1 Introduction and Main Results
Let Ω be a bounded connected open subset of ℝN with smooth boundary and let V∈L∞(Ω) be a nonnegative function. The weak maximum principle ensures that the distributional solution u∈C1(Ω¯) of the Dirichlet problem
(1.1)
{
-
Δ
u
+
V
u
=
f
in
Ω
,
u
=
0
on
∂
Ω
,
satisfies u≥0 on Ω whenever f∈L∞(Ω) is a nonnegative function; see Lemma 2.2 below. From the minimality of u on ∂Ω, the normal derivative of u with respect to the inward unit normal vector n thus verifies ∂u∂n≥0 on ∂Ω. When f≢0, the classical Hopf lemma (see [10, Lemma 6.4.2] or [11, Lemma 3.4]) gives the stronger conclusion
(1.2)
∂
u
∂
n
>
0
on
∂
Ω
.
Boundedness of V is an important element to obtain (1.2) as it allows one to construct a positive minorant of u on Ω with positive normal derivative at any given point on ∂Ω. To understand in what respect this assumption on V can be relaxed, we assume henceforth that
V
∈
L
loc
1
(
Ω
)
and
V
≥
0
almost everywhere on
Ω
,
but we restrict ourselves to the class of nonnegative data f∈L∞(Ω). In this setting, a solution of (1.1) is a function u that belongs to W01,2(Ω)∩L2(Ω;Vdx) and satisfies the equation
-
Δ
u
+
V
u
=
f
in the sense of distributions in
Ω
.
Observe that u is the unique minimizer of the energy functional
E
(
z
)
=
1
2
∫
Ω
(
|
∇
z
|
2
+
V
z
2
)
d
x
-
∫
Ω
f
z
d
x
with z∈W01,2(Ω)∩L2(Ω;Vdx).
As the solution of (1.1) need not be C1, nor even continuous, due to some possible singularity from V, we first need to address the pointwise meaning of the normal derivative ∂u∂n. Since u is the difference between a continuous and a bounded superharmonic function, every x∈Ω is a Lebesgue point and the precise representative of u satisfies the following representation formula in terms of the Green function G of -Δ on Ω:
u
^
(
x
)
=
∫
Ω
G
(
x
,
y
)
(
-
Δ
u
(
y
)
)
d
y
for every
x
∈
Ω
.
Then, from a formal computation, one presumably gets at a point a∈∂Ω:
(1.3)
∂
u
^
∂
n
(
a
)
=
∫
Ω
K
(
a
,
y
)
(
-
Δ
u
(
y
)
)
d
y
,
where K:=∂G∂n denotes the Poisson kernel of -Δ on Ω. This formula can be rigorously justified when V∈L∞(Ω), and then Δu∈L∞(Ω), using standard estimates on G.
There is no reason why (1.3) should remain valid in general as we do not assume any particular behavior of V near ∂Ω. We show nevertheless that, for any fixed V, there is a common property which is shared by all nontrivial solutions of (1.1) with nonnegative f∈L∞(Ω). To this end, let ζ1 be the solution of (1.1) with constant density f≡1 and define the set
𝒩
=
{
a
∈
∂
Ω
:
the classical normal derivative
∂
ζ
1
^
∂
n
exists at
a
and (1.3) is valid with
u
=
ζ
1
}
.
To simplify the notation, we do not explicit the dependence of 𝒩 on V.
We prove:
Theorem 1.
For every nonnegative function f∈L∞(Ω), f≢0, the solution u of (1.1) involving f has a classical normal derivative at a∈∂Ω that satisfies (1.3) if and only if a∈N.
The set 𝒩 thus provides one with a common ground where a normal derivative exists, independently of the solution of (1.1). We can now address the question of whether the Hopf lemma is valid on 𝒩. We rely on the characterization of the set of points a∈∂Ω for which the boundary value problem
(1.4)
{
-
Δ
v
+
V
v
=
0
in
Ω
,
v
=
δ
a
on
∂
Ω
,
involving the Dirac measure δa has a distributional solution in the sense that the function v∈L1(Ω) is such that Vv∈L1(Ω;d∂Ωdx) and satisfies
(1.5)
∂
ζ
∂
n
(
a
)
=
∫
Ω
v
(
-
Δ
ζ
+
V
ζ
)
d
x
for every ζ∈C∞(Ω¯) with ζ=0 on ∂Ω, where d∂Ω:Ω→ℝ+ is the distance to the boundary. When the test function ζ is non-identically zero and satisfies -Δζ+Vζ≥0 on Ω, it follows from (1.5) and the strong maximum principle for the Schrödinger operator with potential in Lloc1 (see [1, Théorème 9], [5, Theorem 1] or [19]) that ∂ζ(a)∂n>0. It is therefore reasonable to expect the validity of the Hopf lemma for (1.1) on the set of points a∈∂Ω for which the boundary value problem (1.4) has a solution. This motivates the following:
Definition 1.1.
The exceptional boundary set Σ associated to -Δ+V is the set of points a∈∂Ω for which the boundary value problem (1.4) with datum δa does not have a distributional solution.
We can now state the Hopf lemma on 𝒩:
Theorem 2.
Let u be the solution of (1.1) for some nonnegative datum f∈L∞(Ω), f≢0. For every a∈N, we have
∂
u
^
∂
n
(
a
)
>
0
if and only if
a
∉
Σ
.
In the case where V∈Lq(Ω) for some q>N, one has 𝒩=∂Ω and Σ=∅. Hence,
∂
u
^
∂
n
(
a
)
>
0
for every
a
∈
∂
Ω
;
see Corollary 7.2 below. As one can expect, the validity of the Hopf lemma depends on the behavior of V near the boundary. For example, under the assumption that
(1.6)
V
≤
C
d
∂
Ω
2
almost everywhere on
Ω
for some constant C≥0, Ancona established in [2] (see also the Appendix in [20]) a beautiful characterization of the set of points where (1.4) has a solution: a∈∂Ω∖Σ if and only if the Poisson kernel of -Δ at a is a supersolution of (1.1). Using his result and the pointwise behavior of K, we can state the following:
Corollary 1.1.
Assume that V satisfies (1.6) and let u be the solution of (1.1) for some nonnegative datum f∈L∞(Ω), f≢0. For every a∈N, we have
∂
u
^
∂
n
(
a
)
>
0
if and only if
∫
Ω
d
∂
Ω
2
(
y
)
|
y
-
a
|
N
V
(
y
)
d
y
<
+
∞
.
Quadratic blow-up of the potential as in (1.6) is a threshold for the validity of the Hopf lemma. More precisely:
Corollary 1.2.
Assume that V satisfies
V
≥
C
′
d
∂
Ω
2
almost everywhere on
Ω
for some C′>0. Then N=∂Ω and, for every solution u of (1.1) with f∈L∞(Ω), we have
∂
u
^
∂
n
=
0
on
∂
Ω
.
Corollary 1.2 is a consequence of our Theorem 1 and a result from Díaz [8] which establishes the existence of a bounded nonnegative eigenfunction u for -Δ+V that satisfies a Dirichlet problem of the type (1.1) and such that ∂u^(a)∂n=0 for every a∈∂Ω.
Although we have introduced the exceptional set Σ by dealing with Dirac masses on ∂Ω, the set Σ allows one to characterize all nonnegative finite Borel measures ν on ∂Ω for which the boundary value problem
(1.7)
{
-
Δ
v
+
V
v
=
0
in
Ω
,
v
=
ν
on
∂
Ω
,
has a distributional solution. This is the content of our next theorem that extends a previous result by Véron and Yarur [20]:
Theorem 3.
The boundary value problem (1.7) associated to a nonnegative finite Borel measure ν on ∂Ω has a distributional solution if and only if ν(Σ)=0.
The proof of Theorem 3 is inspired by the recent paper of Orsina and the first author [17] concerning the failure of the strong maximum principle for the Schrödinger operator -Δ+V in the case where V is merely a nonnegative Borel measurable function. In this respect, we introduce in Section 2 a notion of pointwise normal derivative for solutions of (1.1) that is defined everywhere on ∂Ω, but possibly depends on the potential V. In Section 3, we present a counterpart for (1.7) of the notion of duality solution introduced by Malusa and Orsina [13]. The exceptional boundary set Σ is then identified in Section 4 with the set of boundary points at which all such normal derivatives vanish. Using the tools developed in Sections 3 and 4, we prove Theorem 3 in Section 5. Theorems 1 and 2 are proved in Sections 6 and 7, respectively.
2 Pointwise Normal Derivative Associated to the Schrödinger Operator
A property that is common to all solutions of (1.1) concerns the existence of a distributional normal derivative as an element in L1(∂Ω). This is a general feature that relies on the facts that
u
∈
W
0
1
,
1
(
Ω
)
and
Δ
u
is a finite Borel measure on
Ω
.
Brezis and the first author proved in [7] that in this general setting there exists a function in L1(∂Ω), which is denoted by ∂u∂n and coincides with the classical normal derivative when u is a C2 function, that satisfies
∥
∂
u
∂
n
∥
L
1
(
∂
Ω
)
≤
|
Δ
u
|
(
Ω
)
and
(2.1)
∫
Ω
∇
u
⋅
∇
ψ
d
x
=
-
∫
Ω
ψ
Δ
u
-
∫
∂
Ω
∂
u
∂
n
ψ
d
σ
for every
ψ
∈
C
∞
(
Ω
¯
)
,
where σ=ℋN-1⌊∂Ω is the surface measure on ∂Ω; see [7, Theorem 1.2] or [18, Proposition 7.3]. We recall that n is the inward unit normal vector, which explains the minus sign in front of the second integral in the right-hand side of (2.1). When u≥0 almost everywhere on Ω, one additionally has
∂
u
∂
n
≥
0
almost everywhere on
∂
Ω
;
see [7, Corollary 6.1] or [18, Lemma 12.15]. Since the mapping
{
u
∈
W
0
1
,
1
(
Ω
)
:
Δ
u
∈
L
1
(
Ω
)
}
→
L
1
(
∂
Ω
)
,
u
↦
∂
u
∂
n
is linear, such a property yields a handy comparison principle: if v and w both satisfy (1.1), with possibly different potentials V and data f, and if v≤w almost everywhere on Ω, then
∂
v
∂
n
≤
∂
w
∂
n
almost everywhere on
∂
Ω
.
More specific to solutions of (1.1), we show that there is a notion of pointwise normal derivative that is adapted to the Schrödinger operator -Δ+V and used in the proofs of Theorems 1, 2 and 3. For this purpose, let (Vk) be a nondecreasing sequence of nonnegative functions in L∞(Ω) that converges almost everywhere to V on Ω. The construction of this pointwise normal derivative relies on the main result of this section which is the following:
Proposition 2.1.
Let u be the solution of (1.1) associated to f∈L∞(Ω) and denote by uk the solution of
(2.2)
{
-
Δ
u
k
+
V
k
u
k
=
f
in
Ω
,
u
k
=
0
on
∂
Ω
.
Then:
u
k
→
u
and
V
k
u
k
→
V
u
in
L
1
(
Ω
)
and almost everywhere on Ω,
(
∂
u
k
∂
n
)
is uniformly bounded on
∂
Ω
,
(
∂
u
k
∂
n
)
converges pointwise to a function
g
:
∂
Ω
→
ℝ
such that
g
=
∂
u
∂
n
almost everywhere on
∂
Ω
and, for every
N
<
p
≤
∞
,
∥
g
∥
L
∞
(
∂
Ω
)
≤
C
∥
f
∥
L
p
(
Ω
)
with a constant
C
>
0
depending on
p
and Ω . Moreover, g≥0 on ∂Ω whenever f≥0 almost everywhere on Ω.
Since Vk is bounded, we have uk∈C1(Ω¯) and in particular the classical normal derivative ∂uk∂n is well-defined on ∂Ω. To see why this is true, let w∈C1(Ω¯) be the solution of
{
-
Δ
w
=
|
f
|
in
Ω
,
w
=
0
on
∂
Ω
.
The weak maximum principle implies that |uk|≤w almost everywhere on Ω; thus uk∈L∞(Ω). Since Vk and f are bounded, we have Δuk∈L∞(Ω), hence uk∈W2,p(Ω) for every 1<p<∞; see [11, Theorem 9.15 and Lemma 9.17]. Taking any p>N, it follows from the Morrey–Sobolev embedding theorem that uk∈C1(Ω¯); see [21, Theorem 6.4.4]. In addition, one has the estimate
(2.3)
∥
w
∥
C
1
(
Ω
¯
)
≤
C
∥
Δ
w
∥
L
p
(
Ω
)
=
C
∥
f
∥
L
p
(
Ω
)
for some constant C>0 depending on p and Ω. Since
|
∂
u
k
∂
n
|
≤
∂
w
∂
n
on
∂
Ω
,
one deduces from (2.3) that
(2.4)
∥
∂
u
k
∂
n
∥
L
∞
(
Ω
)
≤
∥
w
∥
C
1
(
Ω
¯
)
≤
C
∥
f
∥
L
p
(
Ω
)
.
Using Proposition 2.1, we then define the pointwise normal derivative of u with respect to -Δ+V as
∂
u
^
∂
n
(
a
)
:=
g
(
a
)
for every
a
∈
∂
Ω
.
At first sight, this definition could depend on the choice of approximation (Vk) like
V
k
=
min
{
V
,
k
}
,
but as we shall see later on it does not; see Remark 5.1. As a consequence of assertion (iii) in Proposition 2.1, ∂u^∂n is a distributional normal derivative of u.
Before proceeding with the proof of Proposition 2.1, we recall standard estimates for solutions of the Dirichlet problem
(2.5)
{
-
Δ
u
+
V
u
=
μ
in
Ω
,
u
=
0
on
∂
Ω
,
where μ∈L1(Ω). By a solution of (2.5), we mean a function u∈W01,1(Ω)∩L1(Ω;Vdx) that satisfies the equation in the sense of distributions in Ω. For all 1≤p<NN-1, the solution exists, is unique and belongs to W01,p(Ω) with
(2.6)
∥
u
∥
W
1
,
p
(
Ω
)
≤
C
∥
μ
∥
L
1
(
Ω
)
for some constant C>0 depending on p and Ω. This can be deduced from elliptic estimates due to Littman, Stampacchia and Weinberger [12, Theorem 5.1] and from the absorption estimate
(2.7)
∥
V
u
∥
L
1
(
Ω
)
≤
∥
μ
∥
L
1
(
Ω
)
.
The latter inequality can be obtained using as test function a suitable approximation of the sign function sgnu; see [4, Proposition 4.B.3] or [18, Proposition 21.5].
The weak maximum principle for (2.5) that is mentioned in the introduction is justified by the following lemma.
Lemma 2.2.
Let u be the solution of (2.5) involving μ∈L1(Ω). If μ≥0 almost everywhere on Ω, then u≥0 almost everywhere on Ω.
The proof of Lemma 2.2 relies on a variant of Kato’s inequality: if w∈L1(Ω), h∈L1(Ω,d∂Ωdx) and ν∈ℳ(∂Ω) satisfy
(2.8)
-
∫
Ω
w
Δ
ζ
d
x
=
∫
Ω
h
ζ
d
x
+
∫
∂
Ω
∂
ζ
∂
n
d
ν
for every
ζ
∈
C
0
∞
(
Ω
¯
)
,
then
(2.9)
-
∫
Ω
w
+
Δ
ζ
d
x
≤
∫
{
w
≥
0
}
h
ζ
d
x
+
∫
∂
Ω
∂
ζ
∂
n
d
ν
+
for every
ζ
∈
C
0
∞
(
Ω
¯
)
,
ζ
≥
0
on
Ω
¯
;
see [14, Lemma 1.5] or [15, Proposition 1.5.9]. Here ℳ(∂Ω) denotes the vector space of finite Borel measures on ∂Ω and
C
0
∞
(
Ω
¯
)
=
{
ζ
∈
C
∞
(
Ω
¯
)
:
ζ
=
0
on
∂
Ω
}
.
When ν=0, the integral identity (2.8) implicitly encodes the fact that w=0 on ∂Ω in an average sense as test functions need not have compact support in Ω; see [18, Proposition 20.2] and also [9] for related questions. To deduce the weak maximum principle, it now suffices to take w=-u, h=Vu-μ and ν=0, and then (2.9) becomes
-
∫
Ω
(
-
u
)
+
Δ
ζ
d
x
≤
0
for every
ζ
∈
C
0
∞
(
Ω
¯
)
,
ζ
≥
0
on
Ω
¯
.
One last ingredient involved in the proof of Proposition 2.1 is the following comparison principle:
Lemma 2.3.
Let V1,V2∈Lloc1(Ω) be two nonnegative functions such that V1≤V2 almost everywhere on Ω, and let ui∈L1(Ω)∩L1(Ω;Vid∂Ωdx), with i∈{1,2}, be two nonnegative functions such that
-
∫
Ω
(
u
2
-
u
1
)
Δ
ζ
d
x
+
∫
Ω
(
V
2
u
2
-
V
1
u
1
)
ζ
d
x
=
0
for every
ζ
∈
C
0
∞
(
Ω
¯
)
.
Then u2≤u1 almost everywhere on Ω.
Lemma 2.3 can be deduced using Kato’s inequality as above by taking w=u2-u1.
Proof of Proposition 2.1.
We assume that f is nonnegative; the general case follows by solving the Dirichlet problem with the positive and negative parts of f, and then conclude using the linearity of the equation in (1.1) and uniqueness of solutions. Hence, by the weak maximum principle, u and uk are nonnegative. Since u satisfies
-
Δ
u
+
V
k
u
=
f
-
(
V
-
V
k
)
u
in the sense of distributions in
Ω
,
we have
-
Δ
(
u
k
-
u
)
+
V
k
(
u
k
-
u
)
=
(
V
-
V
k
)
u
in the sense of distributions in
Ω
.
One deduces from (2.6) applied to uk-u that
∥
u
k
-
u
∥
L
1
(
Ω
)
≤
C
∥
(
V
-
V
k
)
u
∥
L
1
(
Ω
)
.
By Lebesgue’s dominated convergence theorem, the right-hand side of this inequality tends to 0 as k→∞. Hence uk→u in L1(Ω). Since (uk) is nonincreasing as a consequence of Lemma 2.3, the convergence also holds everywhere on Ω. The triangle inequality and the absorption estimate (2.7) applied to uk-u imply that
∥
V
k
u
k
-
V
u
∥
L
1
(
Ω
)
≤
∥
V
k
(
u
k
-
u
)
∥
L
1
(
Ω
)
+
∥
(
V
k
-
V
)
u
∥
L
1
(
Ω
)
≤
2
∥
(
V
-
V
k
)
u
∥
L
1
(
Ω
)
,
and then Vkuk→Vu in L1(Ω).
By comparison of normal derivatives, the sequence (∂uk∂n) is nonincreasing and nonnegative. In particular, it is uniformly bounded on ∂Ω and converges in L1(∂Ω) and everywhere on ∂Ω to some nonnegative bounded measurable function g:∂Ω→ℝ. Let us show that
g
=
∂
u
∂
n
almost everywhere on
∂
Ω
,
where ∂u∂n is the distributional normal derivative of u. For this purpose, we recall that each uk satisfies
(2.10)
∫
Ω
∇
u
k
⋅
∇
ψ
d
x
=
∫
Ω
f
ψ
d
x
-
∫
Ω
V
k
u
k
ψ
d
x
-
∫
∂
Ω
∂
u
k
∂
n
ψ
d
σ
for every ψ∈C∞(Ω¯). By standard interpolation, which in this case follows from an integration by parts, one also has the estimate
∥
∇
u
k
∥
L
2
(
Ω
)
2
≤
∥
u
k
∥
L
∞
(
Ω
)
∥
Δ
u
k
∥
L
1
(
Ω
)
;
see [18, Lemma 5.8]. We claim that the right-hand side of this inequality is bounded. Indeed, as (uk) is nonincreasing, it is bounded from above by u0. On the other hand, we deduce from the triangle inequality and the absorption estimate (2.7) that
∥
Δ
u
k
∥
L
1
(
Ω
)
≤
∥
f
∥
L
1
(
Ω
)
+
∥
V
k
u
k
∥
L
1
(
Ω
)
≤
2
∥
f
∥
L
1
(
Ω
)
,
which validates our claim.
Since (∇uk) is bounded in L2(Ω;ℝN) and uk→u in L1(Ω), we have
∇
u
k
⇀
∇
u
weakly in
L
2
(
Ω
;
ℝ
N
)
.
Taking the limit as k→∞ in (2.10), we obtain
∫
∂
Ω
∂
u
∂
n
ψ
d
σ
=
∫
∂
Ω
g
ψ
d
σ
for every
ψ
∈
C
∞
(
Ω
¯
)
.
Hence ∂u∂n=g almost everywhere on ∂Ω. The estimate
∥
g
∥
L
∞
(
∂
Ω
)
≤
C
∥
f
∥
L
p
(
Ω
)
follows from (2.4) since 0≤g≤∂uk∂n on ∂Ω. ∎
3 Duality Solution with Measure Data on the Boundary
We investigate the boundary value problem (1.7) involving a finite Borel measure ν on ∂Ω by comparing two notions of solution based on different choices of test functions.
Definition 3.1.
A function v∈L1(Ω) is a distributional solution of (1.7) with datum ν∈ℳ(∂Ω) whenever Vv∈L1(Ω;d∂Ωdx) and
∫
Ω
v
(
-
Δ
ζ
+
V
ζ
)
d
x
=
∫
∂
Ω
∂
ζ
∂
n
d
ν
for every
ζ
∈
C
0
∞
(
Ω
¯
)
.
The boundary value problem for this type of solutions has been studied by Véron and Yarur [20] with nonnegative potentials V∈Lloc∞(Ω). In particular, the authors prove that nonnegative measures for which (1.7) has a solution cannot charge Σ; see [20, Theorem 4.4]. Their approach is based on the careful study of some capacity associated to the Poisson kernel of -Δ on Ω. In our case, we rely instead on the concept of duality solution in the spirit of the work of Malusa and Orsina [13] that has its roots in the seminal paper of Littman, Stampacchia and Weinberger [12].
Definition 3.2.
A function v∈L1(Ω) is a duality solution of (1.7) with datum ν∈ℳ(∂Ω) whenever
∫
Ω
v
f
d
x
=
∫
∂
Ω
∂
ζ
f
^
∂
n
d
ν
for every
f
∈
L
∞
(
Ω
)
,
where ζf is the solution of (1.1) with datum f.
Existence of duality solutions is a straightforward consequence of the Riesz representation theorem:
Proposition 3.1.
The boundary value problem (1.7) has a unique duality solution for every datum ν∈M(∂Ω).
Proof.
Let N<p<∞. It follows from Proposition 2.1 that, for every f∈L∞(Ω),
|
∫
∂
Ω
∂
ζ
f
^
∂
n
d
ν
|
≤
∥
ν
∥
ℳ
(
∂
Ω
)
∥
∂
ζ
f
^
∂
n
∥
L
∞
(
∂
Ω
)
≤
C
∥
ν
∥
ℳ
(
∂
Ω
)
∥
f
∥
L
p
(
Ω
)
,
where
∥
ν
∥
ℳ
(
∂
Ω
)
:=
|
ν
|
(
∂
Ω
)
.
Hence, the linear functional
L
∞
(
Ω
)
→
ℝ
,
f
↦
∫
∂
Ω
∂
ζ
f
^
∂
n
d
ν
is continuous on L∞(Ω), endowed with the Lp norm. The Riesz representation theorem implies the existence of a unique v∈Lp′(Ω) such that
∫
Ω
v
f
d
x
=
∫
∂
Ω
∂
ζ
f
^
∂
n
d
ν
for every
f
∈
L
∞
(
Ω
)
,
where p′=pp-1 is the conjugate exponent with respect to p. Hence v is the unique duality solution of (1.7) involving ν. ∎
We now prove that distributional solutions are duality solutions:
Proposition 3.2.
If v is a distributional solution of (1.7) with datum ν∈M(∂Ω), then v is also a duality solution of (1.7) with datum ν.
For the proof of Proposition 3.2, we need a couple of lemmas. We begin with:
Lemma 3.3.
Assume that (1.7) has a distributional solution v with datum ν∈M(∂Ω) and let vk be the distributional solution of
(3.1)
{
-
Δ
v
k
+
V
k
v
k
=
0
in
Ω
,
v
k
=
ν
on
∂
Ω
.
Then vk→v in L1(Ω).
Proof.
First notice that the function v-vk satisfies
∫
Ω
(
v
-
v
k
)
(
-
Δ
ζ
+
V
k
ζ
)
d
x
=
∫
Ω
(
V
k
-
V
)
v
ζ
d
x
for every
ζ
∈
C
0
∞
(
Ω
¯
)
.
Kato’s inequality (2.9) applied to v-vk and -(v-vk) with ν=0 implies that
∫
Ω
|
v
-
v
k
|
(
-
Δ
ζ
+
V
k
ζ
)
d
x
≤
∫
Ω
sgn
(
v
-
v
k
)
(
V
k
-
V
)
v
ζ
d
x
,
and then
∫
Ω
|
v
-
v
k
|
(
-
Δ
ζ
+
V
k
ζ
)
d
x
≤
∫
Ω
ζ
(
V
-
V
k
)
|
v
|
d
x
for every ζ∈C0∞(Ω¯), ζ≥0 on Ω¯. We take as test function the unique solution of
(3.2)
{
-
Δ
θ
=
1
in
Ω
,
θ
=
0
on
∂
Ω
.
Observing that Vkθ≥0 and θ≤∥∇θ∥L∞(Ω)d∂Ω on Ω, we obtain the estimate
(3.3)
∥
v
k
-
v
∥
L
1
(
Ω
)
≤
∥
∇
θ
∥
L
∞
(
Ω
)
∥
(
V
-
V
k
)
v
∥
L
1
(
Ω
;
d
∂
Ω
d
x
)
.
Since 0≤Vk≤V and Vv∈L1(Ω;d∂Ωdx), Lebesgue’s dominated convergence theorem implies that
V
k
v
→
V
v
in
L
1
(
Ω
;
d
∂
Ω
d
x
)
.
The lemma then follows by letting k→∞ in (3.3). ∎
Lemma 3.4.
Assume that V∈Lq(Ω) for some q>N. Then the boundary value problem (1.7) has a distributional solution for every ν∈M(∂Ω). In particular, Σ=∅.
We recall that if v∈L1(Ω), f∈L1(Ω;d∂Ωdx) and ν∈ℳ(∂Ω) satisfy
-
∫
Ω
v
Δ
ζ
d
x
=
∫
Ω
f
ζ
d
x
+
∫
∂
Ω
∂
ζ
∂
n
d
ν
for every
ζ
∈
C
0
∞
(
Ω
¯
)
,
which is the weak formulation of v being a solution of
{
-
Δ
v
=
f
in
Ω
,
v
=
ν
on
∂
Ω
,
then:
for every 1≤p<NN-1, we have v∈Lp(Ω) and there exists a constant C>0 depending on p and Ω such that
(3.4)
∥
v
∥
L
p
(
Ω
)
≤
C
(
∥
f
∥
L
1
(
Ω
;
d
∂
Ω
d
x
)
+
∥
ν
∥
ℳ
(
∂
Ω
)
)
,
for every 1≤p<NN-1 and every ω⋐Ω, we have ∇v∈Lp(ω;ℝN) and there exists a constant C′>0 depending on p and ω such that
(3.5)
∥
∇
v
∥
L
p
(
ω
;
ℝ
N
)
≤
C
′
(
∥
f
∥
L
1
(
Ω
;
d
∂
Ω
d
x
)
+
∥
ν
∥
ℳ
(
∂
Ω
)
)
.
We refer the reader to [15, Theorem 1.2.2] for a proof of these assertions.
Proof of Lemma 3.4.
Let (gk) be a sequence of smooth functions on ∂Ω such that ∥gk∥L1(∂Ω)→∥ν∥ℳ(∂Ω) and gk⇀∗ν weak* in ℳ(∂Ω), i.e.,
lim
k
→
∞
∫
∂
Ω
ϕ
g
k
d
x
=
∫
∂
Ω
ϕ
d
ν
for every
ϕ
∈
C
(
∂
Ω
)
.
Such a sequence can be obtained, for example, from a convolution of ν with a sequence of mollifiers. Denote by vk the distributional solution of (1.7) associated to gk. Given ω⋐Ω, we deduce from (3.4) and (3.5) that
∥
v
k
∥
W
1
,
1
(
ω
)
≤
C
1
(
∥
V
v
k
∥
L
1
(
Ω
;
d
∂
Ω
d
x
)
+
∥
g
k
∥
L
1
(
∂
Ω
)
)
for some constant C1>0 depending on ω. Taking a subsequence if necessary, we have
(3.6)
v
k
→
v
almost everywhere on
Ω
.
On the other hand, since q′<NN-1, we deduce from (3.4) that
∥
v
k
∥
L
q
′
(
Ω
)
≤
C
2
(
∥
V
v
k
∥
L
1
(
Ω
;
d
∂
Ω
d
x
)
+
∥
g
k
∥
L
1
(
∂
Ω
)
)
for some constant C2>0 depending on q and Ω. Hence (vk) is bounded in Lq′(Ω). This, together with (3.6), implies that
v
k
⇀
v
weakly in
L
q
′
(
Ω
)
.
Recalling that V∈Lq(Ω) and taking the limit as k→∞ in the equation
∫
Ω
v
k
(
-
Δ
ζ
+
V
ζ
)
d
x
=
∫
Ω
∂
ζ
∂
n
g
k
d
σ
for every
ζ
∈
C
0
∞
(
Ω
¯
)
,
we get the conclusion. ∎
We now turn to the
Proof of Proposition 3.2.
We first assume that V is bounded. In this case, ζf∈C01(Ω¯) for every f∈L∞(Ω). Since ζf need not be smooth enough to be used as test function for v, we approximate ζf in C1(Ω¯) by a sequence (ζfk) in C0∞(Ω¯), where (fk) is a bounded sequence in L∞(Ω) such that fk→f almost everywhere on Ω. For this purpose, we follow the construction given in [17]: for each k we define the function gk=ρk*g, where g=f-Vζf and (ρk) is a sequence of mollifiers, and we denote by wk∈C0∞(Ω¯) the solution of
{
-
Δ
w
k
=
g
k
in
Ω
,
w
k
=
0
on
∂
Ω
.
Observe that wk=ζfk with fk=gk+Vwk. Moreover, estimate (2.3) ensures that for some fixed N<p<∞,
∥
ζ
f
k
-
ζ
f
∥
C
1
(
Ω
¯
)
≤
C
∥
g
k
-
g
∥
L
p
(
Ω
)
.
Letting k→∞ in this estimate, we have
ζ
f
k
→
ζ
f
uniformly on
Ω
and
∂
ζ
f
k
∂
n
→
∂
ζ
f
∂
n
uniformly on
∂
Ω
.
Since
∫
Ω
v
f
k
d
x
=
∫
Ω
v
(
-
Δ
ζ
f
k
+
V
ζ
f
k
)
d
x
=
∫
∂
Ω
∂
ζ
f
k
∂
n
d
ν
,
taking the limit as k→∞, we obtain
∫
Ω
v
f
d
x
=
∫
∂
Ω
∂
ζ
f
∂
n
d
ν
.
In the general case where V∈Lloc1(Ω), let vk be the solution of
{
-
Δ
v
k
+
V
k
v
k
=
0
in
Ω
,
v
k
=
ν
on
∂
Ω
,
whose existence is ensured by Lemma 3.4. Given f∈L∞(Ω), we denote by zk∈C01(Ω¯) the solution of
{
-
Δ
z
k
+
V
k
z
k
=
f
in
Ω
,
z
k
=
0
on
∂
Ω
.
It follows from the first part of the proof that
∫
Ω
v
k
f
d
x
=
∫
∂
Ω
∂
z
k
∂
n
d
ν
.
Proposition 2.1 implies that (∂zk∂n) is uniformly bounded and converges pointwise to ∂ζf^∂n on ∂Ω. By Lemma 3.3, we obtain
∫
Ω
v
f
d
x
=
∫
∂
Ω
∂
ζ
f
^
∂
n
d
ν
.
∎
4 Pointwise Normal Derivative on the Exceptional Set Σ
In this section, we characterize the exceptional boundary set Σ using the pointwise normal derivative with respect to the Schrödinger operator -Δ+V introduced in Section 2. We prove:
Proposition 4.1.
For every a∈∂Ω, we have a∈Σ if and only if
∂
ζ
f
^
∂
n
(
a
)
=
0
for every
f
∈
L
∞
(
Ω
)
.
As a fundamental property that is used in the proofs of Proposition 4.1 and Theorem 3, we first extend Lemma 3.3 to the case where (1.7) need not have a distributional solution.
Proposition 4.2.
Let ν∈M(∂Ω) be a nonnegative measure and let v be the duality solution of (1.7) associated to ν. We have:
if
v
k
is the distributional solution of (
3.1
), then
v
k
→
v
in
L
1
(
Ω
)
,
there exists a nonnegative measure
λ
∈
ℳ
(
∂
Ω
)
such that
v
is the distributional solution of (
1.7
) associated to
ν
-
λ
.
We recall the following estimate whose proof is sketched for the convenience of the reader:
Lemma 4.3.
If v is a distributional solution of (1.7) with ν∈M(∂Ω), then
∥
v
∥
L
1
(
Ω
)
+
∥
V
v
∥
L
1
(
Ω
;
d
∂
Ω
d
x
)
≤
C
∥
ν
∥
ℳ
(
∂
Ω
)
for some constant C>0 depending on Ω.
Proof.
One deduces using Kato’s inequality (2.9) with w=v and w=-v that
∫
Ω
|
v
|
(
-
Δ
ζ
+
V
ζ
)
d
x
≤
∥
∂
ζ
∂
n
∥
L
∞
(
∂
Ω
)
∥
ν
∥
ℳ
(
∂
Ω
)
for every ζ∈C0∞(Ω¯), ζ≥0 on Ω¯. Take as test function the solution θ of (3.2). As a consequence of the classical Hopf lemma, there exists C1>0 such that θ≥C1d∂Ω on Ω. Therefore, by nonnegativity of V,
∥
v
∥
L
1
(
Ω
)
+
C
1
∥
V
v
∥
L
1
(
Ω
;
d
∂
Ω
d
x
)
≤
∥
∂
θ
∂
n
∥
L
∞
(
∂
Ω
)
∥
ν
∥
ℳ
(
∂
Ω
)
.
∎
Proof of Proposition 4.2.
By a straightforward counterpart of Lemmas 2.2 and 2.3 for distributional solutions of (1.7), the sequence (vk) is nonnegative and nonincreasing. Hence (vk) converges in L1(Ω) to some nonnegative function w. Let f∈L∞(Ω) and let uk be the solution of (2.2). Proposition 3.2 implies that
∫
Ω
v
k
f
d
x
=
∫
∂
Ω
∂
u
k
∂
n
d
ν
.
By Proposition 2.1, the sequence (∂uk∂n) is uniformly bounded and converges pointwise to ∂ζf^∂n on ∂Ω. Taking the limit as k→∞ in the identity above, we deduce from Lebesgue’s dominated convergence theorem that
∫
Ω
w
f
d
x
=
∫
∂
Ω
∂
ζ
f
^
∂
n
d
ν
for every
f
∈
L
∞
(
Ω
)
.
We have thus proved that w is a duality solution of (1.7) involving ν. By uniqueness of duality solutions, we have v=w.
For every k≥1, we have
0
≤
V
k
v
k
≤
V
v
1
almost everywhere on
Ω
.
Since v1 is subharmonic, it is locally bounded on Ω; see [21, Theorem 8.1.5]. Then, Lebesgue’s dominated convergence theorem implies that
V
k
v
k
→
V
v
in
L
loc
1
(
Ω
)
.
Let θ be the unique solution of (3.2). Since 0<θ≤∥∇θ∥L∞(Ω)d∂Ω on Ω, by Lemma 4.3 the sequence (Vkvkθ) is bounded in L1(Ω). Therefore, taking a subsequence if necessary, we may assume that there exist nonnegative finite Borel measures μ on Ω and τ on ∂Ω such that, for every ψ∈C∞(Ω¯),
(4.1)
lim
k
→
∞
∫
Ω
V
k
v
k
θ
ψ
d
x
=
∫
Ω
ψ
d
μ
+
∫
∂
Ω
ψ
d
τ
.
On the other hand, for every φ∈Cc∞(Ω),
lim
k
→
∞
∫
Ω
V
k
v
k
θ
φ
d
x
=
∫
Ω
V
v
θ
φ
d
x
.
Hence μ=Vvθdx. Given ζ∈C0∞(Ω¯), we define γ=ζθ on Ω. Since ∂θ∂n>0 on ∂Ω, the function γ extends continuously to ∂Ω, and
γ
=
∂
ζ
∂
n
1
∂
θ
∂
n
on
∂
Ω
.
Taking ψ=γ in (4.1), we obtain
lim
k
→
∞
∫
Ω
V
k
v
k
ζ
d
x
=
∫
Ω
V
v
ζ
d
x
+
∫
∂
Ω
∂
ζ
∂
n
1
∂
θ
∂
n
d
τ
for every
ζ
∈
C
0
∞
(
Ω
¯
)
.
The result follows with λ=1∂θ∂nτ. ∎
Another ingredient involved in the proof of Proposition 4.1 is the inverse maximum principle for distributional solutions of (1.7); see [6, Lemma 1].
Lemma 4.4.
Let ν∈M(∂Ω) and let h∈L1(Ω;d∂Ωdx). Assume that v∈L1(Ω) satisfies
-
∫
Ω
v
Δ
ζ
d
x
=
∫
Ω
h
ζ
d
x
+
∫
∂
Ω
∂
ζ
∂
n
d
ν
for every
ζ
∈
C
0
∞
(
Ω
¯
)
.
If v≥0 almost everywhere on Ω, then ν≥0 on ∂Ω.
Given a∈∂Ω, we denote by Pa the duality solution of (1.7) associated to the Dirac measure δa, that is,
(4.2)
∂
ζ
f
^
∂
n
(
a
)
=
∫
Ω
P
a
f
d
x
for every
f
∈
L
∞
(
Ω
)
.
One deduces from the definition of duality solution using f=χ{Pa<0} as test function that
P
a
≥
0
almost everywhere on
Ω
.
We apply this simple observation:
Proof of Proposition 4.1.
We first assume that a∈Σ. By Proposition 4.2, Pa is a distributional solution of (1.7) with datum δa-λ for some nonnegative measure λ∈ℳ(∂Ω). Since Pa≥0 almost everywhere on Ω, we deduce from Lemma 4.4 that
δ
a
≥
λ
≥
0
on
∂
Ω
.
Thus, λ=αδa for some 0≤α≤1. If we had α≠1, then Pa1-α would be a distributional solution of (1.4), in contradiction with the assumption that a∈Σ. Hence, α=1 and
∫
Ω
P
a
(
-
Δ
ζ
+
V
ζ
)
d
x
=
0
for every
ζ
∈
C
0
∞
(
Ω
¯
)
.
Taking ζ=θ, where θ satisfies (3.2), we deduce that
∫
Ω
P
a
d
x
=
0
.
Since Pa is nonnegative, we have Pa=0 almost everywhere on Ω. The representation formula (4.2) then implies that
∂
ζ
f
^
∂
n
(
a
)
=
0
for every
f
∈
L
∞
(
Ω
)
.
For the converse, one deduces from the assumption on a and (4.2) applied to f≡1 that
∫
Ω
P
a
d
x
=
0
.
Hence Pa=0 almost everywhere on Ω, so that Pa cannot be a distributional solution of (1.7) involving δa. Since Pa is the only candidate for such a solution due to Proposition 3.2, we conclude that a∈Σ. ∎
5 Proof of Theorem 3
(⇐) Let v be the duality solution of (1.7) associated to ν. Proposition 4.2 implies the existence of a nonnegative measure λ∈ℳ(∂Ω) such that v is a distributional solution of (1.7) involving ν-λ. We claim that λ(∂Ω∖Σ)=0. By Proposition 3.2, v is also a duality solution of (1.7) with datum ν-λ. Hence
∫
∂
Ω
∂
ζ
f
^
∂
n
d
ν
=
∫
Ω
v
f
d
x
=
∫
∂
Ω
∂
ζ
f
^
∂
n
d
(
ν
-
λ
)
for every
f
∈
L
∞
(
Ω
)
,
which implies that
∫
∂
Ω
∂
ζ
f
^
∂
n
d
λ
=
0
for every
f
∈
L
∞
(
Ω
)
.
By Proposition 4.1, we have ∂ζ1^∂n>0 on ∂Ω∖Σ. Since λ is nonnegative, we conclude that λ(∂Ω∖Σ)=0 as claimed. On the other hand, since v≥0 almost everywhere on Ω, we deduce from Lemma 4.4 that
ν
≥
λ
≥
0
on
∂
Ω
.
By assumption, ν(Σ)=0. Hence λ(Σ)=0. We thus have
λ
(
∂
Ω
)
=
λ
(
∂
Ω
∖
Σ
)
+
λ
(
Σ
)
=
0
,
that is, λ=0.
(⇒) Let v be the distributional solution of (1.7) associated to ν. Proposition 3.2 implies that v is also a duality solution of (1.7) involving the same datum. By Proposition 4.1, we have
∫
Ω
v
f
d
x
=
∫
∂
Ω
∂
ζ
f
^
∂
n
d
ν
=
∫
∂
Ω
∂
ζ
f
^
∂
n
d
ν
⌊
∂
Ω
∖
Σ
for every
f
∈
L
∞
(
Ω
)
,
so that v is also a duality solution of (1.7) with datum ν⌊∂Ω∖Σ. The reverse implication in Theorem 3 implies that (1.7) associated to ν⌊∂Ω∖Σ has a unique distributional solution z. But then, Proposition 3.2 ensures that z is also a duality solution of (1.7) with datum ν⌊∂Ω∖Σ. Since duality solutions are unique, we have v=z almost everywhere on Ω. Thus, v is a distributional solution of (1.7) with both ν and ν⌊∂Ω∖Σ, which implies that ν=ν⌊∂Ω∖Σ, and then ν(Σ)=0.
Remark 5.1.
Using Theorem 3 and Proposition 4.1, we can now explain why ∂u^∂n does not depend upon the particular choice of approximating sequence (Vk) in Proposition 2.1. Indeed, we have by Proposition 4.1 that
∂
u
^
∂
n
(
a
)
=
0
for every
a
∈
Σ
,
while Σ is independent of (Vk). When a∈∂Ω∖Σ, it follows from Theorem 3 that Pa is a distributional solution of (1.4), whose definition does not involve (Vk). Thus, by the representation formula (4.2), ∂u^∂n is independent of (Vk) also on ∂Ω∖Σ.
6 Proof of Theorem 1
We deduce Theorem 1 as a consequence of the following proposition:
Proposition 6.1.
Let u be the solution of (1.1) for some nonnegative datum in L∞(Ω) such that u^ has a classical normal derivative at a∈∂Ω which satisfies (1.3). If v is another solution of (1.1) for some nonnegative datum in L∞(Ω), and if v≤u almost everywhere on Ω, then v^ also has a classical normal derivative at a which satisfies (1.3).
We recall that whenever v satisfies (1.1) with datum h∈L∞(Ω), one has
v
^
(
x
)
=
∫
Ω
G
(
x
,
y
)
(
h
-
V
v
)
(
x
)
d
x
for every
x
∈
Ω
.
Using standard estimates on the Green function G, by boundedness of h one shows that
lim
t
↓
0
∫
Ω
G
(
a
+
t
n
,
y
)
t
h
(
y
)
d
y
=
∫
Ω
K
(
a
,
y
)
h
(
y
)
d
y
,
where n=n(a). The main difficulty in the proof of Proposition 6.1 thus consists in getting that
lim
t
↓
0
∫
Ω
G
(
a
+
t
n
,
y
)
t
V
v
(
y
)
d
y
=
∫
Ω
K
(
a
,
y
)
V
v
(
y
)
d
y
.
Proof of Proposition 6.1.
Let (εk) be a nonincreasing sequence of positive numbers converging to 0. We define on Ω
g
k
(
y
)
=
G
(
a
+
ε
k
n
,
y
)
ε
k
.
On the one hand, we have gk(y)→K(a,y) for all y∈Ω. On the other hand, by assumption on u,
∫
Ω
g
k
(
y
)
V
u
(
y
)
d
y
→
∫
Ω
K
(
a
,
y
)
V
u
(
y
)
d
y
or, equivalently,
∥
g
k
u
∥
L
1
(
Ω
;
V
d
x
)
→
∥
K
(
a
,
⋅
)
u
∥
L
1
(
Ω
;
V
d
x
)
.
We thus have pointwise convergence of (gku) and also convergence of norms. Therefore,
g
k
u
→
K
(
a
,
⋅
)
u
in
L
1
(
Ω
;
V
d
x
)
,
which is a special case of the Brezis–Lieb lemma [3]; see [21, Proposition 4.2.6]. Since 0≤v≤u, we deduce from Lebesgue’s dominated convergence theorem that gkv→K(a,⋅)v in L1(Ω;Vdx). The conclusion is now straightforward. ∎
For the proof of Theorem 1, we also need the following:
Lemma 6.2.
Let u be the solution of (1.1) for some nonnegative datum f∈L∞(Ω). Then, for every a∈∂Ω, we have
lim sup
ε
↓
0
u
^
(
a
+
ε
n
)
ε
≤
∂
u
^
∂
n
(
a
)
≤
∫
Ω
K
(
a
,
y
)
(
f
-
V
u
)
(
y
)
d
y
.
Proof.
Let uk satisfy (2.2). By comparison, we have u^≤uk on Ω. Hence,
lim sup
ε
↓
0
u
^
(
a
+
ε
n
)
ε
≤
∂
u
k
∂
n
(
a
)
for every
a
∈
∂
Ω
.
Then, as k→∞ we deduce the first inequality in the statement.
Next, since Δuk is bounded, for every a∈∂Ω we have
∂
u
k
∂
n
(
a
)
=
∫
Ω
K
(
a
,
y
)
(
-
Δ
u
k
(
y
)
)
d
y
=
∫
Ω
K
(
a
,
y
)
(
f
-
V
k
u
k
)
(
y
)
d
y
.
By Proposition 2.1, the sequence (Vkuk) converges almost everywhere to Vu. Fatou’s lemma implies that, for every a∈∂Ω,
∫
Ω
K
(
a
,
y
)
V
u
(
y
)
d
y
≤
lim inf
k
→
∞
∫
Ω
K
(
a
,
y
)
V
k
u
k
(
y
)
d
y
.
Observe that the right-hand side is finite since the sequence (∂uk∂n) is bounded. Hence, for every a∈∂Ω, we have
∂
u
^
∂
n
(
a
)
=
lim
k
→
∞
∂
u
k
∂
n
(
a
)
=
lim
k
→
∞
∫
Ω
K
(
a
,
y
)
(
f
-
V
k
u
k
)
(
y
)
d
y
≤
∫
Ω
K
(
a
,
y
)
(
f
-
V
u
)
(
y
)
d
y
,
which implies the second inequality in the statement. ∎
Proof of Theorem 1.
The reverse implication (⇐) follows from Proposition 6.1 and the fact that
u
≤
ζ
∥
f
∥
L
∞
(
Ω
)
=
∥
f
∥
L
∞
(
Ω
)
ζ
1
almost everywhere on
Ω
.
We now prove the direct implication (⇒). One shows the existence of a constant C>0 such that, for every ε>0, the solution vε of the Dirichlet problem
{
-
Δ
v
ε
+
V
v
ε
=
χ
{
u
C
>
ε
}
in
Ω
,
v
ε
=
0
on
∂
Ω
,
satisfies vε≤uε almost everywhere on Ω; this is a consequence of Kato’s inequality, as explained in [16, proof of Proposition 4.1]. Let (εj) be a nonincreasing sequence of positive numbers converging to 0. Since χ{uC>εj}≤1, by Proposition 6.1 the function vεj^ has a normal derivative at a which satisfies (1.3), which means that
∂
v
ε
j
^
∂
n
(
a
)
=
∫
Ω
K
(
a
,
y
)
(
χ
{
u
C
>
ε
j
}
-
V
v
ε
j
)
(
y
)
d
y
.
By the strong maximum principle for the Schrödinger operator with potential in Lloc1(Ω), we have u>0 almost everywhere on Ω. Hence
χ
{
u
C
>
ε
j
}
→
1
almost everywhere on
Ω
.
The convergence thus holds in L1(Ω), which implies that
v
ε
j
→
ζ
1
in
L
1
(
Ω
)
.
As the sequence (vεj) is nondecreasing, we deduce from Levi’s monotone convergence theorem that
lim
j
→
∞
∂
v
ε
j
^
∂
n
(
a
)
=
∫
Ω
K
(
a
,
y
)
(
1
-
V
ζ
1
(
y
)
)
d
y
.
Since vεj^≤ζ1^ on Ω, we also have, by classical comparison of limits,
lim
j
→
∞
∂
v
ε
j
^
∂
n
(
a
)
≤
lim inf
j
→
∞
ζ
1
^
(
a
+
ε
j
n
)
ε
j
.
Lemma 6.2 implies that
lim inf
k
→
∞
ζ
1
^
(
a
+
ε
k
n
)
ε
k
≤
lim sup
k
→
∞
ζ
1
^
(
a
+
ε
k
n
)
ε
k
≤
∫
Ω
K
(
a
,
y
)
(
1
-
V
ζ
1
(
y
)
)
d
y
.
Combining the inequalities above, we deduce that ∂ζ1^(a)∂n exists and
∂
ζ
1
^
∂
n
(
a
)
=
lim
k
→
∞
∂
v
ε
k
^
∂
n
(
a
)
=
∫
Ω
K
(
a
,
y
)
(
1
-
V
ζ
1
(
y
)
)
d
y
.
Hence, by definition, a∈𝒩. ∎
7 Proof of Theorem 2
We first prove a version of the Hopf lemma in terms of the pointwise normal derivative associated to the Schrödinger operator -Δ+V. In this case, the answer does not involve the set 𝒩.
Proposition 7.1.
Let u be the solution of (1.1) for some nonnegative datum f∈L∞(Ω), f≢0. Then, for every a∈∂Ω, we have
∂
u
^
∂
n
(
a
)
>
0
if and only if
a
∉
Σ
.
Proof.
By Proposition 4.1, we only have to prove that ∂u^∂n>0 on ∂Ω∖Σ. For this purpose, let a∈∂Ω∖Σ. In this case, Pa is both a duality and a distributional solution of (1.7) involving δa. As a distributional solution, it satisfies the strong maximum principle for the Schrödinger operator with potential in Lloc1(Ω). Hence,
(7.1)
P
a
>
0
almost everywhere on
Ω
.
As a duality solution, Pa satisfies the representation formula (4.2). Since f is nonzero, we then deduce from this formula and (7.1) that
∂
u
^
∂
n
(
a
)
>
0
.
∎
We now turn to the
Proof of Theorem 2.
By Theorem 1, the classical normal derivative ∂u^∂n exists on 𝒩 and satisfies (1.3). A direct application of Lemma 6.2 gives, for every a∈𝒩,
∂
u
^
∂
n
(
a
)
≤
∂
u
^
∂
n
(
a
)
≤
∫
Ω
K
(
a
,
y
)
(
f
-
V
u
)
(
y
)
d
y
.
As the integral in the right-hand side equals ∂u^(a)∂n, equality holds everywhere and we get
∂
u
^
∂
n
=
∂
u
^
∂
n
on
𝒩
.
The theorem then follows from Proposition 7.1. ∎
We conclude this section with the following particular case of Theorem 2.
Corollary 7.2.
Assume that V∈Lq(Ω) for some q>N. Then, for every solution u of (1.1) involving a nonnegative datum f∈L∞(Ω), f≢0, the normal derivative of u^ exists at every point a∈∂Ω and satisfies
∂
u
^
∂
n
(
a
)
>
0
.
Proof.
We prove that 𝒩=∂Ω and Σ=∅. Let u be the solution of (1.1) involving some nonnegative datum f∈L∞(Ω). Since u∈L∞(Ω), we have Δu∈Lq(Ω), and then u∈W2,q(Ω). The Morrey–Sobolev embedding theorem ensures that u∈C1(Ω¯), which gives 𝒩=∂Ω. That Σ=∅ follows from Lemma 3.4. We then have the conclusion using Theorem 2. ∎
Acknowledgements
The authors would like to thank Moshe Marcus for discussions on the definition of the weak normal derivative.
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and
Nicolas Wilmet
