2.3 A generalized frequency function
2.3.1 Finding appropriate coordinates
n
X
i=1
bi(x)∂u
∂xj
φ + c(x)uφ dx = 0, (2.65)
where φ is an arbitrary test function from the Sobolev space W01,2(Ω). Via elliptic regularity, it is well-known that such a weak solution u is in the space Wloc2,2(Ω) (cf. Theorem 8.8, [GT01]).
2.3.1 Finding appropriate coordinates
Before we introduce the generalized frequency function, we first make an appropriate coordinate change, tailored along the matrix A(x). This transformation will actually reduce the operator L to an operator with diagonal leading coefficient matrix. We note that such transformations are a standard tool in unique continuation arguments (cf. [GL86], [GL87], [AKS62], etc).
For n at least 3 (the case n = 2 can be handled by an appropriate isothermal coordinate system, but we do not pursue this here), we define the metric {egij}ni,j=1 on the unit ball B1(¯0), whose components are given by:
egij(x) := (det A(x))n−21 aij(x). (2.66) We have
Lemma 2.3.1. The metric eg is Lipschitz whose Lipschitz constant el depends only on n, Γ, Λ.
Moreover,
diveg(∇egu) = (det A)−n−21 div(A∇u), (2.67) where the operatorsdiv,∇ on the right hand side are taken with respect to the Euclidean metric.
Proof. Concerning the Lipschitz property - the determinant det A(x) is a sum of products of bounded Lipschitz functions and the inverse is again term-wise given by cofactor matrices of a similar form. It follows thategij(x) is Lipschitz with Lipschitz constant el, depending only on n, Γ, Λ.
By definition
eg = (det A)n−21 A−1, (2.68)
hence
eg−1= (det A)−n−21 A. (2.69)
Moreover, the exponent n−21 is chosen in such a way, that
|g| = (det A)n−2n det(A−1) = (det A)n−22 . (2.70) Now, recalling the formulae (1.10), (1.8) one concludes the needed claim.
The metriceg already diagonalizes our operator. However, we would also prefer that the geodesic balls around ¯0 are not deformed, i.e. they coincide with the geodesic balls induced by the constant metriceg(¯0). This is achieved through normal coordinates. However, we will take the point of view thateg is conformally deformed - to this end, we introduce an appropriate conformal change. First, we define the first order approximation of the distance function:
r(x) := (egij(¯0)xixj)12, (2.71) where we also apply the Einstein summation convention over repeated indices. The corresponding conformal factor we need is
f (x) :=egij(x)∂r
∂xi
(x) ∂r
∂xj
(x) = 1
r2(x) egij(x)egik(¯0)egjl(¯0)xkxl (2.72)
= (∇r)Teg−1∇r. (2.73)
Lemma 2.3.2. The functionf is a positive Lipschitz function.
Proof. Suppose that the positive numbers κ(x), K(x) are the smallest, resp. largest, eigenvalues of the matrix eg(x). By definition ofeg and the bounds on A(x), it follows that κ, K are uniformly bounded away from 0 in terms of n, η, Γ, Λ. This implies that
f (x) = xT eg(¯0)Teg(x)−1eg(¯0) x xTeg(0)x ≥
1
K(x)|eg(0)x|2 K(0)|x|2 =
1
K(x)heg(0)Teg(0)x, xi
K(0)|x|2 (2.74)
≥ κ(0)2
K(0)K(x)> 0. (2.75)
In a similar way one can also obtain an upper bound for f (x). Furthermore, we can express the difference f (x)− f(y) and use the well-known Lipschitz property of the ordered eigenvalues with respect to the matrix sup-norm (cf. also [HW53], [Wil88]) to derive the Lipschitz continuity of f .
We finally define the required conformal metric on B1(¯0) as
g(x) := f (x)eg(x). (2.76)
As usual, we denote the components of g(x) as gij(x) and{gij(x)}ni,j=1 will represent the inverse matrix. Furthermore, for every vector ξ in Rn the following bounds hold:
τ1|ξ|2≤ |ξ|2g:= ξTgξ = ξT
f (det A)n−21 A−1
ξ≤ τ2|ξ|2, (2.77) where the positive τ1, τ2 depend only on bounds on the matrix A, i.e. on η, Λ, Γ, n (the conformal factor f is also bounded in terms of the A, as we saw in Lemma 2.3.2). Moreover, we remark that τ1, τ2are close to 1 if the matrix A is close to being the identity matrix.
An important statement we will utilize is the following:
Proposition 2.3.1. For any positive number r in the interval (0, 1), the geodesic balls Brg(¯0) and Beg(¯r0)(¯0) induced by g and eg(¯0) respectively, coincide. In particular, if A(¯0) is the identity matrix, then the geodesic ballsBgr(¯0) coincide with the Euclidean balls Br(¯0).
Proof. We will show that the function r(x) defined above actually measures the Riemannian geodesic distance dg(¯0, x) with respect to the metric g. A possible way to achieve this is to determine the radial geodesics through the origin via a Christoffel symbols’ computation. However, we will determine the radial geodesics ad hoc in the spirit of [AKS62].
Let Breg(¯00)(¯0) denote the open ball centered at ¯0 and of radius r0with respect to the metriceg(¯0).
In other words,
Breg(¯00)(¯0) ={x : r(x) ≤ r0}. (2.78) If r0 is sufficiently small (depending only η, Λ), then Breg(¯00)(¯0) ⊆ Br(¯0), where the number r is a fixed number in the interval (0, 1).
Let c0 be a point in ∂Breg(¯00)(¯0) and consider the ODE:
˙c(t) = g−1∇r|c(t), c(r0) = c0, (2.79) where∇ denotes the Euclidean gradient. Using the theory of ODEs with Lipschitz right hand side, it follows that there exists a unique solution c(t) of class C1 which is defined in an open interval containing r0. By the explicit metric construction and the chain rule, we observe
d This implies that r(c(t)) = t + C for some constant C. However,
r0+ C = r(c(r0)) = r(c0) = r0, (2.82) hence C = 0. It follows that one can extend the curve c(t) at least over the interval (0, r0] (the right hand side of the ODE is sufficiently regular on this interval and one can apply the standard extension procedure there).
We can apply this construction to every point p in ∂Breg(¯00)(¯0) to get a family of disjoint simple arcs which sweep out the entire ball Breg(¯00)(¯0). Now, as the r(x) is sufficiently regular (for x6= ¯0), the level set ∂Breg(¯00)(¯0) forms an embedded smooth n− 1-dimensional manifold, upon which we can select local coordinates θ1, . . . , θn−1.
This allows us to introduce the coordinates (r, θ1, . . . , θn) on the product space (0, r0)×∂Beg(¯r00)(¯0).
Moreover, using the ODE (2.79) and the computation (2.80) we observe
gij
This means that the metric g is represented in the new coordinates as
We conclude that the lines θ = const are geodesics and, moreover, bpq(r, θ) is the restriction of the metric g on the concentric hypersurfaces ∂Beg(¯r00)(¯0) up to a factor of r12. In particular, one can also view the coordinates (r, θ) as geodesic normal coordinates.
We also need the following control on the coefficients bpq:
Lemma 2.3.3. The functions{bpq}n−1p,q=1 defined in Proposition 2.3.1 satisfy:
whereB is a positive number that depends only on n, η, Λ.
Proof. This follows essentially from the construction of the matrix b. For complete details we refer to Sections V and VI from [AKS62].
The above estimate on ∂r∂ bpq(r, θ) was also discussed in [GL86], [GL87], [HL].
To finalize the construction of the appropriate coordinates, we check how the operator L transforms.
Proposition 2.3.2. The functionu is a weak solution to Lu = 0 if and only if u is a weak solution of the following operator:
Lgu :=− divg(ω(x)∇gu) + bg· ∇gu + cgu = 0, (2.91) that is, for every test function φ in the Sobolev space W1,2(B1(¯0)) one has
ˆ
B1(¯0)
ω(x)h∇gu,∇gφig+ φbg· ∇gu + cguφ = 0. (2.92)
Hereω(x) = f−n−22 , the vector bgis given by|g|−12(gb) and cgis|g|−12c. In particular, the function ω(x) is a bounded Lipschitz function with
w1≤ ω(x) ≤ w2,
Proof. One computes
p|g| = −fn2(det A)n−21 . (2.94)
Hence, plugging into the formulae (1.10) and (1.8) one obtains:
− divg(ω(x)∇gu) + bg· ∇gu + cgu = (|g|)−12Lu, (2.95) which yields the first claim. The properties of w(x) follow from those of f (x).
These facts also imply
Proposition 2.3.3. The operator Lg satisfies the same assumptions as the operator L after an eventual modifications of the corresponding constants ηg, Λg, Γg.