2.3 A generalized frequency function
2.3.2 Defining the generalized frequency function and obtaining basic estimates
In the spirit of Subsection 2.2 we now define the generalized frequency function.
Definition 2.3.1. For any number r in the interval (0, R) we define the quantities:
Dg(r) :=
ˆ
Bgr(¯0)
ω(x)|∇gu|2gdx, and Hg(r) :=
ˆ
∂Brg(¯0)
ω(x)u2dσ, (2.96)
Ig(r) := Dg(r) + ˆ
Brg(¯0)
(bg· ∇gu)u + cgu2dx, (2.97) where integration, unless otherwise stated, is with respect to the volume form of the metric g.
The generalized frequency functionNg(r) with respect to the metric g is defined as Ng(r) := rIg(r)
Hg(r), (2.98)
wheneverHg(r) is not vanishing.
A few remarks are in order.
First we remind that although the function u is in the space W01,2(Ω) (and Wloc2,2(Ω)) one can still define surface integrals (in particular, Hg(r) is well defined) via the trace operator and, furthermore, integration by parts over appropriate subdomains also holds (divergence theorem). For background, we refer to Section A, Theorems A.3.1, A.3.2. In our case however, the function u is a weak solution to an elliptic PDE and hence, H¨older continuous by the techniques of De Giorgi-Nash-Moser. We refer to Theorem 8.24, [GT01].
Second, in contrast to Subsection 2.2 the sign of Ng(r) is no longer clearly determined. Furthermore, it is not clear whether Hg(r) is not vanishing for a large set of radii r. We address these issues below.
Third, there are other possibilities of defining the frequency function - for example, instead of Ig(r) one may directly consider the energy associated to the operator L, i.e. without undergoing the coordinate transformation above and obtaining the metric g. Furthermore, one could also try
replacing Dg(r) by the Euclidean gradient, as we did in the case of harmonic functions, and use this quantity instead. These options are reasonable. For our purposes, A(x) will be derived from the Laplace operator, and hence, in normal coordinates, A(¯0) = I where I denotes the n× n- identity matrix. This will imply that all of the mentioned options for a definition of the frequency function will be comparable up to constants close to 1 in a sufficiently small neighbourhood around ¯0.
In fact, in order to reduce the amount of technicalities, from now on we make the following:
Assumption 2.3.1. We assume thatA(x) is sufficiently close to the identity, i.e.
kA(x) − IkL∞(B1(¯0))≤ δ, (2.99)
where δ is a small positive number. This means, in particular, that all bounds on the metrics g,eg above are close to the bounds for the identity matrix. The amount of ”closeness”, i.e. the number δ, will be given in the particular context whenever needed.
We now address the vanishing Hg(r). It will turn out that for a sufficiently small radii (depending only n, Λg, Γg, ηg) the quantity Hg(r) is positive and Ng(r) is well-defined. We start with the following comparison between the quantities Dg(r), Hg(r) and Ig(r):
Proposition 2.3.4. Suppose is an arbitrary number in the interval (0, 1). Then there exists a positive number r0, depending on , n, Λg, τ1, w1, so that for any number r in the interval (0, r0)
We begin by first establishing the following form of the Heisenberg uncertainty principle:
Lemma 2.3.4. For any positive numberρ in (0, 1) we have ˆ Proof of Lemma 2.3.4. The idea is to use Fubini’s theorem (or co-area formula) and split the integral in radial/tangential parts, followed by integration by parts:
ˆ
where as usual the subscript ν denotes differentiation along the normalized radial direction. Finally, Young’s inequality with parameter µ gives us
ˆ
Setting η as n2 finishes the proof of the Lemma.
Proof of Proposition 2.3.4. Onwards, using Definition 2.3.1, Young’s inequality with parameter µ and the assumptions on the coefficients (2.63) we have
Dg(r) =
Now, we bound the last expression employing the Heisenberg uncertainty Lemma 2.3.4 to obtain
Dg(r)≤ Ig(r) + 1
where, in the last inequality, we have also used (2.77) and the bounds on ω(x) from Proposition 2.3.2.
First, we take µ to be sufficiently large, i.e. let µ = 1
τ1w1
. (2.114)
Now we choose r sufficiently small, so that the coefficients in front of the integrals are small:
w1
Hence, we conclude from our latter estimate
Dg(r)≤ Ig(r) + ˆ
Bgr(¯0)|∇gu|2g+ ˆ
∂Brg(¯0)
u2, (2.117)
and after elementary algebraic manipulation Dg(r)≤ 1
1− Ig(r) +
1− Hg(r). (2.118)
This gives the first bound in the Proposition. To obtain the second bound, instead of adding the absolute value of the integral in the first inequality after (2.108), we subtract it, in order to obtain a reversed inequality, and proceed further in an analogous way.
One obtains the following immediate Corollaries:
Corollary 2.3.1. There exists a positive number (threshold)t0 in the interval(0, 1) which depends only on n, Λg, Γg, τ1 and has the following property: if the restriction u|Brg(¯0) is not identically vanishing for any choice of radiusr in the interval (0, t0) (see also Remark 2.3.1 below), then one has
Hg(r) > 0, ∀r ∈ (0, t0). (2.119) Proof of Corollary 2.3.1. We fix = 12 and plug it in Proposition 2.3.4 to obtain a corresponding radius t0that depends only on Λ, n, τ1. We will show that t0 is the required threshold. To this end, let us assume the contrary, i.e.
Hg(r) = 0, (2.120)
for some radius r in the interval (0, t0). This implies that u vanishes almost everywhere on
∂Brg(¯0). Furthermore, since u is in Wloc2,2(B1g(¯0)), it is in particular, also in the space W2,2(Brg(¯0)).
Furthermore, our assumptions on the metric g dictate that the leading order coefficients, which appear in front of the partial derivatives ∂iu in the expression ω(x)|∇gu|2g, are Lipschitz, hence absolutely continuous, whose derivatives are also square integrable. This allows for integration by parts which yields
Ig(r) = ˆ
Brg(¯0)
ω(x)|∇gu|2g+ (bg· ∇gu)u + cgu2dx (2.121)
= 0 + ˆ
Bgr(¯0) − divg ω(x)|∇gu|2g + (bg· ∇gu) + cgu udx (2.122)
= 0. (2.123)
The last integral is vanishing, due to the fact that ˆ
Bgr(¯0) − divg ω(x)|∇gu|2g + (bg· ∇gu) + cgu φdx = 0, (2.124) for an arbitrary test function φ in C0∞(Brg(¯0)) and one can find a sequence
{φn}∞n=1⊂ C0∞(Br(¯0)), φn →L2(Brg(¯0))u. (2.125) Now the choice of t0 in combination with the first bound in Proposition 2.3.4 also yields
Dg(r) = 0. (2.126) Using the Heisenberg Uncertainty as in Lemma 2.3.4 this yields
u≡ 0, (2.127)
in the ball Bgr(¯0) - a contradiction with the non-identically vanishing assumption on u.
Remark 2.3.1. We will see that the ”clumsily-formulated” non-vanishing assumption on u in Corollary 2.3.1 can be replaced by requiring thatu is not identically vanishing on Btg0(¯0).
Indeed, with this requirement, let us assume the contrary, i.e. Hg(r) vanishes for some radius r in(0, t0) and suppose that ˆr is the supremum of all such r.
First, the proof of Corollary2.3.1 implies that u identically vanishes on the ball Brgˆ(¯0). Thus, ˆ
r < t0 and, in particular, Hg(r) is positive on (ˆr, t0), so the generalized frequency Ng(r) is well-defined there.
Second, in the statements below, we will establish a doubling condition by means of the frequency function, which tells us thatHg(r1) controls Hg(r2) for radii r1< r2. This will imply thatu vanishes identically onBt0(¯0), a contradiction.
Such arguments seem common in the study of unique continuation principles.
From now on, we will assume that Hg(r) is positive on a given interval (0, t0), where t0 is the threshold from Corollary 2.3.1. For this to hold, as pointed out in Remark 2.3.1, it is sufficient that u is not identically vanishing on Bgt0(¯0).
We also have a bound on the generalized frequency function indicating how small it can become.
It turns out that it is ”almost non-negative”. Namely,
Corollary 2.3.2. Let be an arbitrary number in (0, 1). There exists a number ρ0 in (0, 1) depending only on n, Λg, Γg, such that for any number r in (0, ρ0),
Ng(r)
r ≥ −. (2.128)
Proof. Let r0be the number outputted from Proposition 2.3.4 with respect to . We set
ρ0:= min(r0, t0), (2.129)
where t0 is the number from Corollary 2.3.1. We claim that ρ0 satisfies the required property.
Indeed, the frequency function is well-defined on (0, t0) and the bounds from Proposition 2.3.4 imply
Ng(r)
r = Ig(r)
Hg(r)≥ (1− )Dg(r)− Hg(r)
Hg(r) ≥ −. (2.130)