1. A motivating example
Consider the -Laplace equation
where is a nonempty bounded open subset of , and on . The energy functional associated with the PDE is
For , because of convexity, has a unique minimizer which solves the -Laplace equation.
Now, suppose that for some function as (in certain sense). Does solve any PDE?
First note that by some computations, for ,
Formally, solves . Dividing both sides by , the equation becomes
Letting , should solve .
Moreover, if we define (which is essentially the same as above), then as . So formally converges to a minimizer of . But does the above discussion really make sense? For example, might make no sense because the domains of ‘s are different!
To rigor above discussion, our goal is to develop a theory of functional convergence such that
- converges to which is a minimizer of ; and even better (if possible)
- solves a “limiting” equation.
We will do 1 only. For 2, one may refer to the paper “Extension of functions satisfying Lipschitz conditions” by Gunnar Aronssov.
Definition (-convergence). Let and be a family of topological spaces. A family of functionals (defined on ) -converges to a functional (defined on ) if the following hold:
- (Compactness) If , where , then up to extraction, for some .
- (Lower bound) If for some (), then .
- (Recovery sequence) For any , there exist such that and .
Note that in the above definition, we do not specify what is meant by . It has to be specified in each application. Also, we may replace the in 3 by because of 2.
Now we prove the fundamental theorem, and probably the only theorem of -convergence:
Theorem 1. If -converges to and if is a minimizer of with , then up to extraction, and is a minimizer of .
Proof. By compactness, up to extraction. By lower bound,
We claim that is a minimizer of . Let . Then there exists a recovery sequence such that and . Since minimizes , and so
Therefore minimizes .
3. Returning to our example
Now we go back to our example in the first section and see how -convergence applies.
Let be a Lipschitz function on . We define the following:
- defined on .
- defined on .
- means the weak convergence in for all .
With the above definitions, we obtain
Theorem 2. -converges to as .
Therefore, by Theorem 1, converges to a minimizer of .
Proof. The recovery sequence is very easy to construct. If , then for all (recall that the domain is bounded). Therefore, we can take the recovery sequence to be just the constant sequence .
We now prove the compactness. Suppose with . Fix . By Hölder’s inequality, if ,
Therefore, is bounded in (you might need to think carefully why). Thus, we can extract a subsequence, still called for simplicity, converging weakly to with on . We can repeat this argument for , , so that in each step, we select the weakly convergent sequence as a subsequence of the previous sequence. Taking the diagonal sequence, we conclude that there exists a subsequence such that in for all . This implies for all . Now, by lower semicontinuity,
Taking , . That is, .
It remains to show the lower bound condition holds. Suppose such that for some . That is, in for all . Then same argument as before shows
Let , we get .