Layer potentials beyond singular integral operators

0.24 MB·

by  Andreas Rosén

LAYER POTENTIALS BEYOND SINGULAR INTEGRAL
OPERATORS
ANDREAS ROSE´N1
Abstract. We prove that the double layer potential operator and the gradient
of the single layer potential operator are L2 bounded for general second order
divergenceformsystems. As comparedto earlierresults,our proofis independent
2
of well posedness for the Dirichlet problem. More precisely, we use functional
1
0 calculus of differential operators with non-smooth coefficients to represent the
2 layer potential operators as abstract bounded Hilbert space operators. In the
v presenceofMoserlocalbounds,inparticularforrealscalarequationsandsystems
o that are small perturbations of real scalar equations, these abstract operators are
N shown to be the usual singular integrals. Our proof gives a new construction of
fundamental solutions to divergence form systems, valid also in dimension 2. In
1
general,the layerpotentialoperatorsareshowntodependholomorphicallyonthe

] coefficient matrix A L∞, showing uniqueness of the extension of the operators
P ∈
beyond singular integrals.
A
.
h
t
a
m
1. Introduction
[

This paper concerns the classical boundary value problems for divergence form
2
v second order elliptic systems
2
8 m
5 divAij uj = 0, i = 1,...,m,
7 ∇
. j=1
0 X
1
for a vector valued functions u = (uj)m on the upper half space R1+n := (t,x)
2 j=1 + { ∈
1 R Rn ; t > 0 , n,m 1, with boundary data in L (Rn). In general, we
2
v: onl×y assume that t}he coeffi≥cients A = (Aij)m are uniformly bounded and accre-
i,j=1
Xi tive. Unless otherwise stated, we assume that A(t,x) = A(x) is independent of the
transversal direction t. (Accretivity, or more precisely strict accretivity, is defined
r
a
in (4) below.) However, we do not assume that A is real or symmetric.
By scalar coefficients, or equation, we mean that Aij = 0 for i = j. For technical
6
reasons we consider systems where the functions uj are complex-valued, and thus
Aij(t,x) (C1+n). However, working at the level of systems of equations of arbi-
∈ L
trary size, complex coefficients are no more general than real coefficients. Indeed,
using the relation C = R2 we see that any system of equations with complex coeffi-
cients of size m can be viewed as a system of equations with real coefficients of size
2m.
2010 Mathematics Subject Classification. Primary: 31B10; Secondary: 35J08.
Key words and phrases. Double layer potential, fundamental solution, divergence form system,
functional calculus.
1
Formerly Andreas Axelsson.
Supported by Grant 621-2011-3744from the Swedish researchcouncil, VR.
1
2 ANDREASROSE´N
For an 1 + n-dimensional vector f, we let f denote the normal/vertical part
⊥
(identified with the corresponding scalar component), and write f for the tan-
k
gential/horizontal part. Similarly, we write , div and curl for the differential
∇k k k
operators acting only in the tangential/horizontal variable x. To ease notation, we
use the Einstein summation convention throughout this paper. Sometimes we shall
even suppress indices i,j.
A classical method for solving the Dirichlet problem is to solve the associated
double layer potential equation on the boundary Rn. In our framework, the method
is the following. Let Γ = (Γij )m be the fundamental solution for divA∗
(t,x) (t,x) i,j=1 ∇
δ , i = k,
in R1+n with pole at (t,x), that is div(Aji)∗ Γjk = (t,x) , and let
∇ (t,x) 0, i = k
(
6
∂ Γij := ((Aki)∗ Γkj ) denote its (inward) conormal derivative.
νA∗ (t,x) ∇ (t,x) ⊥
Given a function h : Rn Cm on the boundary, define the function
→
hi(x) := ∂ Γji (0,y),hj(y) dy, (t,x) R1+n,
Dt − νA∗ (t,x) ∈ +
ZRn
(cid:0) (cid:1)
where ∂ is the outward conormal derivative. The function u(t,x) := h(x)
− νA∗ Dt
then solves the equation divA u = 0 in R1+n, and has boundary trace
∇ +
hi(x) := lim ∂ Γji (0,y),hj(y) dy.
D t→0+ZRn − νA∗ (t,x)
(cid:0) (cid:1)
Finding the solution u with Dirichlet data ϕ : Rn Cm on the boundary, then
→
amounts to solving the double layer equation
h = ϕ
D
for h, which then gives the solution u(t,x) = h(x). In the case of smooth coef-
t
D
ficients A, it is well known that the operator is well defined and is 1I plus an
D 2
integral operator. For general systems with non-smooth coefficients, as considered
in this paper, the double layer potential operator is beyond the scope of singular
D
integral theory.
Similarly, the single layer potential is used to solve the Neumann problem. See
Section 7. In this introduction, we focus on the double layer potential and the
Dirichlet problem.
During the last years, new results on boundary value problems for more general
non-smoothdivergence formsystems havebeenproved. Inparticular, therehasbeen
two seemingly different developments, one based on singular integrals (S) and one
based on functional calculus (F). The purpose of this paper is to demonstrate that
the singular integral operators used in (S) actually are special cases of the abstract
operators used in (F).
(S) In the paper [1] by Alfonseca, Auscher, Axelsson, Hofmann and Kim, it was
proved in particular that boundedness and invertibility of the layer potential
operators for coefficients A implies boundedness and invertibility of the
0
layer potential operators for coefficients A whenever A A is small,
0 ∞
k − k
depending on A . Here A and A are assumed to be scalar and complex,
0 0
and such that De Giorgi–Nash local H¨older estimates hold for solutions to
these equations. Boundedness here includes square function estimates. This
boundedness and invertibility result was shown to hold for real symmetric
LAYER POTENTIALS BEYOND SIOS 3
coefficients, and the result was also known for coefficients of block form and
for constant coefficients.
Duringthewritingofthispaper,Hofmann, Kenig,MayborodaandPipher[6]
have proved L well posedness, for some p < depending on A, of the
p
∞
Dirichlet problem for general scalar equations with real and t-independent
coefficients. From this they deduce, in [6, Cor. 1.25], boundedness in L (but
2
not invertibility) of the layer potentials for general real scalar equations and
small complex perturbations of such, by inspection of the proofs in [1].
(F) Auscher, Axelsson and McIntosh [3] proved that the L Dirichlet (and Neu-
2
mann) problem is well posed for systems with coefficients A which are small
L perturbations of Hermitean, constant or block form coefficients. Instead
∞
of the double layer potential operator above, this used an operator on
D D
L (Rn) defined by functional calculus from an underlying differential oper-
2
ator on Rn. More precisely, this used a self-adjoint first order differeential
operator D and a transformed multiplication operator B formed from the
coefficients A, to construct a solution
ui(t,x) = hi(x) := (b (BD)h)i (x), (t,x) R1+n,
Dt t ⊥ ∈ +
e e−tz, Rez > 0,
where the function b (z) := is applied to the operatorBD.
t
0, Rez < 0,
(
Here BD, and therefore b (BD), acts on Cm(1+n)-valued functions h on Rn.
t
Bothworks[1,3]buildonharmonicanalysisdeveloped forthesolutionoftheKato
square root problem by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian [4].
However, the approach (F) is more general. On the one hand (S) uses De Giorgi–
Nash local H¨older estimates, which holds for real scalar equations, and small L
∞
perturbations of such, but not for general A. On the other hand, (F) proves that
t
D
in fact is L -bounded for any t-independent and uniformly bounded and accretive
2
coefficients A; it is only invertibility of := lim which may fail. e
t→0+ t
D D
Unlike , the definition of the double layer potential operator require the ex-
D D
istence of a fundamental solution to dievA∗ . For deivergence form systems, such
∇
fundamenteal solutions were constructed by Hofmann and Kim [7] under the hypoth-
esis that solutions to divA u = 0 and divA∗ u = 0 satisfy De Giorgi–Nash local
∇ ∇
H¨older estimates. That solutions to divA u = 0 satisfy such estimates means that
∇
u(y) u(z) 1/2
(1) ess sup | − | . R−α−(1+n)/2 u 2
y,z∈B(x;R),y6=z y z α | |
| − | (cid:18)ZB(x;2R) (cid:19)
holds whenever u is a weak solution to divA u = 0 in B(x;2R) R1+n, for some
∇ ⊂
α > 0. It is known that (1) is equivalent to the gradient estimate
(2) u 2 . (r/R)n−1+2µ u 2, 0 < r < R,
|∇ | |∇ |
ZB(x,r) ZB(x,R)
for all weak solutions u to divA u = 0 in B(x;R) R1+n, for some µ > 0.
∇ ⊂
It is known that (1), or equivalently (2), holds for all divergence form systems
divA u = 0 where A is real and scalar, and small L -perturbations of such (t-
∞
∇
independence of A is not needed here). Estimates (1) and (2) also imply the Moser
4 ANDREASROSE´N
local boundedness estimate
1/2
(3) ess sup u(y) . R−(1+n)/2 u 2
y∈B(x;R)| | | |
(cid:18)ZB(x;2R) (cid:19)
whenever divA u = 0 in B(x;2R) R1+n. We refer to [7, Sec. 2] for further
∇ ⊂
explanation of these results.
At the8thInternational Conference onHarmonicAnalysis andPartial Differential
Equations at El Escorial 2008, S. Hofmann formulated as an open problem whether
(F) as a special case implies the result (S). Our main result in this paper is that this
is indeed the case, as = whenever is defined. More precisely, we prove the
D D D
following.
e
Theorem 1.1. Let n,m 1, and let A = A(x) L (Rn; (Cm(1+n))) be t-
∞
≥ ∈ L
independent and accretive in the sense that there exists κ > 0 such that
(4) Re (Aij(x)fj(x),fi(x))dx κ f(x) 2dx,
≥ | |
ZRn ZRn
for all f L (Rn;Cm(1+n)) such that curl f = 0.
∈ 2 k k
Assume that whenever u is a weak solution to divA u = 0 in a ball B(x;2R), u
∇
is almost everywhere equal to a continuous function and the Moser local boundedness
estimate (3) holds. Thenthere existsa fundamentalsolution Γ W1 (R1+n;Cm2)
(t,x) ∈ 1,loc
to divA∗ with estimates
∇
Γ (s,y) 2dy . (R+ s t )−n,
(t,x)
|∇ | | − |
Z|y−x|>R
for all R > 0, t,s R and x Rn. Moreover
∈ ∈
(5) ∂ Γji (0,y),hj(y) dy = (b (BD)h)i (x)
− νA∗ (t,x) t ⊥
ZRn
holds for almost all ((cid:0)t,x) R1+n and all sc(cid:1)alar functions h L (Rn;Cm). The
∈ + ∈ 2
right hand side is defined in Section 2. In particular, we here identify h with a
normal vector field h L (Rn;Cm(1+n)).
2
∈
This theorem allows us to transfer known results for the double layer potential
operator
Ahi = hi := (b (BD)h)i , t > 0,
Dt Dt t ⊥
defined through functional calculus, to the double layer potential operator
e e
Ahi = hi := ∂ Γji (0,y),hj(y) dy, t > 0,
Dt Dt − νA∗ (t,·)
ZRn
defined classically as an integral(cid:0)operator. The followi(cid:1)ng is a list of such known
results for A, which therefore also hold for A = A under the hypothesis of
Dt Dt Dt
Theorem 1.1. These results for A follow by inspection of the proof of [3, Thm. 2.3
Dt
and 2.2], ande extends the results for A from [1, 6]. e
Dt
We have estimates e
•
∞
sup h 2 + ∂ h 2tdt+ N ( h) 2 . h 2,
kDt k2 k tDt k2 k ∗ Dt k2 k k
t>0 Z0
for anysystem with bounded andaccretive coefficients A, where themodified
e e e e
non-tangential maximal function N is defined in Section 2. In presence of
∗
e
LAYER POTENTIALS BEYOND SIOS 5
Moser local boundedness estimates of solutions, N can be replaced by the
∗
usual point wise non-tangential maximal function.
For any system with bounded and accretive coeffiecients A, the operators
t
• D
converge strongly in L , that is there exists anL (Rn;Cm) bounded operator
2 2
such that e
D
lim h h = 0, for all h L (Rn;Cm).
t 2 2
e t→0+kD −D k ∈
The map
e e
•
accretive A L (Rn; (Cm(1+n))) A A (L (Rn;Cm))
∞ 2
{ ∈ L } ∋ 7→ D ∈ L
isaholomorphicmapbetweenBanachspaces. Inparticular, A (L (Rn;Cm))
e D ∈ L 2
depends locally Lipschitz continuously on A L (Rn; (Cm(1+n))), and
∞
∈ L
therefore invertibility of A is stable under small L perturebations of A.
∞
D
Theoperator A (L (Rn;Cm))isinvertiblewhenAishermitean, (Aij)∗ =
2
• D ∈ L
Aji, when A is constant, Ae(x) = A, and when A is of block form, Aij = 0 =
⊥k
Aij . e
k⊥
It is also known that A is not invertible for many A, even for real and scalar
• D
(but non-symmetric) coefficients A in the plane, n = 1. A counter example
was found in [8, Thme3.2.1] among the coefficients
1 ksgn(x)
A(x) = .
ksgn(x) 1
(cid:20)− (cid:21)
Note that A = A for all these coefficients by Theorem 1.1. It was shown
D D
in [5] that A is not invertible for these coefficients when k = 1. Moreover,
D
from[5] and[3, Reem. 5.4] it follows that A is invertible for these coefficients
D
when k = 1e, but that the coefficients with k > 1 are disconnected from the
6
identity A = I by the set of coefficients feor which A is not invertible.
D
In the process of proving Theorem 1.1, we also give a new construction of funda-
mental solutions to divergence form systems. As comparedeto [7], this works also in
dimension 2 and it does not involve any limiting argument but constructs the fun-
damental solution directly using functional calculus. Extending this construction to
t-dependent coefficients, we prove the following result. Note that we formulate this
result in dimension n, not 1+n.
Theorem 1.2. Let n 2, m 1. Assume that A L (Rn; (Rn)) are real
0 ∞
≥ ≥ ∈ L
and scalar coefficients, identified with a matrix acting component-wise on f Cmn,
∈
which are accretive in the sense that there exists κ > 0 such that
Re(A (x)f,f) κ f 2, for all f Cmn,x Rn.
0
≥ | | ∈ ∈
Then there exists ǫ > 0 such that whenever A L (Rn; (Cmn)) is such that
∞
∈ L
ess sup A(x) A (x) < ǫ, then there exists a fundamental solution Γ to divA ,
x∈Rn| − 0 | x ∇
δ , i = k,
i.e. a function Γ W1 (Rn;Cm2) such that divAij Γjk = x in dis-
x ∈ 1,loc ∇ x 0, i = k,
(
6
tributional sense, with estimates
(6) Γ (y) 2dy . R2−n,
x
|∇ |
ZR<|y−x|<2R
6 ANDREASROSE´N
for all R > 0 and x Rn.
∈
The main difference between our construction here and the construction in [7], is
that we construct the gradient Γ instead of the potential Γ . An advantage of
x x
∇
this approach is that it works in dimension 2 as well. From the gradient estimate
(6), we deduce pointwise estimates of Γ in Section 6. This section also contains
x
the proof of Theorem 1.2, which builds on the proof of Theorem 1.1, which is in
Section 5. Sections 2, 3 and 4 contains the details of the construction of the funda-
mental solution for t-independent coefficients, which uses the Green’s formula from
Definition 3.1. Half of this identity yields the representation formula (5) for the dou-
ble layer potential operator. By a duality argument we also derive corresponding
results for the gradient of the single layer potential operator in Section 7 .
2. Functional calculus for divergence form equations
In this section, we explain the method of functional calculus (F) for the L Dirich-
2
let problem for the equation divA u = 0 in R1+n. We assume in this section that
∇ +
the coefficients A L (R1+n; (Cm(1+n))) are t-independent and accretive in the
∞
∈ L
sense of (4).
Recall from complex analysis the following two relations between harmonic func-
tions and analytic functions in C = R2: (a) u is harmonic if and only if f = u is
∇
anti-analytic, that is divergence- and curl-free, and (b) u is harmonic if and only if
there exists an analytic function v with Rev = u. In this section, we generalize this
result to solutions to divA u = 0 in R1+n, following [3, 2]. Following the notation
∇
from these papers, we shall suppress indices i,j = 1,...,m in this section.
a
(a) If divA u = 0, write f = [f ,f ]t := [∂ u, u]t, where [a,b]t := .
∇ ⊥ k νA ∇k b
(cid:20) (cid:21)
Decomposing the matrix A as
A (x) A (x)
A(x) = ⊥⊥ ⊥k ,
A (x) A (x)
(cid:20) k⊥ kk (cid:21)
we have the conormal derivative ∂ u := A ∂ u + A u, or inversely ∂ u =
νA ⊥⊥ t ⊥k∇k t
A−1(f A f ). In terms of f, the equation for u becomes
⊥⊥ ⊥ − ⊥k k
∂ f +div A A−1(f A f )+A f ) = 0.
t ⊥ k k⊥ ⊥⊥ ⊥ − ⊥k k kk k
(cid:16) (cid:17)
The condition that f is the conormal gradient of a function u, determined up to
constants, can be expressed as the curl-free condition
∂ f = A−1(f A f ) ,
t k ∇k ⊥⊥ ⊥ − ⊥k k
(curl f = 0(cid:16). (cid:17)
k k
In vector notation, we equivalently have
f 0 div A−1 A−1A f
∂ ⊥ + k ⊥⊥ − ⊥⊥ ⊥k ⊥ = 0,
t f 0 A A−1 A A A−1A f
(cid:20) k(cid:21) (cid:20)−∇k (cid:21)(cid:20) k⊥ ⊥⊥ kk − k⊥ ⊥⊥ ⊥k(cid:21)(cid:20) k(cid:21)
together with the constraint curl f = 0. Define
k k
0 div A−1 A−1A
D := k and B := ⊥⊥ − ⊥⊥ ⊥k ,
0 A A−1 A A A−1A
(cid:20)−∇k (cid:21) (cid:20) k⊥ ⊥⊥ kk − k⊥ ⊥⊥ ⊥k(cid:21)
LAYER POTENTIALS BEYOND SIOS 7
so that the equation becomes
(7) ∂ f +DBf = 0
t
together with the constraint f := R(D) = f L ; curl f = 0 for each fixed
t > 0. (Here and below, R( ) a∈ndHN( ) denote r{ang∈e an2d null kspkace o}f an operator.)
· ·
This equation for f, which is an L (Rn;Cm(1+n)) vector-valued ODE in t, can be
2
viewed as a generalized Cauchy–Riemann system.
Definition 2.1. The conormal gradient of u is the vector field
∂ u
u := νA ,
∇A u
(cid:20)∇k (cid:21)
where ∂ u = (A u) is the (inward relative R1+n) conormal derivative.
νA ∇ ⊥ +
(b) Another Cauchy–Riemann type system related to divA u = 0 is
∇
∂ v +BDv = 0,
t
where D and B have been swapped. Applying D to this equation yields (∂ +
t
DB)(Dv) = 0, so
f := Dv = [div v , v ]
k k −∇k ⊥
is the conormal gradient of a solution u to divA u = 0. Looking at f , we see that
∇ k
we should set
u := v .
− ⊥
Then u = f . Moreover
∇k k
∂ u = ∂ v = (BDv) = (Bf) = A−1(f A u),
t − t ⊥ ⊥ ⊥ ⊥⊥ ⊥ − ⊥k∇k
so that ∂ u = f . Thus the equation
νA ⊥
(8) ∂ v +BDv = 0
t
for v = [ u,v ]t implies that u solves divA u = 0. The vector-valued function v
− k ∇ k
can be viewed as as a set of generalized conjugate functions to u.
Definition 2.2. A conjugate system for u is a vector field v solving ∂ v+BDv = 0
t
such that
u = v .
− ⊥
We now consider the closed and unbounded operators DB and BD in the Hilbert
space L = L (Rn;Cm(1+n)). Here D is a non-injective (if n 2) self-adjoint
2 2
≥
operator with
R(D) = and N(D) = ⊥,
H H
whereas B is a bounded and accretive multiplication operator just like A. Indeed,
in [3] it was noted that the transform
A A A−1 A−1A
A = ⊥⊥ ⊥k Aˆ := ⊥⊥ − ⊥⊥ ⊥k
A A 7→ A A−1 A A A−1A
(cid:20) k⊥ kk(cid:21) (cid:20) k⊥ ⊥⊥ kk − k⊥ ⊥⊥ ⊥k(cid:21)
has the following properties.
ˆ
(i) If A is accretive, then so is A.
(ii) If Aˆ = B, then B = A.
I 0
(iii) If Aˆ = B, then A∗ = NB∗N, where N := − is the reflection operator
b 0 I
(cid:20) (cid:21)
for vectors across Rn.
c
8 ANDREASROSE´N
As B is bounded and accretive, we have
ω := sup arg(Bf,f)] < π/2.
|
f∈H\{0}
The operators DB and BD both have spectrum contained in the double sector
S 0 S ,
ω− ω+
∪{ }∪
where S = λ C 0 ; argλ ω and S := S . There are decom-
ω+ ω− ω+
{ ∈ \ { } | | ≤ } −
positions of L into closed complementary (but in general non-orthogonal) spectral
2
subspaces associated with these three parts of the spectrum. For DB we have
L = E−L B−1 ⊥ E+L
2 A 2 ⊕ H ⊕ A 2
and for BD we have
L = E−L ⊥ E+L .
2 A 2 ⊕H ⊕ A 2
Note that for DB we have R(DB) = = E−L E+L and N(DB) = B−1 ⊥,
whereas for BD we have R(BD)e= BH =AE−e2L⊕ AE+2L and N(BD) = H⊥.
H A 2 ⊕ A 2 H
The proof of the fact that the the projections E± and E± associated with these
A A
splittings are bounded uses harmonic analysis freom the seolution of the Kato square
root problem. e
Important in this paper are the following intertwining and duality relations.
Proposition 2.3. We have well-defined isomorphisms
B : E±L E±L ,
A 2 → A 2
and closed and injective maps with dense domain and range
e
D : E±L E±L .
A 2 → A 2
We also have a duality
(eE∓ ,NE±),
A∗ A
that is the map E∓ L (E±L )∗, mapping g E∓ L to the functional E±L
f (g,Nf) C,Ai∗s a2n→isomoArp2hism. e ∈ A∗ 2 A 2 ∋
7→ ∈
e e
Proof. The intertwining by B is a consequence of associativity B(DB) = (BD)B,
the intertwining by D is a consequence of associativity D(BD) = (DB)D, and the
duality is a consequence of the duality
(g,N(BD)f) = (g, (NBN)DNf) = (( DA∗)g,Nf).
− −
(cid:3)
c
To solve Equation (7) for f , we note that DB restricts to an operator in
∈ H
E±L with spectrum
A 2
σ(DB ) S .
|EA±L2 ⊂ ω±
Thus e−tDBf is well defined for f E+L if t 0 and for f E−L if t 0.
∈ A 2 ≥ ∈ A 2 ≤
The following result was proved in [2]. Here the modified non-tangential maximal
function of a function f in R1+n is the function N f on Rn defined by
± ∗
N f(x) := sup t −(1+n)/2 f , x Rn,
∗ | | k kL2(We(t,x)) ∈
±t>0
where the Whitneey regions are W(t,x) := {(s,y) ; c−01 < s/t < c0,|y −x| < c1|t|},
for some fixed constants c > 1,c > 0. Also, here and below, we write f (x) :=
0 1 t
f(t,x).
LAYER POTENTIALS BEYOND SIOS 9
Proposition 2.4. Let f E±L and define
0 ∈ A 2
f(t,x) := (e−tDBf )(x), t > 0,x Rn.
0
± ∈
Then
(i) f = [∂ u, u]t for a weak solution u to divA u = 0 in R1+n, unique up
νA ∇x ∇ ±
to constants,
(ii) (0, ) t f L is continuous, with lim f = f and lim f =
t 2 t→0± t 0 t→±∞ t
± ∞ ∋ 7→ ∈
0 in L sense, and
2
(iii) we have estimates
f 2 sup f 2 ∂ f(t,x) 2tdtdx N (f) 2dx.
k 0k2 ≈ ±t>0k tk2 ≈ ZZR1±+n| t | ≈ ZRn| ∗ |
Conversely, if u is any weak solution to divA u = 0 in Re1+n, with estimate
∇ ±
N (f) < , or sup f < , of the conormal gradient f = [∂ u, u],
k ∗ k2 ∞ ±t>0k tk2 ∞ νA ∇x
then there exists f E±L such that f(t,x) = (e−tDBf )(x) almost everywhere in
0 ∈ A 2 0
Re1+n.
±
Similar results apply to Equation (8). The following result was proved in [2].
Proposition 2.5. Let v E±L and define
0 ∈ A 2
v(t,x) := (e−tBDv )(x), t > 0,x Rn.
0
e ± ∈
Then
(i) u := v is a weak solution u to divA u = 0 in R1+n,
− ⊥ ∇ ±
(ii) (0, ) t v L is continuous, with lim v = v and lim v =
t 2 t→0± t 0 t→±∞ t
± ∞ ∋ 7→ ∈
0 in L sense, and
2
(iii) we have estimates
v 2 sup v 2 u(t,x) 2tdtdx N (v) 2dx.
k 0k2 ≈ ±t>0k tk2 ≈ ZZR1±+n|∇ | ≈ ZRn| ∗ |
Conversely, if u is any weak solution to divA u = 0 in Re1+n, with estimate
∇ ±
u(t,x) 2tdtdx < , then there exists v E±L and a constant c Cm
R1±+n|∇ | ∞ 0 ∈ A 2 ∈
such that u(t,x) = (e−tBDv ) (x)+c almost everywhere in R1+n.
RR − 0 ⊥ e ±
3. Green’s formula on the half space
Recall that for the Laplace operator, that is the special case A = I and m = 1,
we have the fundamental solution
−1 t2 + x 2 −(n−1)/2, n 2,
Φ(t,x) = (n−1)σn | | ≥
(21π ln√(cid:0)t2 +x2, (cid:1) n = 1,
with pole (0,0), where σ denotes the area of the unit sphere in R1+n. We note that
n
1 (t,x)
Φ(t,x) = ,
∇ σ (t2 + x 2)(n+1)/2
n
| |
for n 1.
≥
In this section, we construct a fundamental solution to more general divergence
form operators divA using functional calculus. We assume in Sections 3, 4 and 5
∇
that the coefficients A L (R1+n; (Cm(1+n))) are t-independent, accretive in the
∞
∈ L
10 ANDREASROSE´N
sense of (4) and that solutions to divA u = 0 satisfy the Moser local boundedness
∇
estimate (3).
To explain the definition, we start with the following formal calculation. Assume
that Γ = (Γij (t,x))m is a fundamental solution to divA∗ , that is
(t0,x0) i,j=1 ∇
δ , i = j,
div(Aki)∗ Γkj = (t0,x0) .
∇ (t0,x0) 0, i = j
(
6
Assume that t > 0 and that u solves divA u = 0 in R1+n. With appropriate
0 ∇ +
estimates of u and Γ∗, Green’s formula shows that
ui(t ,x ) = Γji (0,x),∂ uj(0,x) ∂ Γji (0,x),uj(0,x) dx,
0 0 (t0,x0) νA − νA∗ (t0,x0)
ZRn
(cid:16) (cid:17)
where the conormal(cid:0)derivative is ∂ uj = (A(cid:1)jk (cid:0)uk) . Now let v be a c(cid:1)onjugate
νA ∇ ⊥
system for u so that uj = vj and ∂ uj = div vj. Then by integration by parts,
− ⊥ νA k k
we obtain
ui(t ,x ) = Γji (0,x),Nvj(0,x) dx,
0 0 − ∇A∗ (t0,x0)
ZRn
where the conormal gradient is (cid:0)Γji = [∂ Γji, Γji]t. Mor(cid:1)e generally, it follows
∇A∗ νA∗ ∇k
in this way that if v E+L , then
0 ∈ A 2
(e−t0BDv )i (x ), t > 0,
Γji e(0,x),Nvj(x) dx = 0 ⊥ 0 0
∇A∗ (t0,x0) 0 0, t < 0,
ZRn ( 0
(cid:0) (cid:1)
and if v E−L , then
0 ∈ A 2
0, t > 0,
e Γji (0,x),Nvj(x) dx = 0
ZRn ∇A∗ (t0,x0) 0 (−(e−t0BDv0)i⊥(x0), t0 < 0.
(cid:0) (cid:1)
We now reverse this argument, taking these four formulae as definition. From the
Moser local boundedness estimate (3), it follows that
u(t ,x ) . N u(x) for x x < c t /2.
0 0 ∗ 0 1 0
| | | − | | |
Thus
(e−t0BeDv ) (x ) . t −n/2 v ,
| 0 ⊥ 0 | | 0| k 0k2
uniformly for v E±L and t > 0. Proposition 2.4 and the duality from Propo-
0 ∈ A 2 ± 0
sition 2.3 enable us to make the following construction.
Definition 3.1. Foer (t ,x ) R1+n and i = 1,...,m, let Γi = (Γji ) be
0 0 ∈ + (t0,x0) (t0,x0) j
the, unique up to constants, weak solution to divA∗ Γi = 0 in R1+n such that
∇ (t0,x0) −
Γji (0,x),Nvj(x) dx = (e−t0BDv )i (x ),
∇A∗ (t0,x0) 0 0 ⊥ 0
ZRn
(cid:0) (cid:1)
for all v E+L .
0 ∈ A 2
For (t ,x ) R1+n and i = 1,...,m, let Γi = (Γji ) be the, unique up to
0 0 ∈ − (t0,x0) (t0,x0) j
constants, weeak solution to divA∗ Γi = 0 in R1+n such that
∇ (t0,x0) +
Γji (0,x),Nvj(x) dx = (e−t0BDv )i (x ),
∇A∗ (t0,x0) 0 − 0 ⊥ 0
ZRn
(cid:0) (cid:1)
for all v E−L .
0 ∈ A 2
e