Regularity of the minimizers in the composite membrane problem in R^2

Optimal configuration and symmetry breaking phenomena in the composite membrane problem with fractional Laplacian

We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain $Ω\subset \mathbb{R}^n$, $α>0$ and $0<A<|Ω|$, find a subset $D\subset Ω$ of area $A$ such that the first Dirichlet eigenvalue of the operator $(-Δ)^s+αχ_D$ is as small as possible. The solution $D$ is called as an optimal configuration for the data $(Ω,α,A)$. Looking at the well-known extension definition for the fractional Laplacian, in the case $s=1/2$ this is essentially the composite membrane problem for which the mass is concentrated at the boundary as one …

Conformal Geometry and The Composite Membrane Problem

We consider smooth bounded surfaces with a smooth boundary and a prescribed background metric g_0. We now consider all metrics g conformal to g_0 which have a prescribed volume M. We now minimize the first eigenvalue of the Laplace operator of g over the metrics conformal to g_0 and having the prescribed volume. We show that this problem is equivalent to the study of the Composite Membrane Problem, a free boundary problem studied earlier by the author and his collaborators in all dimensions. Thus complete answers, existence of the limit metric, regularity of the minimizing eigenfunction and va…

Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes

We consider the following eigenvalue optimization problem: Given a bounded domain $Ω\subset\R^n$ and numbers $α\geq 0$, $A\in [0,|Ω|]$, find a subset $D\subsetΩ$ of area $A$ for which the first Dirichlet eigenvalue of the operator $-Δ+ αχ_D$ is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than $Ω$; on the other hand, for convex $Ω$ reflection symmetries are preserved. Also, we present numerical…

On approximation of Ginzburg-Landau minimizers by $\mathbb S^1$-valued maps in domains with vanishingly small holes

We consider a two-dimensional Ginzburg-Landau problem on an arbitrary domain with a finite number of vanishingly small circular holes. A special choice of scaling relation between the material and geometric parameters (Ginzburg-Landau parameter vs hole radius) is motivated by a recently dsicovered phenomenon of vortex phase separation in superconducting composites. We show that, for each hole, the degrees of minimizers of the Ginzburg-Landau problems in the classes of $\mathbb S^1$-valued and $\mathbb C$-valued maps, respectively, are the same. The presence of two parameters that are widely se…

Existence of varifold minimizers for the multiphase Canham-Helfrich functional

We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of single- and multiphase minimizers under area and enclosed-volume constrains. Additionally…

Error estimate for a homogenization problem involving the Laplace-Beltrami operator

In this paper we prove a concentration result and an error estimate for a model of heat conduction in composite materials having a microscopic structure arranged in a perodic array and thermally active membranes separating the heat conductive phases.

Strong solutions, global regularity and stability of a hydrodynamic system modeling vesicle and fluid interactions

In this paper, we study a hydrodynamic system modeling the deformation of vesicle membranes in incompressible viscous fluids. The system consists of the Navier-Stokes equations coupled with a fourth order phase-field equation. In the three dimensional case, we prove the existence/uniqueness of local strong solutions for arbitrary initial data as well as global strong solutions under the large viscosity assumption. We also establish some regularity criteria in terms of the velocity for local smooth solutions. Finally, we investigate the stability of the system near local minimizers of the elast…

拉梅问题中的褶皱：一个Γ-收敛方法

Wrinkling in the Lamé problem: a $Γ$-convergence approach

本文基于拉梅问题分析，研究薄弹性环在径向拉伸下的褶皱模式，通过Γ-收敛方法分析渐进行为，并建立重新缩放后的能量函数的Γ-收敛结果，同时证明了极限功能的极小值的存在性和性质。

具有相干与非相干界面的两相自由边界问题极小值的存在性

Existence of minimizers for a two-phase free boundary problem with coherent and incoherent interfaces

本文引入一种描述两相连续介质形态的变分模型，允许相干与非相干界面相互作用，并在变分法框架下证明了平面能量极小值的存在性。

一种用于两相复合优化的加速重排方法

An accelerated rearrangement method for two-phase composite optimization

本文提出了一种加速重排方法用于解决两相复合优化问题，通过引入动量加速机制和重启变体，实现了更高效的收敛性。

抛物算子在加权Lebesgue空间中的全局梯度估计

Global gradient estimates in weighted Lebesgue spaces for parabolic operators

本文研究了抛物算子在加权Lebesgue空间中的正则性问题，通过Calderón-Zygmund型理论得出弱解空间梯度的正则性结果。

Regularity for the optimal compliance problem with length penalization

We prove some regularity results for a connected set S in the planar domain O, which minimizes the compliance of its complement O§, plus its length. This problem, interpreted as to find the best location for attaching a membrane subject to a given external force f so as to minimize the compliance, can be seen as an elliptic PDE version of the average distance problem/irrigation problem (in a penalized version rather than a constrained one), which has been extensively studied in the literature. We prove that minimizers consist of a finite number of smooth curves meeting only by three at 120 deg…

Uniform Regularity close to Cross Singularities in an Unstable Free Boundary Problem

We introduce a new method for the analysis of singularities in the unstable problem $$Δu = -χ_{\{u>0\}},$$ which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of "supercharacteristic" growth of the solution, i.e. points at which the solution grows faster than the characteristic/invariant scaling of the equation would suggest. At such points the classical theory is doomed to fail, due to incompatibility of the invariant scaling of the equation and the scaling of the solution. In the case of two dimensions our result shows that in a neig…