In this short note we prove that, given a $\mathbb{C}$-elliptic operator $\mathcal{A}$ and a map $u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V)$ satisfying \( \nabla_{\mathrm{ap}}u\in L^1(\Omega;V\otimes\mathbb{R}^d), \) then $u\in \mathrm{BV}(\Omega;V)$. The result is quantitative and follows from the Korn-type estimate \[ |Du|(\Omega) \leq C_{d,\mathcal{A}}\left( \|\nabla_{\mathrm{ap}}u\|_{\mathrm{L}^1(\Omega)} + |\mathcal{A}u|(\Omega) \right), \] valid for every such $u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V)$. As a consequence, we obtain the characterization \[ \mathrm{BV}^{\mathcal{A}}(\Omega;V)\setminus \mathrm{BV}(\Omega;V) = \left\{ u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V): \nabla_{\mathrm{ap}}u\notin L^1(\Omega;V\otimes\mathbb{R}^d) \right\}. \] Generative AI has been exploited. The usage is detailed in a specific Section.
We study the Stokes operator with no-slip boundary conditions on the spaces $\mathrm{L}_{\sigma,n}(\Omega)$ and ${\mathrm{L}^1(\Omega,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})}$, where $\Omega\subset\mathbb{R}^d$ is an arbitrary bounded $\mathrm{C}^{1,\alpha}$-domain. We show that the Stokes operator o...
Let $u:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ be a smooth nonconstant harmonic map. We prove that its Morse index is equal to $ 2n+2$ if and only if \begin{equation*} u = h \circ \pi \circ \Xi, \end{equation*} where $\Xi:\mathbb{S}^{2n+1}\to\mathbb{S}^{2n+1}$ is an isometric transformation, $\pi:\mathbb{S}^{2n+1}\to\mathbb...
For a matrix $\gamma\in\mathrm{SL}_2({\mathbb Z})$, we define \[ {\mathcal R}(\gamma)=a_1^2+a_2^2+a_3^2+a_4^2, \qquad \text{where} \ \gamma=\begin{pmatrix}a_1&a_2\\ a_3&a_4\end{pmatrix}, \] and let $S_{\mathrm{sq}}(X)$ count the number of matrices $\gamma$ with $\|\gamma\|_\infty = \max\{|a_1|, |a_2|,|a_3|,|a_4|\} \leq...
Let $\mathrm{D}$ be a quaternion division algebra over a non-Archimedean local field $\mathrm{F}$ of characteristic zero, and let $\mathrm{G}_n=\mathrm{GL}_n(\mathrm{D})$. Let $\mathrm{H}_{1,n-1}=\{\mathrm{diag}(g_1,g_2)\in\mathrm{G}_1,\ g_2\in\mathrm{G}_{n-1}\}$ and, for $s \in \mathbb{R}$, define $\chi_s(\mathrm{diag...
Let $\mathcal{X}$ be a complex manifold and $X$ a compact complex manifold with boundary in $\mathcal{X}$. For a complex Lie group $G$ and a regularity class $$\mathfrak{r}\in \big\{\mathcal{C}^k|\ k\in\mathbb{N}\cup\{\infty\}\big\}\cup\big\{\Lambda^r_{\rm loc}|\ r\in (0,\infty)\big\} $$ we define the sheaf of groups $...
We characterize the standard modules of $\GL_{2m}(\C)$ that are distinguished by $\GL_m(\HH)$. Let $\delta_1,\dots, \delta_{2m}$ be characters of $\mathbb{C}^\times$. Assume that $S = \delta_1 \times \cdots \times \delta_{2m}$ is a standard module of $\GL_{2m}(\mathbb{C})$. For each $i$, define ${\delta_i^*} (z) = \del...
A new computational approach reveals how cells communicate to assemble into tissues. This could accelerate stem cell engineering, improve disease models, and inform therapies.
MIT News · Artificial Intelligence· news.mit.eduOct 15, 2026
