11
$\begingroup$

As the title says, what exactly is a degenerate pde? I think it's a pde where it has a coefficient which shoots off to infinity at a point. Why is it that people use weighted Sobolev spaces to show existence? Can I get a Reference to where such a existence theorem is given in weighted Sobolev spaces? The most basic one.

I would appreciate a parabolic setting as opposed to just an elliptic case. Thanks.

$\endgroup$

1 Answer 1

25
$\begingroup$

To every second order elliptic PDE $$Lu = f$$on $\Omega \subset \mathbb{R}^n$, where $L$ is an elliptic operator and $f$ a measurable function, is associated a positive-definite, bounded, symmetric coefficient matrix $$Q(x) = [q_{ij}(x)], i,j = 1,\ldots n.$$ That is, the quadratic form $$\mathcal{Q}(x,\xi) = \xi^\top Q(x) \xi$$ satisfies, for some $c,C > 0$ $$c|\xi|^2 \leq \mathcal{Q}(x,\xi) \leq C|\xi|^2$$ for almost every $x \in \Omega$ and every $\xi \in \mathbb{R}^n$. It has been shown by various authors (but most fundamentally De Giorgi, Nash and Moser) that in the presence of local Sobolev and Poincare inequalities, and with the existence of an accumulating sequence of Lipschitz cutoff functions, that weak solutions to $Lu = f$ exist and are Holder continuous. As is well known, weak solutions live in Sobolev spaces, hence their importance to the subject.

Now, suppose we relax the condition on the quadratic form. Call $Lu = f$ degenerate elliptic if we have only that $$0 \leq \mathcal{Q}(x,\xi) \leq C|\xi|^2.$$ In particular, we say that $Q$ degenerates at $x \in \Omega$ if there exists $\xi \neq 0$ such that $\mathcal{Q}(x,\xi) = 0$. As it turns out, allowing that the quadratic form vanish causes major difficulties in adapting the theory of weak solutions. Weak solutions still exist, but the do NOT live, any longer, in Sobolev spaces. Instead, as has been proved by Sawyer and Wheeden (2009) and Rodney (2012), they reside in degenerate Sobolev spaces, which are quite a bit more difficult to deal with.

These spaces are defined with reference to the particular matrix $Q$ with which we are working. So given such a matrix $Q$, define the (possibly infinite) norm $$||w||_{QH^{1,p}(\Omega)} = \left( ||w||_p^p + \int_\Omega |\nabla w^\top Q(x) \nabla w|^\frac{p}{2} dx\right)^\frac{1}{p}$$ on $Lip_{loc}(\Omega)$, the space of locally Lipschitz functions. Note that the gradients of such functions exist almost everywhere by the Rademacher-Stepanov theorem. We define the degenerate Sobolev space $QH^{1,p}(\Omega)$ as the completion of the $$\{w \in Lip_{loc}(\Omega) : ||w||_{QH^{1,2}(\Omega)} < \infty\}$$ analogously to the (secondary) definition of classical Sobolev spaces, but the the gradient norm weighted to the matrix $Q$.

As a remark, when you are reading about this subject in Sawyer and Wheeden's paper, for example, they call the spaces as defined this way $W^{1,p}_Q (\Omega)$, even though this goes against the names given to the classical spaces.

$\endgroup$
2
  • $\begingroup$ That was a really nice comment. I lack basic intuition on elliptic operators and Sobolev spaces. I know the embedding theorems and regularity results of Ch. 5 and 6 in Evans but have difficulty telling the story. Do you have the time to share some thoughts on that? Thanks a lot once again. $\endgroup$
    – snar
    Commented Nov 13, 2013 at 6:33
  • $\begingroup$ Wonderful answer, and I apologse for taking so long to see it. $\endgroup$
    – soup
    Commented Dec 12, 2013 at 21:51

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .