOUNDARY
R. G. Salahudinov
Abstract
The main goal of the paper is to prove that Lp-norms of dist(x,9G) and dist-1(x,dG) are decreasing functions of p, where G is a domain in Rn(n > 2). We also obtain a sharp estimation of the rate of decreasing for these norms using Lp -norms of the distance function for a consistent ball. We prove a new isoperimetric inequality for Lp-norms of dist(x,9G), this inequality is analogous to the inequality of Lp -norms of the conformal radii (see Avkhadiev F.G., Salahudinov R.G. // J. of Inequal.& Appl. 2002. V. 7, No 4. P. 593 601).
Note that L2-norm of dist(x,9G) plays an important role to investigate the torsional rigidity in Mathematical Physics (see. for instance. Avkhadiev F.G. // Sbornik: Math. 1998.
V. 189, No 12. P. 1739 1748: Banuelos R., van den Berg M.. Carroll T. // J. London Math. Soc. 2002. V. 66, No 2. P. 499 512). As a consequence we get new inequalities in the torsional rigidity problem.
Also we generalize the n-dimensional isoperimetric inequality.
Introduction
Let G be a domain in Rn(n > 2). Let us consider the following geometrical functional [1, 2]
I(p, G) = J distp(x, dG) dA. (1)
G
Here dist(x, dG) denotes the distance function from x e G to the boundary dG, dA is the volume element dxi,..., dxn, and p > —1. In [1] it was noted, that it might be justified to call I(p, G) the p-order eulidean moment of G with respect to its boundary.
First we remark some applications of (1). and further we note connection of (1) with a new conception of isoperimetrical monotonicity.
It is clear that 0-order moment is the volume of G. Also, as a limit case of (1),
we can get d(G) = maxdist(x, dG). We also found that for a wide class of domains
xEG
(—1) G
G
elastic torsion problem. This is the second order eulidean moment or, otherwise, the eulidean moment of inertia with respect to the boundary. Consider the boundary value problem
Am = — 1 in G, u = 0 on dG,
where —A is the Dirichlet Laplacian, and let
P(G) := 4^ u(x) dA,
G
(G) G
call P(G) the torsional rigidity of G, even if n > 2 and/or G is not simply connected.
We begin from the case n = 2. In 1995 F.G. Avkhadiev [1] proved the two-sided inequality
1(2, G) <P(G) < 6^ 1(2, G). (2)
Later in [4] the left-side hand of the inequality was improved to 31(2, G) < 2P(G).
n
-I(2,G)<P(G)<CgI(2,G), (3)
n
G
constant (see [6]), where Co is a functional on G. In particular, the two-sided inequalities (2), (3) answered the question:“ When is the torsional rigidity of a simply connected
domain in Rn bounded?’, however for n > 3 the question is still open.
Another application of (1) was discovered by F.G. Avkhadiev in [2], this is also n
JJ If |q(x) dA
1/q
KP,q{G) = sup ■ f ec~(o)
Jj |gradf |p(x) dA
1/p'
Kp,q(G) appears in the Poincare - Sobolev’s inequality. In [2] it was proved the two sided estimations for Kp,q(G) using (1).
The first property of isoperimetric monotonicity was conjectured by J. Hersch [7], and it was proved by M.-Th. Kohler-Jobin [8]. We shall consider the following boundary value problem [9]
Av + pv + 1 = 0 in G, v = 0 on dG ( — to < p < A1(G)) (4)
and the corresponding functional
Q(P) := J vdA. o
In particular, we have Q(0) = P(G)/^ Q(Ai) = nj4/ (8A1 (G)^, and (—P)Q(P) —^
A(G) (see, for example, [12]).
By Q(P) we denote the radius of a ball in Rn with same Q(P).
Theorem A [8, 10]. Let G be a bounded domain and not a ball in Rn, then Q{j3) is a decreasing function on /3. If G is a ball, then Q(f3) is a constant function.
This monotonicity property contains several well-known plane isoperimetric inequalities (see [12]), which include functionals A^G), P(G^^d A(G). On the other hand, existence theorems for P(G^^d A1 (G) are expressing in terms of I(a, G) (see [1, 11]). Further, other monotonicity properties were discovered by C. Bandle [13], and by J. Hersch [12]. All of these isoperimetric monotonicity properties were connected with the solutions of the differential equations (4). So, the conjecture on the isoperimetric monotonicity property of (1) is a geometrical analog of Theorem A. On the other hand, this work was intended as an attempt to bind continuously the well-known geometrical quantities of a domain in Rn such as the surface area, the volume, the radius of the largest ball contained in the domain.
1. Main results and corollaries
Let a > n — 1. Further it would be suitable to use a constant
m -n \-i-i r(n /2)r(a + 1) , .
Ca,« := la - n, Si) = ,,---------- 1ff, v (5)
2n”/2 I(a — n + 1)i(n)
where B1 is a unit bal 1 in R^^d r(^) is Euler’s Gamma function.
Theorem 1. Let a > p > n — 1, and G is a domain in Rn such that I(a — n, G) <
< to . Then
[ dista+l3-n{x,dG)dA< Ca’nC/3’n [ dist“~n(x, <9G) cMx o o
x J dist^-”(x, dG) dA (6)
o
The equality holds iff G is a ball in R”.
This kind of inequality was first proved in [14] for conformal moments of a simply connected plane domain. Also, in [14] a chain of plane isoperimetric inequalities was obtained in order to get sharp lower bound in the torsional problem. This chain is similar to ‘discrete isoperimetric monotonicity.' Note that the inequality (6) contains
G
Corollary 1. Let n = 2 and a = p = 1 in Theorem 1, then inequality (6) turns to the plane classical isoperimetric inequality
In the next two assertions we prove isoperimetric monotonicity properties for integral functionals which depend on dist(x, dG).
Theorem 2. i) Let | dist(x,dG)| p < to, then
||dist(x, dG)\p> ||dist(x, £>£>!)||p/
||dist(x, dG)\p» ~ ||dist(x, dDi)\p» ’ ‘
where p' > p > p" > ^ p' > p", and D1 is a ball such that | dist(x, dD1) | p :=
| dist(x, dG)| p. Equality holds only for a ball in R”.
ii) Let | dist-1(x, dG) | p < to, then
||dist-1(x,dG)\p> ^ ||dist_1(x,9_D2)||p'
||dist 1(x,dG) ||p// ||dist 1{x,dD2)\p»’
where 0 < p' < p < p'' < 1, p' < p'', and D2 is a ball such that | dist-1(x, dD2) |p := | dist-1(x, dG) | p. Equality holds iff G a ball in R”.
p = p = 0
(G) p = p = 0 p = 1
p = p = 0
p >0
Using a result from [4] we can get the following lower estimations for the torsional rigidity of a plane domain.
Corollary 2. Let G be a simply connected plane domain, and let P(G) is bounded, then we have for a > 4
where a > n — 1. In the Lebesgue’s sense In-1(G) does not exist, however we give a meaningful definition for the case a = n — 1. In that case we suppose that G has a smooth boundary dG, which belongs to C 1(G), and we put
Therefore, d(G) < to is the necessary condition to apply our theorems. Further we prove that the evaluations given in (9) Mid (10) are well defined.
Theorem 3. Let G is a domain in Rn, and suppose that Iao (G) is bounded for some ao > n — 1. Then
i) If G is not a ball in Rn, then Ia(G) is a strictly decreasing function of a for a > a0 and ITO(G) = d(G).
ii) If G is a ball in Rn, then Ia(G) is the radius of the ball G for all a > n — 1.
Corollary 3. Let G is a domain in Rn such th at Ia(G) < to for a e [n — 1,n), then
Rn
n
inequality, which was proved by E. Schmidt [17], moreover, our assertion is a more sharper result.
2. Proofs of theorems
(a, G)
a > —1.
(a — n, G)
(9)
dG
where dS is the volume element of dG. On the other hand, we have
lim Ia(G) =d(G).
(10)
In(G) < Ia(G), and Ia(G) < In-i(G).
Throughout the proof we will use the following notations G^(a) := {x e G | dist“(x, dG) > A} ,
rA(a) := {x e G |dist“(x, dG) = A}
(ID
G
and denote by [^s the Schwarz symmetrization of a domain in R”, and of a function over the domain. We will usually fix a parameter a in our proof, and in those cases, for brevity, we will drop a, for example, GA := GA(a).
Lemma 1. Let a > 0, and G be an unbounded doma,in in R” such that I(a, G) <
< to. Then: 1) a(GA) < to for 0 < A < d“(G), in particular, d(G) < to; 2) Aa(GA) —► 0.
A^0
A>0
I(a, G) > / dist“(x, dG) dA >a(GA) inf dist“(x, dG) = Aa(GA),
J x£Gj
Ga
so a(GA) < A-11(a, G) < to .
Further, note that
I(a, G) = f dist“(x, dG) dA + f dist“(x, dG) dA.
Ga G\Ga
Using the definition of integral by Lebesgues, and integration by parts, we obtain
s(Ga) d“(G)
J dist“(x, dG) dA = J A(a) da = Aa(GA)+ J a(Gt) dt.
Ga 0 A
The last integral bounded by I(a,G). Hence a(GA) is integrable on [0, d“(G)] because a(GA) > 0 for all admissible A. Therefore
A
0 < lim A a (Ga ) < lim / a(Gt) dt = 0.
A^0 A^0 J
□
In the sequel we will use some well-known properties of the level sets (see, for example, [7]), in particular, we will frequently make use of the relations
d'3 (G)
J dist^(x, dG) dA = J a(GA(^))dA for ^> 0,
g 0
f dist^(x, dG) dA = a(G)d^(G) + f a(GA(^)) dA for ^< 0, (12)
g d3 (g)
^(Ga(1)) _
dA
rA(i)
dS.
Lemma 2. Let G be not a balI in R”, and [^]S is the Schwarz symmetrization of
GA
1) f dist“(x, dG) dA < f dist“(x, d[G]S) dA for a> 0;
Ga [Ga]s
2) J dist“(x, dG) dA > J dist“(x, d[G]S) dA for — 1 <a< 0.
[Ga]s
g
A
Proof. Let x e GA, then it is easy to check that
dist(x, dG) = dist(x, dGA) + A1/a. (13)
GA
have [dist(^ dGA)]S (x) < dist(x, d[GA]S) for x e [GA]^^d a([GA]S) < a(([G]S)A) (see [7] ). Using (13) we obtain
[dist(-, dG)]S (x) = dist(•, dGA) + A1/a (x) = [dist(^ dGA)]S (x) + A1/a <
< dist(x, d[GA]S) + A1/a < dist(x, d ([G]S)A) + A1/a = dist(x, d[G]S).
a>0
J dist“(x,dG) dA = J ([dist(^ dG)]S (x))“ dA < J dist“(x, d[G]S) dA.
Ga [Ga]s [Ga]s
—1 < a < 0
□
We now define a ball D(c R”), to the domain G. From (5) we
can conclude that I(a, D), ^^^xed a, is a strictly increasing function of the radius of D, ^^d rans from 0 to to with the radius of D. Therefore there is exactly one ball
D, ^o an Euclidean motion, such that I(a, D) = I(a, G). Note that D depends of
aG
According to Lemma 2 we shall distinguish two cases: 1) a > 0, and 2) —1 < a < 0. a > 0 D
1) 1(7, G) < I(y,D) /or y > a; (14)
2)I(y,G) > I(y,D) for 0 < y < a. (15)
G
Proof. To prove the assertion we apply the M.-Th. Kohler-Jobin symmetrization with a slight modification. This method was introduced in [8. 10]. and it was applied to study isoperimetric properties of the solutions of (4), A1(G), and the first eigenfunction of (4) (see [3, 8. 10, 12]).
a=0
A = 0. Therefore let us fix a > 0. Further note that the case y = 0 also is another
A=0
I(a, D) = I(a, G) < I(a, [G]S),
and using (5), we obtain 1(0, D) = a(D) < a([G]S) = a(G) = 1(0, G).
Thus we can suppose y > 0.
Set
i(A) = J dist“(x, dG) dA — Aa(Ga).
Ga
The corresponding value for D we denote by i*(A*). From (12) we obtain
cT(G)
i(A) = J a(Gt) dt.
A
and since i(d“(G)) = 0, we have
di(A)
dA
An analogous computation for DA* leads to
di*(A
= a(GA). (16)
dA*
(DA
Now let us define a correspondence between GA and DA* % requiring that i(A) = = i* (A*). By the definition of D, for A = A* = 0 we have i(0) = I(a, G) = I(a, d) = = **(0).
Because GA is bounded, by Lemma 1, applying Lemma 2 to GA and DA*, we obtain a(DA*) < a(GA). Let A(i) be the inverse of i(A). Hence by the defined correspondence we must have the inequality
d\*(i) d\(i)
di di
Again, using A*(I(a, D)) = 0 = A(I(a, G)), we obtain by integration from i0(> 0) to I(a, G)
A*(i) > A(i). (17)
In particular, we have the inequality d(D) > d(G), which, indeed, follows immediately
D
From the definition (11) easily follows GA([) = GA„/^, where [ > 0. Using Lemma 1, (11), (12), and the equality —di(A) = a(GA)dA, by (16), we get
a(G) dY (G) dY (G)
*t(Y)) «t = I a(Gta/T) dt
J dist7(x, dG^A = J A(a)da = J a(Gt(Y)) dt = J a(Gt
G 0 0 0
d“(G) d“(G) I(a,G)
= ^ [ iT,/“-1a(Gt)dt =-- f tl/a-ldi{t) = ^ [ X'>/a-1{i)di. (18)
a J a J a J
In the both cases y > and 0 < y < a, using (17), we obtain (14), and (15), but
in the second case we suppose that I(y,G) < to. The cases of equality immediately
follows from Lemma 2. □
Proposition 2. Let —1 <a< 0, y < 0, an d D is the ball defined in the same
way like above, then
1) I(y,G) < I(y, D) for 0 > Y > a;
0
2)I(y,G) > I(y, D) for a>Y.
Proof. We will use same ideas as in Proposition 1, but in quite different situation. First, we note that if we fix a, then I(a, D) is an increasing function of the radius D
(DA) a A
D
a(G)=a([G]s) <a(D). (19)
As in Proposition 1 the case y = 0 is the particular case of Lemma 2 with A = 0. Therefore let y < 0.
Fig. 1
Sot
i(A) = J a(Gt) dt.
where
(GA)
a(G) for 0 < A <d“(G).
&(Ga) for A > d“(G).
In particular, using (12) wo got
i(0)=a(G)d“(G) + f a(Gt) dt = I(a,G).
da(G)
For tho convonionco of tho roador wo givo an illustration for the plane case (see Fig. 1). The corresponding value for D we denote by i*(A*).
Further, almost everywhere we have
-^=a(GA). (20)
An analogous computation for DA* leads to
di*( AH dX*
a(DA*).
Let us define a correspondence between GA and DA* by requiring that i(A) = i*(A*). By the definition of D, for A = A* = 0 we have i(0) = I(a, G) = I(a, D) **(0)•
LP
159
Applying Lemma 2 to GA and DA*, the note at the beginning of the proof, and (19), we obtain a(DA*) > a(GA). Let A(i) be the inverse of i(A). Hence by the defined correspondence we must have the inequality
dA*(i) dA(i)
^----
di di
Using A*(I(a, D)) = 0 = A(I(a, G)), we obtain by integration from i0(> 0) to I(a, G)
A*(i) < A(i). (21)
From the definition (11) easily follows GA([) = GA„/^, where [ < 0. Using
Lemma 1, (11), (12), and the equality —di(A) = a(GA)dA, by (20), we get
Idist7(x, dG) dA = a(G)d7(G) + j a(Gt(Y)) dt = a(G) (d“(G))7/“ +
dY (G)
d“(G)
+ J a(Gtc/-,) dt = — J t1'/a~1a(G) dt + — J #7/“_1a(Gt) dt =
dY(G) 0 d“(G)
I(a?G)
= 7 [tr//°‘-1!i(Gt)dt. = -- [ty/*-1 di(t.) = - [ {X{i))'l/a-1 di. (22)
a J ay a J
0 0 0
Let a < y < 0, then Y/a — 1 < 0. From (21) and (22) we easy obtain
I(a,G) I (a,D)
1(7, G) = — f (A(*))7/“-1d* < 1 f (A*(*))7/“_1 di = 1(7, _D),
aa
00
which is the first inequality of our proposition. If we interchange a Mid Yj then the second inequality follows from the first.
This completes the proof of Proposition 2. □
1
l(a,D)= l(a,G). ' ' □
Proof of Theorem 2. To prove the first part we apply Proposition 1 for the case I(p, D) = I(p, D1). The second part follows from Proposition 2 for the case I(p, D) = = I(p,D2). □
Proof of Theorem 3. First note that the isoperimetric inequality Ig(G) > (G)
for [ < n < S follows easily from Lemma 2, and moreover, we have the same lower and the upper bound In(G) for Ig(G) and Ia (G) respectively. Now we prove that the definition (9) is well-defined for domains with smooth boundaries. For the brevity we will use the denotation lim := lim . Indeed, using (5), (11), and (12), and
a a^(n-1) + 0
G
applying a integration by parts, we obtain
lim(Ia(G))a = limea n f dist“-”(x, dG) dA =
a a ' J
a(G)
d(G)
limca n [ A(a(GA(l)))“-nda(GA(l)) =lim—^— / [ dSdXa-,l+1
a ’ 7 a a — n+17 7
0 rA(1)
Um r(a + l)r(„/2) ;;
a 27Tn/2r(o! — n + 2)T(n)
d(G) /
d(G)a-n+1 J dS J Aa-n+1d
^(G}(1) 0
dS
Va(1)
T(»/2)
2yrn/2
dS.
dG
Note that, from (5) and (8) we can see Ia(D) = R for all a > n — 1, where R is D
Let G is not a bal 1 in R”, and fix a(> a0^en Ia(G) is bounded, by Proposition 1 and Proposition 2. Like above we can show that there is exactly one ball D(a) in R”, up to an Euclidean motion, such that Ia(G) = Ia(D(a)). Further we conclude from (5) and (6) that IY(D(a)) = I^(D(a)) for all admissible ^d S. Therefore for a small e > 0, from Proposition 1 and Proposition 2 foilow again that Ia+e(G) is bounded, and the ball D(a) gives a larger Ia+e (•) domain G, that is
Ia+e(G) < Ia+e(D(a)) = Ia(D(a)) = Ia(G),
(23)
which is the desired conclusion.
To complete the proof we have to prove (10). From the monotonic property of Ia(G) with respect to the domain we get the inequality Ia (G) > d(G). Thus it would be enough for us to establish the reverse inequality. Let Ia0 (G) < to for some a0 > n. Then, applying (18) for a > a0, we can write the equality
Ia(G) = (c a,^ (a — n))1/a = /
I
V
/
V
I(a0—”,G)
1/a
^l(G)
(a - n)Catr
a0 — n
I(ao —”,G)
A(a—ao)/(ao—n) (*) d*
A
a
(G)
(a—ao)/(ao—”)
1/a
( i) di
(a0 — n) ^l(G)]-ao
0
where A corresponds to the level sets of distao—”(x, dG). Thus we easily get Ia(G) < d(G)
(a0 — n) [ (G)]
a Ia(G) < (G)
This finishes the proof of the theorem.
□
g
0
0
In the conclusion we have to make a remark about reverse inequalities in the theorems. Note that, using Proposition 1 we also can ’go back’ from Ia(G) to Ia—e (G), but
we need to know that Ia_e(G) is bounded. We now give a simple example of domain G on the plane, such that it has Ia(G) = to and IY(G) is bounded for 7 > a. In particular, that means, we cannot prove the reverse inequality Ia(G) > cIa (G) for a > S without additional restrictions on G, here c > 0 is a universal constant.
Without loss of generality we suppose a > 0, and consider the domain placed
between curves x = 1, y = 0, and y = x_1/a. There exists X e R such that
dist(z, dG) = y/2, where z = x + iy, and x > X. Denote by GX the subdomain
of G between x = 1 and x = X, we obtain
Xo
dx
-S' TO,
[ dist (x, dG) dAm [ dist (x, dG) dA+ - [ —
J J 2 J x Xo^^
G Gx X
nevertheless
dx
dista+e{x,dG)dA< j dista+e{x,dG)dA + j
< TO.
x1+e/Q
G Gx X
where £ > 0. The generalization to an n-dimensional case is obvious.
The author is thankful to Professor F.G. Avkhadiov for interesting discussions on the mathematical physics and useful advices.
The work was also supported by Russian Foundation of Basic Research (grant No. 05-01-00523).
Резюме
Р.Г. Салахудинов. Изопериметрические неравенства для Lp-HopM функции расстояния до границы области.
Доказана изопериметрическая монотонность евклидовых степенных моментов области относительно своей границы. Доказанное свойство, эквивалентно изоперимет-рическим неравенствам для Lp-HopM функции расстояния до границы области для различных значений p.
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Поступила в редакцию 21.03.06
Салахудинов Рустем Гумерович кандидат физико-математических паук, доцент. старший научный сотрудник НИИ математики и механики им. Н.Г. Чеботарева Казанского государственного университета.
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