• 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 Shahgholian38

• 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 Pérez28

• 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 Shahgholian380

• 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 Admin6

• 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 Admin10

• Differential Geometry
• Least-area soap film

In the context of a realistic model for soap films (Almgren's minimal sets), prove the existence of one of least area with a given boundary. ...

Open

Posted online: 2021-11-15 20:57:12Z by Frank Morgan24

• Differential Geometry
• Convergence problem of $a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$

let us moving to telescopic sum using exponent ,Assume we have this sequence: $a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$ with $n\geq1$ , This sequence can be written as power of sequences :${x_n} ^ {{{y_n}^{C_n}}^{......}}$ such that all them value are in $(0,1)$, We want to know if the titled sequence should converge to $1$ ? ...

OpenYear of origin: 2001

Posted online: 2021-07-11 01:00:10Z by Rafik Zeraoulia585

• Dynamical Systems
• Mathematical Physics
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