• Minkowski inequality for mean convex domains

The classical Minkowski inequality states that, $\forall \Omega\subset \mathbb R^{n+1}$ convex with $C^2$ boundary, \begin{align} \int_{\partial \Omega} H\ge C_n |\partial \Omega|^{\frac{n-1}{n}}, \label{MI} \end{align} where $H$ is the mean curvature, $c_n$ is a dimensional constant. Equality holds if and only if $\Omega$ is a round ball.

\medskip ...

Open

Posted online: 2022-10-12 18:51:50Z by Pengfei Guan17

• Analysis of PDEs
• Differential Geometry
• Gradient estimate for a locally constrained flow in $\mathbb H^{n+1}$

The following hypersurface flow in $\mathbb H^{n+1}$ was introduced by Brendle-Guan-Li: \begin{align} X_t=\Big(\cosh(\rho) \frac{\sigma_{k}}{\sigma_{k+1}}(\kappa)-\frac{u}{c_{n,k}}\Big)\nu, \label{flow} \end{align} where $\nu$ the outer normal, $\kappa$ the principal curvature vector. The flow preserves $k$-th quermassintegral $\mathcal{A}_{k}$ and decreases $k+1)$-th quermassintegral $\mathcal{A}_{k+1}$. All regularity estimates and convergence would follow if one can establish the gradient estimate (or preservation of starshapedness) for solution of the flow. In turn, Alexandrov-Fenchel inequality for quermassintegrals in hyperbolic space can be established for general starshaped $k+1$-domains. ...

Open

Posted online: 2022-10-12 18:55:53Z by Pengfei Guan4

• Analysis of PDEs
• Differential Geometry
• Lipschitz regularity in vectorial nonlinear transmission problems

Consider nonlinear transmission systems, $$\hbox{div} (A(\nabla u)\chi_{D^c} + B(\nabla u)\chi_D) = 0,\tag{1}$$ where $u:B_1\subset {\Bbb R}^n \to {\Bbb R}^m$, and both $A$ and $B$ are strongly elliptic, nonlinear operators. It is well-known that nonlinear systems do not have Lipschitz solutions, in general, even if $A = B$ and the dependence on $\nabla u$ is smooth. This remains true even for minimisers of a nonlinear functional, see [2]. It is also known that the boundary regularity fails for nonlinear systems, even if the boundary data is smooth, see e.g., [3]. However, if we assume that $u$ is Lipschitz up to $\partial D$, then the Lipschitz regularity may have some chances of propagating to the other side, in some small neighborhood, depending on the geometry of $\partial D$. This is because the governing system yields a matching condition of the normal derivatives of $u$ on $\partial D$: formally, $$A_i^\alpha (\nabla u|_{D^c})\nu_\alpha + B_i^\alpha (\nabla u|_D)\nu_\alpha = 0,$$ whenever the outward normal $\nu$ is defined on $\partial D$. This may leave us in a better situation than a Dirichlet boundary problem, since for the latter problem the normal derivatives of the solution does not need to match those of the boundary data. ...

OpenYear of origin: 2022

Posted online: 2022-05-08 17:45:53Z by Henrik Shahgholian54

• Analysis of PDEs
• Constant mean curvature spheres in homogeneous 3-manifolds

Here are questions on constant mean curvature (CMC) spheres which arose from the study of the isoperimetric problem in three-dimensional, simply connected, non-compact homogeneous manifolds $X$ diffeomorphic to $\mathbb{R}^3$. ...

OpenYear of origin: 2021

Posted online: 2022-01-28 19:38:50Z by Joaquín Pérez34

• Differential Geometry
• Open problems in Free boundaries

Free Boundary Problem refers to an a priori unknown interface, along which a possible phase transition or a qualitative change in the given equations occurs. The subject area has developed in the last 50 years, and has found new branches of directions.

Recently several new research directions have arrived, with many new and challenging problems. ...

OpenYear of origin: 2012

Posted online: 2022-01-25 11:17:43Z by Henrik Shahgholian401

• Analysis of PDEs
• On the Hodge spectra of lens spaces

In [1], R. Miatello suggests the following problems:

1) Construct congruence lattices which are norm$_1$ and norm$_1*$- isospectral in all dimensions (see [2]).

2) Are there families of $p$-isospectral lens spaces for all $p$, with more than two elements? ...

OpenYear of origin: 2021

Posted online: 2022-01-25 05:15:32Z by SciLag Admin9

• Differential Geometry
• Isoperimetric Problem in ${\Bbb C}P^2$

In the open problem notes, [1], Problem 15, Frank Morgan proposed the following problem.

Statement: Prove that geodesic spheres provide the least-perimeter way to enclose prescribed volume in $\mathbb{C} P^2$.

For further references see [2]. ...

UnconfirmedYear of origin: 2021

Posted online: 2022-01-24 07:26:33Z by SciLag Admin15

• Differential Geometry
1. 1
2. ...
3. 16
4. 17