ESTIMATING THE RESOLVENT OF ELLIPTIC SECOND-ORDER PARTIAL DIFFERENTIAL OPERATORS
DIMITRI MUGNAI - GIORGIO TALENTI
A Franco Guglielmino con affetto
Sharp estimates for the resolvent of a linear elliptic second-order partial differential operator under Dirichlet homogeneous boundary conditions are derived via a symmetrization technique.
1. Introduction.
In this paper we are concerned with partial differential equations having either of the following forms
(1) −
n
�
i, j =1
∂
∂xi
�ai j(x )∂u
∂xj
�
−
n
�
i=1
∂
∂xi�bi(x )u� +�
c(x ) + λ�
u = f (x),
(2) −
n
�
i, j =1
∂
∂xi
�
ai j(x )∂u
∂xj
� +
n
�
i=1
bi(x )∂u
∂xi +
�
c(x ) + λ�
u = f (x).
Entrato in Redazione il 16 aprile 1997.
We suppose the coef�cients and the right-hand side are real-valued measurable functions de�ned in an open subset G of Rn. We suppose λ is a positive constant parameter and the equations are uniformly elliptic, that is
(3)
n
�
i, j =1
ai j(x )ξiξj ≥
n
�
i=1
ξi2 for all x ∈ G and for all ξ ∈ Rn
and
(4) ai j∈ L∞(G).
Moreover, either
(5)
��
G
��n
i=1
b2i�k/2
dx
�1/k
≤ B or
(6) ess sup��n
i=1
bi2�1/2
≤ B,
and
(7) c ∈ Ln/2(G) and c(x ) ≥ 0 for all x ∈ G,
(8) f ∈ Lp(G).
Here B is some nonnegative constant, n < k < ∞, and p = 2n
n + 2 or p > 1 according to whether n > 2 or n = 2.
We are interested in Dirichlet problems with zero boundary data; thus we look for functions u which satisfy either equation (1) or (2) and obey
(9) u = 0 on the boundary ∂G of G.
We deal with weak solutions belonging to Sobolev space W01,2(G).
The present paper parallels earlier ones where estimates for solutions to boundary value problem for elliptic second-order partial differential equations
in divergence form are obtained via inspection of level sets, isoperimetric in- equality and rearrangements of functions. See for instance Alvino-Diaz-Lions- Trombetti [2], Alvino-Ferone-Kawohl-Trombetti [3], Alvino-Lions-Trombetti [4], [5], Alvino-Matarasso-Trombetti [6], Alvino-Trombetti [7], [8], Bandle [9], [10], Chiti [13], [12], Diaz [14], [15], [16], Maderna [18], Maderna-Pagani- Salsa [19], Maderna-Salsa [20], [21], [22], Pacella-Tricarico [24], Talenti [25], [27], [28], Trombetti-Vazquez [31], Weinberger [33] and the references quoted in [28].
It is closely related to [2], [4], [5], [6], [7], [8], [15], [25], [27] and [31].
We provide some variants and re�nements. Comparison results for solutions to both problems (1)(9) and (2)(9) are established. Estimates are obtained as a corollary, where the dependence upon parameter λ is explicitly displayed. In other words, we obtain estimates of the resolvent operator the main goal of this paper.
2. Notations.
Rn n-dimensional euclidean space G open subset of Rn
∂G boundary of G
Lp(G) Lebesgue space W1,2(G) Sobolev space
W01,2(G) closure of C0∞(G) in W1,2(G)
|G| n-dimensional measure of G
G� ball centered at the origin having the same measure as G µu distribution function of u
u∗ decreasing rearrangement of u u� symmetric rerrangement of u
Cn measure of the n-dimensional unit ball κn isoperimetric constant
Recall that µu is de�ned by
µu(t) = |{x ∈ G : |u(x)| > t}|,
u∗ is the right-continuous decreasing function from [0, +∞) into [0, +∞]
equidistributed with u, and u� is the function from Rn into [0, +∞] de�ned by u�(x ) = u∗(Cn|x|n).
Finally recall
κn = 1 nCn1/n
,
the smallest constant which makes the following isoperimetric inequality (measure of E )1−1/n ≤ κn(perimeter of E )
true for every measurable subset E of Rnhaving �nite measure.
For the properties of the objects de�ned above see [17], [25], [26], [27] and [30].
3. Main results.
Theorem 1. Let u belong to W01,2(G) and satisfy equation (1). Suppose (5) holds. Then a measurable function g from [0, +∞) into [0, +∞] exists such that
(10) �
nCn
� nκn|G|1/n 0
g(r)krn−1dr�1/k
≤ B,
and the solution w belonging to W01,2(G�) to the following problem
(11)
−�w −∂ F
∂ �(�, �)
n
�
i=1
∂
∂xi
� xi
|x|g(|x|)w
� +
+∂ F
∂ �(�, �)�
λw − f�(|x|)�
= 0 in G�
w = 0 on ∂ G�,
satis�es (12)
� s 0
u∗(t) dt ≤
� s 0
w∗(t) dt for every s such that 0 ≤ s ≤ |G|.
Here
� = |x|n−1g(|x|)w(x),
� = −λ
� Cn|x|n 0
w∗(t) dt +
� Cn|x|n 0
f∗(t) dt, F(�, �) = max{�/2, � + �}.
Theorem 2. Let u be as in Theorem 1. Suppose that (6) holds; then the solution wof the following problem
(13)
−�w − B∂ F
∂ �(�, �)
n
�
i=1
∂
∂xi
�xi
|x|w
� +
+ ∂ F
∂ �(�, �)�
λw − f�(|x|)�
= 0 in G�
w = 0 on ∂ G�,
satis�es (12) for every s such that 0 ≤ s ≤ |G|.
Here
� = B|x|n−1w(x ) and � is as in Theorem 1, as well as F .
Theorem 3. Let u belong to W01,2(G) and satisfy equation (2). Suppose that (5) holds and g is as in Theorem 1. Then the following problem
(14)
−�w +
n
�
i=1
g(|x|)xi
|x|
∂ w
∂xi + λw = f� in G�
w = 0 on ∂ G�,
has a solution in W01,2(G�); such solution satis�es
� s 0
exp�
−
� nκnt1/n 0
g(r) dr�
u∗(t) dt ≤ (15)
≤
� s 0
exp�
−
� nκnt1/n 0
g(r) dr�
w∗(t) dt for every s such that 0 ≤ s ≤ |G|.
Theorem 4. Let u be as in Theorem 3, and suppose that (6) holds; then the following problem
(16)
−�w + B
n
�
i=1
xi
|x|
∂ w
∂xi + λw = f� in G�
w = 0 on ∂ G�,
has a solution in W01,2(G�), and such solution satis�es (17)
� s 0
exp�
−nκnBt1/n�
u∗(t) dt ≤
� s 0
exp�
−nκnBt1/n�
w∗(t) dt for every s such that 0 ≤ s ≤ |G|.
Theorem 1 is demonstrated in Section 5. Theorem 2 can be proved quite in the same way as Theorem 1. Theorem 3 is a relative of Theorem 4, in the same way as Theorem 1 is a relative of Theorem 2. Theorem 3 can be proved by the same arguments involved in Theorem 1 plus Gronwalls Lemma. Theorem 4 is a special case of [4], Theorem 1. The proofs of Theorems 2, 3 and 4 will be omitted.
4. Some applications.
Theorem 5. Consider the problem made up by equation (2) and boundary condition (9) under hypotheses (3), (4), (6), (7) and (8). Assume |G| < +∞
and p > n/2. Then its weak solution obeys
(18) ess sup |u| ≤ D� f �Lp(G). Constant D is de�ned as follows:
a = n − 1 2
�1 − B
√B2+ 4λ
� ,
b = nκn|G|1/n�
B2+ 4λ,
I(y) = exp��1
2 − B
√B2+ 4λ
� y�
×
�
U (a, n − 1, y) − U (a, n − 1, b)
M(a, n − 1, b)M(a, n − 1, y)� ,
p� = p p − 1,
(19) D = �(a)(B2+ 4λ)n/(2 p)−1 n1/ pCn1/ p(n − 2)!
�� b 0
I(y)p�yn−1d y
�1/ p�
.
Here U and M are Kummers functions. D is the smallest constant that makes (18) true under the assumptions above.
Proof. The very de�nition of u∗ gives
(20) ess sup |u| = u∗(0),
and inequality (17) gives
(21) u∗(0) ≤ w∗(0).
Observe that the solution w to problem (16) is radially symmetric and radially decreasing, in other words
w(x ) = w∗(Cn|x|n).
Therefore (16) reads:
(22)
−d2w d|x|2 −
�n − 1
|x| − B
�dw
d|x| + λw = f∗(Cn|x|n), w(x ) = 0 if |x| = nκn|G|1/n.
Two linearly independent solutions of the homogeneous equation d2w
dr2 −
�n − 1 r − B
�dw
dr + λw = 0, whose wronskian is −(n − 2)!
�(a) β2−nr1−ne(β−2α)r, are
w1(r) = e−αrM(a, n − 1, βr) and w2(r) = e−αrU (a, n − 1, βr).
Here
α = −B +√
B2+ 4λ
2 and β =�
B2+ 4λ.
Recall that M(a, c, z) and U (a, c, z) are the solutions of Kummers equation zd2w
dz2 + (c − z)dw
dz − aw = 0 given by
M(a, c, z) = �(c)
�(a)
∞
�
k=0
�(a + k)
�(c + k) zk k!,
if c is not a negative integer, and U (a, c, z) = π
sin π c
� M(a, c, z)
�(c)�(a + 1 − c) − z1−cM(a + 1 − c, 2 − c, z)
�(a)�(2 − c)
� . For more informations see [1] and [23].
The solution to problem (22) such that
�
G�
�dw d|x|
�2
dx < +∞
is then
(23) w(x ) =�a1(|x|) + A�w1(|x|) + a2(|x|)w2(|x|), where
a1(r) = �(a) (n − 2)!βn−2
� nκn|G|1/n r
tn−1e(α−β)tU (a, n − 1, βt) f∗(Cntn) dt, a2(r) = �(a)
(n − 2)!βn−2
� r 0
tn−1e(α−β)tM(a, n − 1, βt) f∗(Cntn) dt,
A = − �(a)
(n − 2)!βn−2γ
� nκn|G|1/n 0
tn−1e(α−β)tM(a, n − 1, βt) f∗(Cntn) dt and
γ = U (a, n − 1, b) M(a, n − 1, b). Thus
w(0) = �(a) (n − 2)!βn−2
� nκn|G|1/n 0
tn−1e(α−β)t�U (a, n − 1, βt) −
−γ M(a, n − 1, βt)� f∗(Cntn) dt.
H¨older inequality and (20) and (21) give (18). �
Theorem 6. Constant D given in Theorem 5 has the following properties:
D
λn/(2 p)−1 −→ 1
n1/ pCn1/ p2n/2−1�(n/2)× (24)
×
�� ∞
0
t(np−2n+2)/2 p−2[Kn/2−1(t)]p�dt
�1/ p�
as λ → +∞;
D = λn/(2 p)−1{the r.h.s. of (24)} if |G| = +∞ and B = 0.
Here Kν(z) is the modi�ed Bessel function of second kind and order ν (see, e.g., [1], [23], [32]).
Proof. The proof consists of an inspection, based on formula (19) and formula [1], Chapter 13, Section 13.6, 13.6.21 relating Kummer and Bessel functions.
� Theorem 7. Let u belong to W01,2(G) and satisfy (1), let w belong to W01,2(G�) and satisfy (11) or (13), according to whether (5) or (6) is in force. Then (25) �u�Lq(G)≤ �w�Lq(G�)
for every q larger than or equal to 1.
Proof. Since a function and its decreasing rearrangement are equidistribuited, (25) follows from (12) and the lemma appearing in [29]. �
Theorem 8. If u and w are as in the previous theorem, m(G) = +∞ and q > p > 1, then
(26) �u�L( p,q)≤ �w�L( p,q).
Here L( p, q) stands for Lorentz space. For its de�nition we refer to [34].
Miscellaneous estimates, which easily follow from the previous theorems, are listed below. (Standard integrals of Bessel functions, and Bliss inequality see [11] are involved.)
• If |G| = +∞, B = 0 and p > n/2, then
�u�Lp(G) ≤ 1
λ� f �Lp(G).
• If |G| = +∞, B = 0 and q > p > 1, then
�u�L( p,q)≤ K1/q
λ � f �Lp(G), where
K = p
q( p − 1)
� �(qp/(q − p))
�( p/(q − p))�(p(q − 1)/(q − p))
�q/ p−1
.
5. Proof of Theorem 1.
We need the following lemmas.
Lemma 1. For every measurable function f and for every measurable set E we have
(27)
�
E| f (x)| dx ≤
� |E |
0
f∗( f ) ds.
Moreover if E equals a level set {x ∈ G : | f (x)| > t} and t is positive, then equality holds in (27).
Lemma 2. Let u ∈ W01,2(G) and let µ be the distribution function of u. Then (28) κn2µ(t)2/n−2[−µ�(t)]�
−d dt
�
{x∈G:|u(x)|>t}|∇u|2dx�
≥ 1,
for almost every t from 0 to ess sup |u|. As usual, ∇ stands for gradient, so that
|∇u|2=
n
�
i=1
u2xi.
Lemma 1 is a special case of Hardy-Littlewood theorem, see [17], The- orem 378 or [26], Theorem 1.A. Lemma 2 is proved in [25], via Fleming- Rishel coarea formula and the isoperimetric inequality. An alternative form of Lemma 2 appears in [26], Lemma 1.E.
Proof of Theorem 1. Suppose u is a weak solution of problem (1) and (9), that is u ∈ W01,2(G) and
(29)
�
G
n
�
i,k=1
aikuxiφxk +
n
�
i=1
biuφxi + cuφ
dx =
�
G
f φ dx
for every test function φ from W01,2(G). (All the integrals in (29) converge because of the hypothesis made on the coef�cients, and Sobolev embedding theorem.)
We proceed as in [27]. We take φ = S ◦ u, where S is the continuous function de�ned as follows. Let 0 ≤ s < t ; then S(u) = 0 if |u| < s, S(u) = sign u if |u| > t and S is linear otherwise. Plugging such φ into (29) then passing to the limit as s ↑ t results in the following inequality
−d dt
�
{x∈G:|u(x)|>t}|∇u|2dx + λ
�
{x∈G:|u(x)|>t}|u| dx + (30)
− t
�
− d dt
�
{x∈G:|u(x)|>t}
n
�
1
bi2dx
�1/2� d dt
�
{x∈G:|u(x)|>t}|∇u|2dx
�1/2
≤
≤
�
{x∈G:|u(x)|>t}| f | dx for almost every positive t .
We use the following notations:
ξ =�
−d dt
�
{x∈G:|u(x)|>t}|∇u|2dx�1/2
,
ζ = t 2
�
−d dt
�
{x∈G:|u(x)|>t}
n
�
i=1
b2i dx�1/2
and
h =
� µ(t ) 0
�
f∗(r) − λu∗(r)� dr.
Inequality (30) and Lemma 1 imply that ξ (ξ − 2ζ ) ≤ h. Hence ξ ≤ ζ +�
ζ2+ h.
If we let d = κnµ(t)1/n−1√
−µ�(t), Lemma 2 gives ξ ≥ 1/d , so
(31) 1
d − ζ ≤� ζ2+ h.
The left-hand side of (31) can be positive or not. In the former case we obtain 1 ≤ 2ζ d + hd2. In any case we have
(32) 1 ≤ max�ζ d, 2ζ d + hd2� . Thus we have proved that
1 ≤ max� κn
2 t�
− d dt
�
{x∈G:|u(x)|>t}
n
�
i=1
b2i dx�1/2
µ(t)1/n−1�
−µ�(t) , (33)
κnt�
− d dt
�
{x∈G:|u(x)|>t}
n
�
i=1
bi2dx�1/2
µ(t)1/n−1�
−µ�(t) +
+ κn2µ(t)2/n−2[−µ�(t)]
� µ(t ) 0
�f∗(r) − λu∗(r)� dr
�
for almost every t from (0, ess sup |u|).
As shown in [27], the function g de�ned by (34) g(nκns1/n) =
� d ds
�
{x∈G:|u(x)|>u∗(s)}
n
�
1
bi(x )2dx
�1/2
is such that (35)
�
−d dt
�
{x∈G:|u(x)|>t}
n
�
1
bi2dx
�1/2
=�
−µ�(t)g�nκnµ(t)1/n� for almost every t from (0, ess sup |u|), and
(36)
� nCn
� nκn|G|1/n 0
g(r)krn−1dr
�1/k
≤
��
G
�
n
�
i
bi(x )2�k/2
dx
�1/k
. Thus (33) becomes
1 ≤ [−µ�(t)] max� κn
2 tg(nκnµ(t)1/n)µ(t)1/n−1, (37)
κntg(nκnµ(t)1/n)µ(t)1/n−1− λκn2µ(t)2/n−2
� µ(t ) 0
u∗(r) dr +
+ κn2µ(t)2/n−2
� µ(t ) 0
f∗(r) dr
�
for almost every t from (0, ess sup |u|).
We claim that
−du∗
ds (s) ≤ max� κn
2s1/n−1g(nκns1/n)u∗(s), (38)
κns1/n−1g(nκns1/n)u∗(s) − λκn2s2/n−2
� s 0
u∗(r) dr +
+ κn2s2/n−2
� s 0
f∗(r) dr
�
for almost every s from (0, |G|).
Indeed, the very de�nition of u∗ ensures that u∗=
� +∞
0
χ[0,µ(t )]dt,
where χ stands for characteristic function. Hence
� |G|
0
u∗(s)φ�(s)ds =
� +∞
0
φ(µ(t))dt for every φ in C0∞�(0, |G|)�; consequently
� |G|
0
φ(s)�
−du∗ ds (s)�
ds =
� +∞
0
φ(µ(t)) dt, since u∗ is absolutely continuous (see [27]). Suppose φ ≥ 0. Then
� |G|
0
φ(s)�
−du∗ ds (s)�
ds ≤
� +∞
0
φ(µ(t)){the braces in (37)}[−µ�(t)] dt ≤
≤
� +∞
0
φ(µ(t)){...}[−dµ(t)] =
� |G|
0
φ(s){the right-hand side of (38)} ds.
Inequality (38) follows, for φ is arbitrary.
If we put
U (s) =
� s 0
u∗(t) dt, h(s) = κn2
� s 0
f∗(t) dt,
F(s, y, t) = max�κn
2 s1/n−1g(nκns1/n)t, (39)
κns1/n−1g(nκns1/n)t − λκn2s2/n−2y + s2/n−2h(s)� , (38) gives
(40) −U��(s) ≤ F�
s, U (s), U�(s)�
for almost every s belonging to (0, |G|). Observe that U satis�es also
(41) U (0) = 0, U�(|G|) = 0.
Let v be the solution to the following boundary value problem (42) � −v��(s) = F�s, v(s), v�(s)�,
v(0) = 0, v�(|G|) = 0.
We want to show that U (s) ≤ v(s) for every s between 0 and |G|.
If we let ω = v − U , we have
� −ω��(s) ≥ F�s, v(s), v�(s)� − F�s, U(s), U�(s)�, ω(0) = 0, ω�(|G|) = 0.
But F(s, · , ·) is convex, thus
F(s, v, v�) − F(s, U, U�) ≥ FU(s, U, U�)(v − U) + FU�(s, U, U�)(v�− U�), where� FU(s, · , ·), FU�(s, · , ·)� is a subgradient of F . Observe that FU ≤ 0, as (39) shows.
In other words, ω satis�es
(43) � −ω��(s) ≥ −α(s)ω�(s) − β(s)ω(s), ω(0) = 0, ω�(|G|) = 0,
where β is a nonnegative function.
Lets show that any suf�ciently smooth solution ω to problem (43) is nonnegative.
Let us consider the function z de�ned by
z(s) = −ω��(s) + α(s)ω�(s) + β(s)ω(s), which is nonnegative. If y(s) = exp�
−�s
0 α(r) dr�
, ω minimizes the following functional
J (φ) =
� |G|
0 y{(φ�)2+ βφ2− 2zφ}ds
in the following function class �φ ∈ W1,2�(0, |G|)� : φ(0) = φ�(|G|) = 0�.
Observe that J is strictly convex, and J (|φ|) ≤ J (φ) for every competing function φ. Consequently the minimizer is unique and nonnegative. We conclude that ω is nonnegative, as claimed.
A straightforward inspection shows that (11) and (12) imply
� s 0
w∗(t) dt = v(s) for every s belonging to [0, |G|].
Theorem 1 is thus completely proved. �
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Dimitri Mugnai, Dipartimento di Matematica Leonida Tonelli, Via Buonarroti 2, 56127 Pisa (ITALY), e-mail: mugnai@mail.dm.unipi.it Giorgio Talenti, Dipartimento di Matematica Ulisse Dini, Viale Morgagni 67/A, 50134 Firenze (ITALY) e-mail: talenti@udini.math.uni�.it