Average-distance problem with curvature penalization for data parameterization: regularity of minimizers

自由间断问题局部极小化子的对称性破缺

Symmetry breaking for local minimizers of a free discontinuity problem

本文研究二维自由间断问题中局部极小化子的对称性破缺，通过改编校准方法证明存在水平集非简单条纹的奇异极小化子。

多相最优设计问题极小化子的Ahlfors正则性

Ahlfors-regularity for minimizers of a multiphase optimal design problem

本文证明多相最优设计问题中极小化子的最优界面并集具有Ahlfors正则性，采用惩罚方法和能量衰减估计，为界面规则性提供理论保证。

弹性曲线的非图形障碍问题

A nongraphical obstacle problem for elastic curves

本文研究边界固定的开放平面曲线带长度惩罚弹性弯曲能的障碍问题，分析了不同长度惩罚参数下极小值是否接触障碍物的条件，并给出显式阈值与弹性曲线解。

The average distance problem with an Euler elastica penalization

We consider the minimization of an average distance functional defined on a two-dimensional domain $Ω$ with an Euler elastica penalization associated with $\pd Ω$, the boundary of $Ω$. The average distance is given by \begin{equation*} \int_Ω \dist^p(x,\pd Ω)\d x \end{equation*} where $p\geq 1$ is a given parameter, and $\dist(x,\pd Ω)$ is the Hausdorff distance between $\{x\}$ and $\pd Ω$. The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve ${\pd Ω}$, which is proportional to the integrated squared curvature def…

An Optimization Problem in Heat Conduction With Volume Constraint and Double Obstacles

We consider the optimization problem of minimizing $\int_{\mathbb{R}^n}|\nabla u|^2\,\mathrm{d}x$ with double obstacles $ϕ\leq u\leqψ$ a.e. in $D$ and a constraint on the volume of $\{u>0\}\setminus\overline{D}$, where $D\subset\mathbb{R}^n$ is a bounded domain. By studying a penalization problem that achieves the constrained volume for small values of penalization parameter, we prove that every minimizer is $C^{1,1}$ locally in $D$ and Lipschitz continuous in $\mathbb{R}^n$ and that the free boundary $\partial\{u>0\}\setminus\overline{D}$ is smooth. Moreover, when the boundary of $D$ has a pl…

The average distance problem with perimeter-to-area ratio penalization

In this paper we consider the functional \begin{equation*} E_{p,\la}(Ω):=\int_Ω\dist^p(x,\pd Ω)\d x+\la \frac{\H^1(\pd Ω)}{\H^2(Ω)}. \end{equation*} Here $p\geq 1$, $\la>0$ are given parameters, the unknown $Ω$ varies among compact, convex, Hausdorff two-dimensional sets of $\R^2$, $\pd Ω$ denotes the boundary of $Ω$, and $\dist(x,\pd Ω):=\inf_{y\in\pd Ω}|x-y|$. The integral term $\int_Ω\dist^p(x,\pd Ω)\d x$ quantifies the "easiness" for points in $Ω$ to reach the boundary, while $\frac{\H^1(\pd Ω)}{\H^2(Ω)}$ is the perimeter-to-area ratio. The main aim is to prove existence and $C^{1,1}$-regu…

Global minimizers for axisymmetric multiphase membranes

We consider a Canham-Helfrich-type variational problem defined over closed surfaces enclosing a fixed volume and having fixed surface area. The problem models the shape of multiphase biomembranes. It consists of minimizing the sum of the Canham-Helfrich energy, in which the bending rigidities and spontaneous curvatures are now phase-dependent, and a line tension penalization for the phase interfaces. By restricting attention to axisymmetric surfaces and phase distributions, we extend our previous results for a single phase (arXiv:1202.1979) and prove existence of a global minimizer.

凸性约束下形状优化的正则性

Regularity in shape optimization under convexity constraint

本文研究凸性约束下等周问题形状最优解的正则性，通过定义适配凸场景的拟极小解、结合切割方法与惩罚法，证明周长加微扰项的极小解及带体积约束情形均具C^{1,1}正则性，涵盖多类PDE型微扰例子并给出正则性上限反例。

A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem

We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is $1/2$ and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.

Average-distance problem for parameterized curves

We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite positive compactly supported measure $μ$, for $p \geq 1$ and $λ>0$ we consider the functional \[ E(γ) = \int_{\mathbb{R}^d} d(x, Γ_γ)^p dμ(x) + λ\,\textrm{Length}(γ) \] where $γ:I \to \mathbb{R}^d$, $I$ is an interval in $\mathbb{R}$, $Γ_γ= γ(I)$, and $d(x, Γ_γ)$ is the distance of $x$ to $Γ_γ$. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure $\mathcal H^1$, and to (regularized) prin…

Interfacial energy as a selection mechanism for minimizing gradient Young measures in a one-dimensional model problem

Energy functionals describing phase transitions in crystalline solids are often non-quasiconvex and minimizers might therefore not exist. On the other hand, there might be infinitely many gradient Young measures, modelling microstructures, generated by minimizing sequences, and it is an open problem how to select the physical ones. In this work we consider the problem of selecting minimizing sequences for a one-dimensional three-well problem $\mathcal{E}$. We introduce a regularization $\mathcal{E}^\varepsilon$ of $\mathcal{E}$ with an $\varepsilon$-small penalization of the second derivatives…

数据驱动多孔介质问题的热力学一致解的存在性

Existence of thermodynamically consistent solutions for data-driven porous media problems

针对数据驱动多孔介质问题，提出变分DDCM框架将热力学第二定律作为硬约束，通过补偿紧性论证确立极小解存在性，分析拉格朗日乘子与惩罚方案，数值实验验证其可恢复热力学一致性。