Degenerate elliptic operators
Ugo Boscain,[email protected]
We study some degenerate elliptic operators. These operators have a geometric origin explained in the last section.
1. First type of degeneracy
Consider the following degenerate heat equation onR2×[0,∞[
(1) ∂tφ(x, y, t) = ((∂x)2+x2(∂y)2)φ(x, y, t)
1.1. Numerical implementation. Consider the initial condition φ(x, y,0) = 1B(x
0,ε)
where 1B(x
0,ε) is a function equal to 1 on the ε-ball centered at x0 and zero otherwise.
Q1.1: For x0 = (0,0) and x0 = (1,0), by using the explicit method of finite differences, computeφ(x, y, t) for (x, y)∈[−10,10]×[−10,10], for small values oftand ε, and plot it.
2. Second type of degeneracy
LetL2[1x,R] be the space of functions ξ: (0,∞)→R for which Z ∞
0
|ξ(x)|21
xdx <∞.
We study solutions of the degenerate heat equation
∂tφ(x, t) = ( ∂2
∂x2 − 1 x
∂
∂x)φ(x, t), (2)
which belong toL2[1x,R] for everyt∈[0,∞).
Q2.1: Let φ(x, t) be a function belonging to C∞([0,∞)×[0,∞),R) which is a solution of (2) on (0,∞)×[0,∞) and belong toL2[1x,R].
Find the Taylor developement ofφ(x, t) (w.r.t. x)1in a right-neighborhood ofx= 0. Say which terms vanish.
Q2.2: In L2[1x,R], find the eigenfunctions for the operator ∂x∂22 −1x∂x∂ with the boundary conditionφ(1) = 0. Draw some of them. 2
1i.e., in the formφ(x, t) =Pn
j=0aj(t)xj+O(xn+1)
2Suggestion: look for Bessel functions. The Bessel functions of first and second kind Jn(x) andYn(x) are by definition a system of solutions of the differential equation.
x2d2y dx2 +xdy
dx+ (x2−n2)y= 0 Bessel functions are implemented in scilab.
1
2
3. Second type of degeneracy: another equation Q3.1: Consider the equation
∂tφ(x, t) = ( ∂2
∂x2 −cot(x) ∂
∂x)φ(x, t).
on the space of function belonging toC∞([0, π]×[0,∞),R) and such thatRπ
0 |φ(x, t)|sin(x)1 dx <∞for every t∈[0,∞). By using a Taylor expansion similar to the one studied in Q2.1, study if the quantity Rπ
0 |φ(x, t)|sin(x)1 dxis conserved.
By using the implicit and explicit methods of finite differences, study numerically the conservation of the quantityRπ
0 |φ(x, t)|sin(x)1 dx for the initial conditionφ(x,0) = sin(x)2.
4. Second type of degeneracy for the Schroedinger equation In this section we make a similar study as the one made in Section 2, but for the Schroedinger equation (for which the solution is a complex valued function). LetL2[1x,C]) be the space of functionsξ : (0,∞)→Cfor which
Z ∞
0
|ξ(x)|21
xdx <∞.
We study solutions of the degenerate Schroedinger equation
−i∂tψ(x, t) = ( ∂2
∂x2 −1 x
∂
∂x)ψ(x, t).
(3)
which belong to L2[x1,C] for every t ∈ [0,∞). Also we assume that the derivative with respect to x of the solution belongs to this space for every t∈[0,∞).
Q4.1: Let ψ(x, t) be a solution of (3) for which in addition we have
|ψ∂xψ|/x→0 for x→0 and x→ ∞. Prove that the L2 norm with respect to the measure x1dx, (i.e. R∞
0 |ψ|2 1xdx) is conserved.
5. The Laplace Beltrami operator A Riemannian metric on an open set Ω⊂R2 is a smooth map (4) Ω3(x, y)7→g(x, y) =
a(x, y) b(x, y) b(x, y) c(x, y)
∈R2×2
where the symmetric matrix is positive definite.
If v is a vector belonging to the tangent space to Ω in a point (x, y), by definition its Riemannian norm is |v|=p
vT g(x, y)v.
Ifγ : [0, T]→Ω is a smooth curve, its Riemannian length is by definition
`(γ) = Z T
0
q
˙
γ(t)Tg(γ(t)) ˙γ(t)dt.
3
The simplest example of Riemannian manifold is a surface in R3 with the metric induced by the embedding.
With a Riemannian metric one can define the Riemannian gradient of a smooth function defined in Ω:
gradφ= (∂xφ, ∂yφ)g−1.
To a Riemannian metric, one can associate naturally a measure of volume (called Riemannian volume) as dV = √
detg dxdy. The divergence of a vector field is by definition how the flow of the vector field increases or decreases the volume and by definition is
div
X1(x, y) X2(x, y)
= 1
√detg∂x(p
detgX1(x, y))+ 1
√detg∂y(p
detgX2(x, y)).
The Laplace-Beltrami operator in Riemannian geometry plays the role of the standard Laplacian in an Ecuclidean space and it is defined as the divergence of the gradient:
∆LBφ(x, y) = div(grad(φ(x, y)).
The generalization to any dimension is straightforward.
Q5.1: on (0,∞)×R⊂R2 find the Laplace Beltrami operator for the Riemannian metric
(5) g(x, y) =
1 0 0 x12
.
One could also define a generalized Laplace-Beltrami operator by com- puting the gradient with respect to a Riemannian metric and the divergence with respect to a volume that is not the Riemannian one. In this case√
detg is replaced with another smooth and positive function.
Q5.2: Prove that the operator ∂x∂22 − x1∂x∂ is the generalized Laplace Beltrami operator in (0,∞) where the gradient is computed with respect to the standard Euclidean metric and the volume is x1dx.
Prove that the operator ∂x∂22−cot(x)∂x∂ is the generalized Laplace Beltrami operator in (0, π) where the gradient is computed with respect to the standard Euclidean metric and the volume issin(x)1 dx.
This should clarify the geometric origin of the degenerate operators stud- ied in the previous sections.
