Joint Event: Math Department Colloquium & SASA Seminar
- David Cruz-Uribe
- Professor and Chair
- University of Alabama
- Date: Sep. 04, Friday, 2026
- Time: 15:30-16:30
- Host: Bingyang Hu
- Room: STEM Ag 2122
- Abstract: In the background of our talk are the classical results due to Nash, Moser, de Georgi, Trudinger, and others: if \(Q\) is a uniformly elliptic matrix and \(f \in L^2(\Omega)\), then there exists a weak solution \(u\) of the Dirichlet problem, $$ \begin{cases} -\operatorname{Div}\left(Q\nabla u\right) = f, & x\in\Omega, \\ u = 0, & x\in\partial\Omega. \end{cases} $$
Moreover, if we further assume that \(f\in L^q(\Omega)\), \(q>\frac{n}{2}\), then solutions are bounded functions. They also satisfy a Harnack inequality, and as a consequence, the solutions are locally Hölder continuous. Corresponding results hold if we include lower order terms in the differential equation.
Since the 1970s, a number of authors have attempted to extend these results to degenerate elliptic equations: that is, equations when the matrix \(Q\) is no longer uniformly elliptic. Important work was done by Fabes, Kenig, and Serapioni (1982), Chanillo and Wheeden (1986), Franchi, Lu and Wheeden (1995), Sawyer and Wheeden (2010), and Korobenko, Rios, Sawyer and Shen (2021).
In this talk we will discuss our ongoing project to systematically develop the theory of degenerate elliptic equations
$$ \begin{cases} Lu = f + v^{-1}\operatorname{Div}(v \mathbf{e} h), & x \in \Omega, \\ u = 0, & x \in \partial \Omega, \end{cases} $$where
$$ Lu = v^{-1} \operatorname{Div}(Q\nabla u) + \mathbf{b} \cdot \nabla u + v^{-1}\operatorname{Div}(v\mathbf{c} u) + du, $$and \(Q\) satisfies the degenerate ellipticity condition
$$ w(x)|\xi|^2 \leq \langle Q\xi,\xi\rangle \leq v(x) |\xi|^2, \qquad \xi \in \mathbb{R}^n. $$We will first briefly review the history of the problem, and then give sufficient conditions on the weights \(v\) and \(w\), the matrix \(Q\), and the coefficients for there to exist (unique) solutions to this equation. At the heart of our results is the assumption of a global degenerate Sobolev inequality: for instance, an inequality of the form
$$ \bigg( \int_\Omega |\varphi|^{2\sigma} v\,dx\bigg)^{\frac{1}{2\sigma}} \leq \bigg( \int_\Omega |\sqrt{Q} \nabla \varphi|^2\,dx\bigg)^{\frac{1}{2}}. $$This approach, while more abstract, makes clear what is needed to establish our results.
We will then discuss further properties of solutions, restricting the operator \(L\) to the second order term. We first discuss the boundedness of solutions. We show how the assumption of a global Sobolev inequality lets us adapt de Georgi iteration to prove given sufficient integrability on the right-hand side, solutions are bounded.
We will then discuss the problem of proving the continuity of solutions, including some counter-examples and a conjecture by de Georgi. We will conclude our talk with some very recent work where we have established a degenerate Harnack inequality in the special case when \(Q=v\tilde{Q}\), where \(\tilde{Q}\) is a uniformly elliptic matrix. Using this, we have proved the partial regularity of solutions, and as a consequence, shown that we can considerably weaken the hypotheses of the original work of Fabes, Kenig and Serapioni.
This research is in collaboration with Scott Rodney, Cape Breton University, Sydney, Canada; Yusuf Zeren, Yıldız Technical University, Istanbul, Türkiye; and our students Şeyma Çetin and Feyza Elif Dal.