Related papers: Radially Symmetric Solutions To The Graphic Willmo…

We study the axisymmetric self-similar solutions $(\mathbf{u},\mathbf{B})$ to the stationary MHD equations without magnetic diffusion, where $\mathbf{B}$ has only the swirl component. Our first result states that in…

In this paper, we establish symmetry results for solutions of the mean field equation \[ \frac{\alpha}{2} \Delta u + e^u - 1 = 0 \] on \( \mathbb{S}^2 \) for $\frac{1}{3}\leq \alpha < \frac{1}{2}$.The proofs utilize the Sphere Covering…

The Gauss map of non-degenerate surfaces in the three-dimensional Minkowski space are viewed as dynamical fields of the two-dimensional O(2,1) Nonlinear Sigma Model. In this setting, the moduli space of solutions with rotational symmetry is…

We study rotational surfaces in Euclidean 3-space whose Gauss curvature is given as a prescribed function of its Gauss map. By means of a phase plane analysis and under mild assumptions on the prescribed function, we generalize the…

The Willmore energy plays a central role in the conformal geometry of surfaces in the conformal 3-sphere \(S^3\). It also arises as the leading term in variational problems ranging from black holes, to elasticity, and cell biology. In the…

This paper is devoted to the study of semi-stable radial solutions $u\notin H^1(B_1)$ of $-\Delta u=f(u) \mbox{in} \overline{B_1}\setminus \{0\}=\{x\in \mathbb{R}^N : 0<\vert x\vert\leq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We…

Of interest in this note is the following geometric interesting equation $\Delta^2 u + u^{-q} = 0$ in $\mathbb R^3$. It was found by Choi-Xu (J. Differential Equations 246, 216-234) and McKenna-Reichel (Electron. J. Differential Equations…

This paper is devoted to the study of stable radial solutions of $-\Delta u=f(u) \mbox{ in } \mathbb{R}^N\setminus B_1=\{ x\in \mathbb{R}^N : \vert x\vert\geq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We prove that such solutions…

We consider the three-dimensional radial Stefan problem which describes the evolution of a radial symmetric ice ball with free boundary \begin{equation*} \left\{\begin{aligned} &\partial_{t}u-\partial_{rr}u-\frac{2}{r}\partial_{r}u=0 \quad…

We prove that given any smooth metric $\gamma$ and smooth positive function $H$ on $S^{2}$, there is a constant $\lambda > 0$, depending on $(\gamma, H)$, and an asymptotically flat solution $(M, g, u)$ of the static vacuum Einstein…

We consider radially symmetric solutions for a class of resonant problems on a unit ball $B \subset R^n$ around the origin \[ \Delta u+\la _1 u +g(u)=f(r) \s \mbox{for $x \in B$}, \s u=0 \s \mbox{on $\partial B$} \,. \] Here the function…

We consider embedded ring-type surfaces (that is, compact, connected, orientable surfaces with two boundary components and Euler-Poincar\'{e} characteristic zero) in ${\bold R}^3$ of constant mean curvature which meet planes $\Pi_1$ and…

We consider a free boundary problem for the Willmore functional. Given a smooth domain $\Omega$ in ${\mathbb R}^3$, we construct Willmore disks wich are critical in the class of surfaces meeting $\partial \Omega$ orthogonally along their…

Due to the isotropy of $d$-dimensional hyperbolic space, one expects there to exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. The $R$-radius hyperboloid model of hyperbolic geometry…

A smooth end of a Bryant surface is a conformally immersed punctured disc of mean curvature 1 in hyperbolic space that extends smoothly through the ideal boundary. The Bryant representation of a smooth end is well defined on the punctured…

The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schr\"odinger equation $$ i u_t +\Delta u+|x|^{-b}|u|^2 u = 0, $$ where $0<b<1/2$. Let $Q$ be the ground state solution of $-Q+\Delta Q+ |x|^{-b}|Q|^{2}Q=0$ and…

We study the structure of the branch set of solutions to Plateau's problem in metric spaces satisfying a quadratic isoperimetric inequality. In our first result, we give examples of spaces with isoperimetric constant arbitrarily close to…

We study stable solutions of the following nonlinear system $$ -\Delta u = H(u) \quad \text{in} \ \ \Omega$$ where $u:\mathbb R^n\to \mathbb R^m$, $H:\mathbb R^m\to \mathbb R^m$ and $\Omega$ is a domain in $\mathbb R^n$. We introduce the…

In this paper we study the Plateau problem for disk-type surfaces contained in conic regions of $\mathbb{R}^{3}$ and with prescribed mean curvature $H$. Assuming a suitable growth condition on $H$, we prove existence of a least energy…

In this paper, the spectrum of the following fourth order problem \begin{equation*} \begin{cases} \Delta^2 u+\nu u=-\lambda \Delta u &\text{in } D_1,\newline u=\partial_r u= 0 &\text{on } \partial D_1, \end{cases} \end{equation*} where…