Daniela De Silva, Ovidiu Savin
We investigate regularity of minimizers in two dimensions for certain classes of non-smooth convex functionals. In particular our results apply to the surface tensions that appear in recent works on random surfaces and random tilings of Kenyon, Okounkov and others
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Suppose that $n\ge3$ and $H(p)\in C^0(\mathbb{R}^n)$ is a locally strongly convex and concave Hamiltonian. We obtain the everywhere differentiability of all absolute minimizers for $H$ in any domain of $\mathbb{R}^n$.
We prove uniqueness for minimizers of the weighted least gradient problem \[\inf \left\lbrace \int_Ω a|Du|: \ \ u\in BV(Ω), \ \ u|_{\partial Ω}=f \right\rbrace.\] The weight function $a$ is assumed to be continuous and it is allowed to vanish in certain subsets of $Ω$. Existence is assumed a priori. Our approach is motivated by the hybrid inverse problem of imaging electric conductivity from interior knowledge (obtainable by MRI) of the magnitude of one current density vector field.
We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous works of Carlier-Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations. 58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of b…
We study the solvability of second boundary value problems of fourth order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and monopolist's problem in economics. The right hand sides of our Abreu type equations are quasilinear expressions of second order; they are highly singular and a priori just measures. However, our analysis in particular shows that minimizers of t…
We study a $p$-Laplacian equation involving a parameter $λ$ and a concave-convex nonlinearity containing a weight which can change sign. By using the Nehari manifold and the fibering method, we show the existence of two positive solutions on some interval $(0,λ^*+\varepsilon)$, where $λ^*$ can be characterized variationally. We also study the asymptotic behavior of solutions when $λ\downarrow 0$.
Local Lipschitz continuity of the minimizers of nonuniformly convex functionals under the Lower Bounded Slope Condition
本文在$g$有界且$ϕ$满足下有界斜率条件的情况下，证明了非一致凸泛函极小值点的局部Lipschitz正则性，并作为副产品证明了允许$f$非凸时该类泛函局部Lipschitz极小值点的存在性。
We establish local higher integrability and differentiability results for minimizers of variational integrals $$ \mathfrak{F}(v,Ω) = \int_Ω /! F(Dv(x)) \, dx $$ over $W^{1,p}$--Sobolev mappings $u \colon Ω\subset {\mathbb R}^n \to {\mathbb R}^N$ satisfying a Dirichlet boundary condition. The integrands $F$ are assumed to be autonomous, convex and of $(p,q)$ growth, but are otherwise not subjected to any further structure conditions, and we consider exponents in the range $1<p \leq q < p^{\ast}$, where $p^{\ast}$ denotes the Sobolev conjugate exponent of $p$.
We prove that if the given compact set $K$ is convex then a minimizer of the functional $$ I(v)=\int_{B_R} |\nabla v|^p dx+\text{Per}(\{v>0\}),\,1<p<\infty, $$ over the set $\{v\in H^1_0(B_R)|\,\, v\equiv 1\,\,\text{on}\,\, K\subset B_R\}$ has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.
For any $n\ge 2$, $Ω\subset\rn$, and any given convex and coercive Hamiltonian function $H\in C^{0}(\rn)$, we find an optimal sufficient condition on $H$, that is, for any $c\in\mathbb R$, the level set $H^{-1}(c)$ does not contains any line segment, such then any absolute minimizer $u\in AM_H(Ω)$ enjoys the linear approximation property. As consequences, we show that when $n=2$, if $u\in AM_H(Ω)$ then $u\in C^1$; and if $u\in AM_H(\rr^2)$ satisfies a linear growth at the infinity, then $u$ is a linear function on $\rr^2$. In particular, if $H$ is a strictly convex Banach norm $\|\cdot\|$ on $…
Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones
本文研究非凸锥中由Hardy--Sobolev不等式生成的$p$-Laplacian Neumann问题，证明在球截面第一Neumann特征值满足一定条件时径向解不是极值函数，从而极小化子必为非径向，并由此得到多重解。
We consider minimizers of linear functionals of the type $$L(u)=\int_{\p Ω} u \, d σ- \int_Ω u \, dx$$ in the class of convex functions $u$ with prescribed determinant $\det D^2 u =f$. We obtain compactness properties for such minimizers and discuss their regularity in two dimensions.
We study the planar least gradient problem with respect to an anisotropic norm $ϕ$ for continuous boundary data. We prove existence of minimizers for strictly convex domains $Ω$. Furthermore, we inspect the issue of uniqueness and regularity of minimizers only in terms of the modes of convexity of $ϕ$ and $Ω$. The results are independent from the regularity of $ϕ$.
We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are $C^{1, 1/2}$, and that near non-degenerate points the fracture set is $C^{1, α}$, for some $α\in (0, 1)$.
Generalized minimizers for the capillarity energy outside convex obstacles
针对凸障碍物外毛细性能量可能不存在体积约束极小元的问题，通过凸障碍物Hausdorff收敛下的能量收敛结果，证明存在广义极小元，描述其渐近行为并应用于研究等周轮廓与障碍物渐近几何的影响。
We study the question whether Lipschitz minimizers of $\int F(\nabla u)\,dx$ in $\mathbb{R}^n$ are $C^1$ when $F$ is strictly convex. Building on work of De Silva-Savin, we confirm the $C^1$ regularity when $D^2F$ is positive and bounded away from finitely many points that lie in a $2$-plane. We then construct a counterexample in $\mathbb{R}^4$, where $F$ is strictly convex but $D^2F$ degenerates on the intersection of a Simons cone with $S^3$. Finally we highlight a connection between the case $n = 3$ and a result of Alexandrov in classical differential geometry, and we make a conjecture abou…
Scalar minimizers with maximal singular sets and lack of Meyers property
本文提出一种通用构造方法，构建凸标量变分问题，其极小元具有最大维数的奇异集且缺乏更高可积性，即Meyers性质失效。
We consider a class of convex integral functionals composed of a term of linear growth in the gradient of the argument, and a fidelity term involving $L^2$ distance from a datum. Such functionals are known to attain their infima in the $BV$ space. Under the assumption that the domain of integration is convex, we prove that if the datum is in $W^{1,1}$, then the functional has a minimizer in $W^{1,1}$. In fact, the minimizer inherits $W^{1,p}$ regularity from the datum for any $p \in [1, +\infty]$. We also obtain a quantitative bound on the singular part of the gradient of the minimizer in the …
We construct a singular minimizing map ${\bf u}$ from $\mathbb{R}^3$ to $\mathbb{R}^2$ of a smooth uniformly convex functional of the form $\int_{B_1} F(D{\bf u})\,dx$.
We study a large family of axisymmetric Riesz-type singular interaction potentials with anisotropy in three dimensions. We generalize some of the results of our recent work in two dimensions to the present setting. For potentials with linear interpolation convexity, their associated global energy minimizers are given by explicit formulas whose supports are ellipsoids. We show that for less singular anisotropic Riesz potentials, the global minimizer may collapse into one or two dimensional concentrated measures which minimize restricted isotropic Riesz interaction energies. Some partial aspects…
It has recently been established byWang and Xia [WX] that local minimizers of perimeter within a ball subject to a volume constraint must be spherical caps or planes through the origin. This verifies a conjecture of the authors and is in contrast to the situation of area minimizing surfaces with prescribed boundary where singularities can be present in high dimensions. This result lends support to the more general conjecture that volume-constrained minimizers in arbitrary convex sets may enjoy better regularity properties than their boundary-prescribed cousins. Here, we show the importance of …
We establish an $\varepsilon$-regularity result for the derivative of a map of bounded variation that minimizes a strongly quasiconvex variational integral of linear growth, and, as a consequence, the partial regularity of such BV minimizers. This result extends the regularity theory for minimizers of quasiconvex integrals on Sobolev spaces to the context of maps of bounded variation. Previous partial regularity results for BV minimizers in the linear growth set-up were confined to the convex situation.
Conservation laws are usually studied in the context of sufficient regularity conditions imposed on the flux function, usually $C^{2}$ and uniform convexity. Some results are proven with the aid of variational methods and a unique minimizer such as Hopf-Lax and Lax-Oleinik. We show that many of these classical results can be extended to a flux function that is not necessarily smooth or uniformly or strictly convex. Although uniqueness a.e. of the minimizer will generally no longer hold, by considering the greatest (or supremum, where applicable) of all possible minimizers, we can successfully …
Sharp mean Hadamard inequalities and polyconvex integrands that give rise to convex functionals
研究二维均值Hadamard不等式的实例，证明了在混合Dirichlet和Neumann边界条件下，多凸积分元的泛函最小解的唯一性，并通过计算实验验证了最小解的行为。
Let $a>0,b>0$ and $V(x)\geq0$ be a coercive function in $\mathbb R^2$. We study the following constrained minimization problem on a suitable weighted Sobolev space $\mathcal{H}$: \begin{equation*} e_{a}(b):=\inf\left\{E_{a}^{b}(u):u\in\mathcal{H}\ \mbox{and}\ \int_{\mathbb R^{2}}|u|^{2}dx=1\right\}, \end{equation*} where $E_{a}^{b}(u)$ is a Kirchhoff type energy functional defined on $\mathcal{H}$ by \begin{equation*} E_{a}^{b}(u)=\frac{1}{2}\int_{\mathbb R^{2}}[|\nabla u|^{2}+V(x)u^{2}]dx+\frac{b}{4}\left(\int_{\mathbb R^{2}}|\nabla u|^{2}dx\right)^{2}-\frac{a}{4}\int_{\mathbb R^{2}}|u|^{4}dx…
We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property. The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower Semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradi…
We study the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our proof uses an almost monotonicity formula for the Alt-Caffarelli-Friedman functional restricted to the convex domain. This requires a variant of the classical Friedland-Hayman inequality for geodesically convex subsets of the sphere with Neumann boundary conditions. To apply this inequality, in addition to convexity, we require a Dini condition governing the…
Higher regularity for minimizers of very degenerate convex integrals
本文研究一类高度退化凸积分泛函的极小化子，证明对任意在单位球上为零的连续函数K，复合函数K(x,Du)在区域内部连续，从而获得更高正则性。
In this paper we collect some new observations about periodic critical points and local minimizers of a nonlocal isoperimetric problem, arising in the modeling of diblock copolymers. In the main result, by means of a purely variational procedure, we show that it is possible to construct (locally minimizing) periodic critical points whose shape resemble that of any given strictly stable constant mean curvature (periodic) hypersurface. Along the way, we establish several auxiliary results of independent interest.
This paper is concerned with equilibrium configurations of one-dimensional particle system with non-convex nearest-neighbour and next-to-nearest-neighbour interactions and its passage to the continuum. The goal is to derive compactness results for a $Γ$-development of the energy with the novelty that external forces are allowed. In particular, the forces may depend on Lagrangian or Eulerian coordinates. Our result is based on a new technique for deriving compactness results which are required for calculating the first-order $Γ$-limit: instead of comparing a configuration of $n$ atoms to a glob…
We consider the class of semi-stable solutions to semilinear equations $-Δu=f(u)$ in a bounded smooth domain $Ω$ of $R^n$ (with $Ω$ convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions $n \leq 4$, we establish an priori $L^\infty$ bound which holds for every positive semi-stable solution and every nonlinearity $f$. This estimate leads to the boundedness of all extremal solutions when $n=4$ and $Ω$ is convex. This result was previously known only in dimensions $n\leq 3$ by a result of G. Nedev. In dimensions $5 \leq n \leq 9$ the boun…
We consider the functional $$J(v) = \int_Ω[f(|\nabla v|) - v] dx,$$ where $Ω$ is a bounded domain and $f:[0,+\infty)\to \mathbb{R}$ is a convex function vanishing for $s\in [0,σ]$, with $σ>0$. We prove that a minimizer $u$ of $J$ satisfies an equation of the form $$\min(F(\nabla u, D^2 u), |\nabla u|-σ)=0$$ in the viscosity sense.
We study a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse-supported configurations are the global minimizers based on linear convexity arguments. Moreover, for certain anisotropic parts, we prove that for large values of the parameter the global minimizer is only given by ver…
A PDE Framework of Consensus-Based Optimization for Objectives with Multiple Global Minimizers
本文提出一种简单的CBO变体以解决具有多个全局极小值的非凸优化问题，证明相应非线性Fokker--Planck方程解的存在性及向多分量极小值集合的指数集中，并表明对应的Euler-Maruyama离散化算法趋于收敛到多个全局极小值。
We prove an a priori estimate for the second derivatives of local minimizers of integral functionals of calculus of variation with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.
In this paper we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics \cite{Silling2000} or nonlocal diffusion models \cite{Rossi}. We derive nonlocal versions of the Euler-Lagrange equations under two sets of growth assumptions for the integrand. Existence of minimizers is shown for integrands with joint convexity (in the function and nonlocal gradient components). By using the convolution structure we show regularity of solutions for certain Euler-Lagrange equations. No growth assumptions are needed for the exist…
We prove higher summability for the gradient of minimizers of strongly convex integral functionals of the Calculus of Variations with (p,q)-Growth conditions in low dimension. Our procedure is set in the framework of Fractional Sobolev Spaces and renders the desired regularity as the result of an approximation technique relying on estimates obtained through a careful use of difference quotients.
Critical concave-convex problems in Carnot groups
研究Carnot群中具有凹凸和临界非线性的Dirichlet问题，利用变分Perron方法和Sobolev不等式的极小函数估计，证明存在两个正解。
We consider a class of integral functionals with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space W^{1;q}. We prove a result of higer differentiability for the minimizers. We also infer a result of Lipschitz regularity of minimizers if q > n, and a result of higher integrability for the gradient if q = n. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.
Regularity for Minimizers of non Autonomous Singular Functionals with Anisotropic Growth
研究非自治凸积分泛函在次二次各向异性增长条件下局部最小值的更高可微性，通过指数间隙和能量密度振荡的Sobolev假设得到结果。
The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morrey's inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the inequalities. Then, as main application of our inequalities, we establish new $L^q$ and $W^{1,q}$ estimates for semi-stable solutions of $-Δu=g(u)$ in a bounded domain $Ω$ of $\mathbb{R}^n$. These estimates lead to an $L^{2n/(n-4)}(Ω)$ bound for the extremal solution of $-Δu=λf(u)$ when $n\geq 5$ and the domain is convex. We recall that extremal solutions are known to…
An epsilon-regularity result for Griffith almost-minimizers in the plane
本文研究脆性材料 Griffith 断裂能量极小化子的裂纹集正则性，在平面设定下证明了 epsilon 正则性定理，表明裂纹在零 Hausdorff 测度奇异集外局部为 $C^{1,1/2}$ 曲线，且无需对裂纹施加拓扑约束，结果也适用于几乎极小化子。
On the singular set of $\operatorname{BV}$ minimizers for non-autonomous functionals
本文研究非自治凸变分泛函的BV极小化子的正则性，在椭圆性和Hölder连续条件下，证明了梯度更高可积性并给出了奇异集的Hausdorff维数上界。
We prove regularity results for the unique minimizer of the total variation functional, currently used in image processing analysis since the work by L. Rudin, S. Osher and E. Fatemi. In particular we show that if the source term $f$ is locally, respectively globally, Lipschitz, then the solution has the same regularity with local, respectively global, Lipschitz norm estimated accordingly. The result is proved in any dimension and for any (regular) domain. So far we extend a similar result proved earlier by V. Caselles, A. Chambolle and M. Novaga for dimension $N\leq 7$ and (in case of the glo…
We provide a geometric characterization of the minimal and maximal minimizer of the prescribed curvature functional $P(E)-κ|E|$ among subsets of a Jordan domain $Ω$ with no necks of radius $κ^{-1}$, for values of $κ$ greater than or equal to the Cheeger constant of $Ω$. As an application, we describe all minimizers of the isoperimetric profile for volumes greater than the volume of the minimal Cheeger set, relative to a Jordan domain $Ω$ which has no necks of radius $r$, for all $r$. Finally, we show that for such sets and volumes the isoperimetric profile is convex.
In this note we prove that any $W^{1,2}$ mapping $u$ in the plane that minimizes an appropriate quasiconvex energy functional subject to the Jacobian constraint ${\rm det} \na u=1$ a.e., are necessarily Lipschitz. Furthermore we show that the minimizers corresponding to uniformly convex energy are affine and give an example of non-affine minimizers subject to affine boundary data corresponding to a convex energy. We also discuss the regularity issues in dimension greater than or equal to 3.
In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form \[ \mathfrak{F}( v, Ω)= \int_Ω \! F(x, Dv(x)) \, dx, \] where, for ${n > 2}$ and $N\ge 1$, $Ω$ is a bounded open set in $\mathbb{R}^n$, $u \in W^{1,1}(Ω, \mathbb{R}^N)$ and the energy density $F:Ω\times \mathbb{R}^{N \times n}\to \mathbb{R}$ satisfies the so called variable growth conditions. The main novelty of the paper is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the $x$ variable.
This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity. We also show that, under some additional assumptions, the DuBois-Reymond necessary condition still holds in the discontinuous case. Finally, we apply these results to deduce that the value function of the Bolza problem is locally …
We consider the functional $$I_Ω(v) = \int_Ω[f(|Dv|) - v] dx,$$ where $Ω$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, G. Crasta [Cr1] has shown that if $I_Ω$ admits a minimizer in $W_0^{1,1}(Ω)$ depending only on the distance from the boundary of $Ω$, then $Ω$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general…
Monotone approximation of differentiable convex functions with applications to general minimization problems
本文在极小增长与强制假设下研究非自治能量的极小元，证明其满足相关 Euler--Lagrange 方程或不等式，核心工具是凸 $C^1$-能量的延拓结果。
We study $H^1$ versus $C^1$ local minimizers for functionals defined on spaces of symmetric functions, namely functions that are invariant by the action of some subgroups of $\mathcal{O}(N)$. These functionals, in many cases, are associated with some elliptic partial differential equations that may have supercritical growth. So we also prove some results on classical regularity for symmetric weak solutions for a general class of semilinear elliptic equations with possibly supercritical growth. We then apply these results to prove the existence of a large number of classical positive symmetric …
We prove global Lipschitz regularity for a wide class of convex variational integrals among all functions in $W^{1,1}$ with prescribed (sufficiently regular) boundary values, which are not assumed to satisfy any geometrical constraint (as for example bounded slope condition). Furthermore, we do not assume any restrictive assumption on the geometry of the domain and the result is valid for all sufficiently smooth domains. The result is achieved with a suitable approximation of the functional together with a new construction of appropriate barrier functions.
In this paper we prove a $\mathcal C^{1,α}$ regularity result for minimizers of the planar Griffith functional arising from a variational model of brittle fracture. We prove that any isolated connected component of the crack, the singular set of a minimizer, is locally a $\mathcal C^{1,α}$ curve outside a set of zero Hausdorff measure.
The existence of a global minimizer for a variational problem arising in registration of diffusion tensor images is proved, which ensures that there is a regular spatial transformation for the registration of diffusion tensor images.
We consider a variational model describing the shape of liquid drops and crystals under the influence of gravity, resting on a horizontal surface. Making use of anisotropic symmetrization techniques, we establish existence, convexity and symmetry of minimizers for a class of surface tensions admissible to the symmetrization procedure. In the case of smooth surface tensions, we obtain uniqueness of minimizers via an ODE characterization.
We consider the functional $\int_Ωg(\nabla u+\textbf X^\ast)d\mathscr L^{2n}$ where $g$ is convex and $\textbf X^\ast(x,y)=2(-y,x)$ and we study the minimizers in $BV(Ω)$ of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on $g$, there exists a unique minimizer which is also Lipschitz continuous.
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。
We show that if $(u,K)$ is a global minimizer for the Mumford-Shah functional in $R^N$, and if K is a smooth enough cone, then (modulo constants) u is a homogenous function of degree 1/2. We deduce some applications in $R^3$ as for instance that an angular sector cannot be the singular set of a global minimizer, that if $K$ is a half-plane then $u$ is the corresponding cracktip function of two variables, or that if K is a cone that meets $S^2$ with an union of $C^1$ curvilinear convex polygones, then it is a $P$, $Y$ or $T$.
Minimization of the buckling load of a clamped plate with perimeter constraint
本文研究任意维度下带周长约束的屈曲载荷最小化问题，证明二维最小化板为凸集，高维最优集合为开连通集，并证明高阶特征值在指定周长的凸集族中存在最小化子。
Existence and uniqueness of Monge minimizers for a Multi-marginal Optimal Transport problem with intermolecular interactions cost
本文研究源自密度泛函理论强相互作用极限解离模型的多边际最优传输问题，在至少两个边际绝对连续条件下证明了Monge解的存在唯一性，并说明该正则性条件是必要的。
Microstructures and anti-phase boundaries in long-range lattice systems
本文研究一维非凸晶格系统中短程振荡与凸性 M-近邻长程有序作用的竞争，证明双势阱情形下会产生由界面能驱动、形状普适的 M-周期极小元。
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