Nonlinear Stein theorem for differential forms
A talk in the Oberseminar Numerik series by
Swarnendu Sil
| Abstract: | via ZOOM-Conference ID 926 5310 0938 Password: 1928 Abstract: Stein (1981) proved the borderline Sobolev embedding result which states that for $n \geq 2$, $u \in L^1(\mathbb{R}^n)$ and $\nabla u \in L^{(n, 1)}(\mathbb{R}^n;\mathbb{R}^n)$ implies $u$ is continuous. Coupled with standard Calderon-Zygmund estimates for the Laplacian in Lorentz spaces, this implies $u \in C^1(\mathbb{R}^n)$ if $\Delta u \in L^{(n, 1)}(\mathbb{R}^n)$. In this talk, we shall discuss a nonlinear analogue of this result for differential forms. In particular, we can prove that if $u$ is an $\mathbb{R}^N$-valued $W^{1,p}_{loc}$ differential $k$-form, then \begin{equation} d^{\ast} ( a(x) \vert du \vert^{p-2}du ) \in L^{(n,1)}_{loc} \Rightarrow du \textrm{ is continuous,} \end{equation} where $N \geq 1, n \geq 2, 0 \leq k \leq n - 1, 1 \le p \le 1$, and $a$ is an uniformly positive, bounded, Dini continuous scalar function in a domain of $\mathbb{R}^n$: The main dificulty is that for $1 \leq k \leq n - 1$, the system has an infinite dimensional kernel and thus, strictly speaking, not elliptic. |