Section 3.1 Ellipticity and uniform ellipticity
acting on functions via
Here as always the summation convention from Notation 2.1 is in force, so that repeated indices are understood to summed from through
Throughout this chapter we assume that the coefficients are bounded functions on Moreover, we assume that so that the matrix is symmetric.
Definition 3.1. Ellipticity.
for all is uniformly elliptic if (3.2) holds with for some constant independent of We call an ellipticity constant for
Comparing with Definition 2.30 and Corollary 2.29, we see that is elliptic if and only if the symmetric matrix is positive definite for all and uniformly elliptic if the minimum eigenvalue of is for all
Exercises Exercises
1. (PS4) Checking uniform ellipticity.
(a)
Show that is a uniformly elliptic operator for any open
Solution.
We have and hence
for any Thus is uniformly elliptic with ellipticity constant
(b)
For any show that is not elliptic.
Hint.
One option is to work with (3.2) directly. Another option is to calculate the eigenvalues and get (3.2) from Corollary 2.29.
Solution.
We have and hence
which is but vanishes for instance when Thus is not elliptic.
Alternatively, we could check that one of the eigenvalues of the matrix
is which is not positive. A slick way to do this is to just take the determinant and recall how this is related to the eigenvalues.
Comment.
Since when acting on functions, many authors will abbreviate the operator in this problem as
This does not mean that they consider the non-symmetric matrix
as this would violate our assumption Perhaps more importantly, calculating the eigenvalues of this non-symmetric matrix will no longer provide you with any information about the ellipticity of
(c)
Let be bounded functions on Show that
is uniformly elliptic.
Hint.
Same as for previous part.
Solution.
In this case we have, for any
Thus is uniformly elliptic with ellipticity constant
Alternatively, one can simply calculate the eigenvalues of the relevant matrix
by brute force. They are and which again implies uniform ellipticity with
(d)
Show that is elliptic on but not uniformly elliptic.
Solution.
The eigenvalues of the matrix
are clearly and both of which are strictly positive for any fixed Thus is elliptic. On the other hand, as the second eigenvalue and so this operator cannot be uniformly elliptic.
Comment 1.
To show that is not uniformly elliptic it is not enough to show that the smallest eigenvalue of is or equivalently that Indeed, these properties also hold for the Laplacian which is uniformly elliptic on To disprove uniform ellipticity need the reverse inequality that the smallest eigenvalue of is or alternatively that for we get
Comment 2.
Note that, because of our choice of the coefficients of are indeed bounded functions.
2. (PS4) Ellipticity after shifting and scaling.
Consider a differential operator
acting on functions in for some and Given define by
as in Exercise 2.3.1.
(a)
Find a second-order differential operator so that
Hint.
Keep Notation 2.14 in mind! Calculate in terms of derivatives of using the chain rule and/or Exercise 2.3.1.
Solution.
Comment.
In problems like this it is important to keep track of where the functions and coefficients are being evaluated. This year, several students took
but this operator has
which is not quite what we want.
(b)
Suppose that is uniformly elliptic with ellipticity constant Show that is also uniformly elliptic. What is the ellipticity constant?
Solution.
Let and Then and so by our above formulas and the ellipticity of we have
Since and were arbitrary, we conclude that is uniformly elliptic with ellipticity constant (The boundedness and symmetry of the coefficients of easily follow from the corresponding properties of the coefficients of )
3. Divergence form operators.
Show that the operator appearing in Exercise 2.9.3 can be written in the form (3.1) for an appropriate choice of and Recall that the coefficients are assumed to be
4. The symmetry assumption.
Consider an operator where the coefficients are bounded functions on but is not necessarily symmetric. Show that for any we have where is an operator with symmetric second-order coefficients