We study the Calder\'on problem for the equation $-\Delta_\infty u+q(x)u=0$ in a bounded convex domain with $C^2$ boundary. We prove that a positive potential $q$ is uniquely determined and explicitly reconstructible from the measurements $\Lambda_q\bigl(t(e\cdot x+b)|_{\partial\Omega}\bigr)$, where $e\in\mathbb S^{n-1}$, the offset $b>\sup_{\overline\Omega}|x|$ is fixed, and $t\to\infty$. The first potential-dependent term in the large amplitude asymptotics determines weighted integrals of $q$ over the chords parallel to $e$. Combining the measurements in the directions $e$ and $-e$ gives the X-ray transform of the zero extension of $q$, and the Fourier slice identity yields reconstruction and uniqueness.
Let $\Omega\subset\mathbb R^2$ be a smooth bounded domain containing the origin and invariant under reflection across the coordinate axes, and let $0<\lambda<\lambda_1(\Omega)$, where $\Lambda_1 (\Omega)$ is the first eigenvalue for $-\Delta$ on $\Omega$ under Dirichlet boundary conditions. For every fixed integer $k\g...
M. del Pino, Ignacio A. Guerra, M. Musso· 0 citations
Let $1 \leq q<p$ and let $\lambda_1$ be the first eigenvalue of the $p$-Laplacian in a bounded domain $\Omega$. We study the energy functional $$ E_\lambda(u)=\frac{1}{p}\left(\int_\Omega|\nabla u|^p\,dx -\lambda\int_\Omega|u|^p\,dx\right)-\mathcal{F}(u), \quad u\in W_0^{1,p}(\Omega), $$ where $\mathcal{F}$ is positive...
Let $\Omega$ be an open bounded connected subset of $\mathbb{R}^N$, $N \geq 2$, of class $C^{2,\alpha}$, for $\alpha \in (0,1)$. Let $p \geq 2$ and $\beta>0$. We prove the symmetry of the solution of the $p$-torsion problem with Robin boundary condition subject to the natural overdetermined condition coming from a shap...
In this work, we conduct a comprehensive study of problem \begin{equation*} \begin{cases} -\Delta_1 u + g(u)|Du| = h(u)f&\text{in }\Omega, u=0&\text{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^N$ is a bounded Lipschitz domain, $f\in L^1(\Omega)$ is a nonnegative datum, and $g,h$ are...
We prove a H\"ormander multiplier theorem for the Dunkl transform associated with an arbitrary finite reflection group. Uniform $H^\sigma(\mathbb{R}^N)$ bounds for the normalised dyadic pieces of a measurable symbol, with $\sigma>\mathbf N/2$ and $\mathbf N$ the homogeneous dimension, imply $L^p(d\omega)$ boundedness f...
Der-Chen Chang, Ji Li, Chao-Jie Wen et al.· 2 citations· ⚡1
We study the following curvature equation with two singular sources on a flat torus $$ \Delta u+e^u=8\pi(\delta_p+\delta_{-p})\quad\text{on } E_\tau:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau), $$ where $\delta_p$ denotes the Dirac measure at $p$. The case $\wp'(2p)=0$ was solved by Kuo (J. Differential Geom. 2026). In thi...
Zhi-Jie Chen, Zhen Song· 0 citations
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