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For $n\ge 3$, $00$. As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation $u_t=\frac{n-1}{m}\Delta u^m$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$ with initial value $u_0$ satisfying $f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x)$, $\forall x\in\mathbb{R}^n\setminus\{0\}$, which satisfies $U_{\lambda_1}(x,t)\le u(x,t)\le U_{\lambda_2}(x,t)$, $\forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0$, for some constants $\lambda_1>\lambda_2>0$. We also prove the asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $t\to\infty$ when $n=3,4$ and $\frac{n-2}{n+2}\le m<\frac{n-2}{n}$ holds. Asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $t\to\infty$ is also obtained when $3\le n<8$, $1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right)$, and $u(x,t)$ is radially symmetric in $x\in\mathbb{R}^n\setminus\{0\}$ for any $t>0$ under appropriate conditions on the initial value $u_0$., Comment: 34 pages

We demonstrate quantum dot (QD) formation in three-dimensional Dirac semimetal Cd$_{3}$As$_{2}$ nanowires using two electrostatically tuned p$-$n junctions with a gate and magnetic fields. The linear conductance measured as a function of gate voltage under high magnetic fields is strongly suppressed at the Dirac point close to zero conductance, showing strong conductance oscillations. Remarkably, in this regime, the Cd$_{3}$As$_{2}$ nanowire device exhibits Coulomb diamond features, indicating that a clean single QD forms in the Dirac semimetal nanowire. Our results show that a p$-$type QD can be formed between two n$-$type leads underneath metal contacts in the nanowire by applying gate voltages under strong magnetic fields. Analysis of the quantum confinement in the gapless band structure confirms that p$-$n junctions formed between the p$-$type QD and two neighboring n$-$type leads under high magnetic fields behave as resistive tunnel barriers due to cyclotron motion, resulting in the suppression of Klein tunneling. The p$-$type QD with magnetic field-induced confinement shows a single hole filling. Our results will open up a route to quantum devices such as QDs or quantum point contacts based on Dirac and Weyl semimetals.

In this paper, we prove the existence and uniqueness of $W^{2,p}$ ($n

We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and H\"{o}lder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is H\"{o}lder continuous., Comment: 19 pages

We prove the natural weighted Calder\'{o}n and Zygmund estimates for solutions to elliptic and parabolic obstacle problems in nondivergence form with discontinuous coefficients and irregular obstacles. We also obtain Morrey regularity results for the Hessian of the solutions and H\"{o}lder continuity of the gradient of the solutions., Comment: 18 pages

In this paper, we prove local $C^{1}$ regularity of free boundaries for the double obstacle problem with an upper obstacle $\psi$, \begin{align*} \Delta u &=f\chi_{\Omega(u) \cap\{ u< \psi\} }+ \Delta \psi \chi_{\Omega(u)\cap \{u=\psi\}}, \qquad u\le \psi \quad \text { in } B_1, \end{align*} where $\Omega(u)=B_1 \setminus \left( \{u=0\} \cap \{ \nabla u =0\}\right)$ under a thickness assumption for $u$ and $\psi$., Comment: 21 pages