Daniela De Silva, Ovidiu Savin
We consider almost minimizers to the one-phase energy functional and we prove their optimal Lipschitz regularity and partial regularity of their free boundary. These results were recently obtained by David and Toro, and David, Engelstein, and Toro. Our proofs provide a different method based on a non-infinitesimal notion of viscosity solutions.
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In this paper we introduce a notion of almost minimizers for certain variational problems governed by the fractional Laplacian, with the help of the Caffarelli-Silvestre extension. In particular, we study almost fractional harmonic functions and almost minimizers for the fractional obstacle problem with zero obstacle. We show that for a certain range of parameters, almost minimizers are almost Lipschitz or $C^{1,β}$-regular.
We consider almost minimizers to the thin-one phase energy functional and we prove optimal regularity of the solution and partial regularity of the free boundary. We thus recover the theory for energy minimizers. Our methods are based on a noninfinitesimal notion of viscosity solutions.
In this paper we study the local regularity of almost minimizers of the functional \begin{equation*} J(u)=\int_Ω|\nabla u(x)|^2 +q^2_+(x)χ_{\{u>0\}}(x) +q^2_-(x)χ_{\{u<0\}}(x) \end{equation*} where $q_\pm \in L^\infty(Ω)$. Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see \cite{AC}, \cite{ACF}, \cite{CJK}, \cite{W}). Nevertheless we succeed in proving that they are locally Lipschitz, which is the optimal regularity for minimizers.
In this paper we study vector-valued almost minimizers of the energy functional $$ \int_D\left(|\nabla\mathbf{u}|^2+2|\mathbf{u}|\right)\,dx . $$ We establish the regularity for both minimizers and the "regular" part of the free boundary. The analysis of the free boundary is based on Weiss-type monotonicity formula and the epiperimetric inequality for the energy minimizers.
Large scale regularity of almost minimizers of the one-phase problem in periodic media
该研究针对周期介质中的单相自由边界能量泛函，证明其极小元与几乎极小元满足大尺度Lipschitz估计及自由边界平坦蕴含Lipschitz估计的性质，采用De Silva和Savin提出的均匀介质相关技术推导。
We study almost-minimizers of anisotropic surface energies defined by a Hölder continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally Hölder continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument.
The aim of this note is to prove that almost-minimizers of the perimeter are Reifenberg flat, for a very weak notion of minimality. The main observation is that smallness of the excess at some scale implies smallness of the excess at all smaller scales.
In this paper, we study almost minimizers for the parabolic thin obstacle (or Signorini) problem with zero obstacle. We establish their $H^{σ,σ/2}$-regularity for every $0<σ<1$, as well as $H^{β,β/2}$-regularity of their spatial gradients on the either side of the thin space for some $0<β<1$. A similar result is also obtained for almost minimizers for the Signorini problem with variable Hölder coefficients.
We study almost minimizers for the thin obstacle problem with variable Hölder continuous coefficients and zero thin obstacle and establish their $C^{1,β}$ regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.
We prove that, given~$p>\max\left\{\frac{2n}{n+2},1\right\}$, the nonnegative almost minimizers of the nonlinear free boundary functional $$ J_p(u,Ω):=\int_Ω\Big( |\nabla u(x)|^p+χ_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz continuous.
Uniform concentration property for Griffith almost-minimizers
本文证明 Griffith 几乎极小元的 Hausdorff 极限保持该性质，提出不依赖余面积公式的新方法，并应用于通过爆破极限构造全局极小元。
We study vector-valued almost minimizers of the energy functional $$\int_D\left(|\nabla\mathbf{u}|^2+\frac2{1+q}\left(λ_+(x)|\mathbf{u}^+|^{q+1}+λ_-(x)|\mathbf{u}^-|^{q+1}\right)\right)dx,\quad0<q<1.$$ For Hölder continuous coefficients $λ_\pm(x)>0$, we take the epiperimetric inequality approach and prove the regularity for both almost minimizers and the set of "regular" free boundary points.
Free boundary regularity for almost minimizers of the parabolic Signorini problem
本文针对抛物型Signorini问题中的几乎极小化子，通过建立Weiss型单调性公式和Almgren型频率公式，结合epiperimetric不等式，证明了正则自由边界点的优化增长及正则集的正则性。
In this paper we study the free boundary regularity for almost-minimizers of the functional \begin{equation*} J(u)=\int_{\mathcal O} |\nabla u(x)|^2 +q^2_+(x)χ_{\{u>0\}}(x) +q^2_-(x)χ_{\{u<0\}}(x)\ dx \end{equation*} where $q_\pm \in L^\infty(\mathcal O)$. Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [AC], [ACF], [CJK], [W]). Nevertheless we succeed in proving that, under a non-degeneracy assumption on $q_\pm$, the free boundary is uniformly rectifiable. Furthermore, when $q_-\equiv 0$, and $q_+$ is Hölder continuous we s…
Regularity for almost minimizers of a one-phase Bernoulli-type functional in Carnot Groups of step two
本文研究二阶Carnot群中单相Bernoulli型泛函几乎极小元的正则性，证明其非负几乎极小元具有内在Lipschitz连续性。
Good access to the crack and integrability of the full gradient for Griffith almost-minimizers in the plane
研究平面上Griffith能量几乎极小值函数，通过证明裂纹补集被有限John域覆盖及位移满足估计，推导全梯度可积性与迹线有限性，得出二维中此类函数局部属于SBV^2空间的结论。
Lipschitz regularity of almost-minimizers in one-phase problems with generalized Orlicz growth
研究了单相问题中几乎极小值解的Lipschitz正则性，针对具有广义Orlicz增长的Alt-Caffarelli型泛函建立了最优局部正则性结果。
We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt- Caffarelli-Friedman type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis- Spolaor-Velichkov on the structure of branch points in the free boundary of stationary solutions.
Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane
本文证明平面中Griffith断裂能极小化子及几乎极小化子的裂纹集一致可修正，进而得出奇异集维数严格小于1，并证明梯度更高可积性，肯定回答De Giorgi猜想类似物。
We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their $C^{1,β}$ regularity on the either side of the thin manifold, the optimal growth away from the free boundary, the $C^{1,γ}$ regularity of the regular part of the free boundary, as well as a structural theorem for the singular set. The analysis of the free boundary is based on a successful adaptation of energy methods such as a one-parameter family of Weiss-type monotonicity formulas, Almgren-type frequency formula, and the epiperimetric and logarithmic epiperim…
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。
Two-phase almost minimizers for a fractional free boundary problem
针对分数阶 Alt-Caffarelli-Friedman 型泛函，研究两相几乎极小元，建立其最优 $C^{0,s}$ 正则性、正负自由边界不相交性，并由平坦性推出自由边界的 $C^{1,\gamma}$ 正则性。
Almost minimizers to a transmission problem for $(p,q)$-Laplacian
本文研究了 $(p,q)$-Laplacian 传输问题泛函的近似极小子，证明了局部 $(1+ε)$-极小子在 $ε$ 普遍小时的 Hölder 正则性，以及当 $|p-q|\ll 1$ 且 $ε\ll 1$ 时的近 Lipschitz 正则性。
Epsilon-regularity for Griffith almost-minimizers in any dimension under a separating condition
本文证明在分离条件下，Griffith几乎极小化子的奇异集在稍小球内具有C^{1,α}正则性，推广了二维结果至任意维度。
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 $α\in (0, 1)$.
Motivated by the construction of time-periodic solutions for the three-dimensional Landau-Lifshitz-Gilbert equation in the case of soft and small ferromagnetic particles, we investigate the regularity properties of minimizers of the micromagnetic energy functional at the boundary. In particular, we show that minimizers are regular provided the volume of the particle is sufficiently small. The approach uses a reflection construction at the boundary and an adaption of the well-known regularity theory for minimizing harmonic maps into spheres.
Regularity of two-phase free boundary minimizers in periodic media
本文研究周期介质中两相能量泛函极小元的正则性，建立了大尺度Lipschitz估计，并通过非退化极小元的平坦性改进，最终得到Liouville性质。
Continuity up to the boundary for minimizers of the one-phase Bernoulli problem
该研究针对边界数据连续及赫尔德连续的情况，证明了单相Alt-Caffarelli泛函极小元的新边界正则性结果，并将相关唯一性与正则性结论推广至连续函数族。
Given a global 1-homogeneous minimizer $U_0$ to the Alt-Caffarelli energy functional, with $sing(F(U_0)) = \{0\}$, we provide a foliation of the half-space $\R^{n} \times [0,+\infty)$ with dilations of graphs of global minimizers $\underline U \leq U_0 \leq \bar U$ with analytic free boundaries at distance 1 from the origin.
Boundary regularity of weakly coupled vectorial almost-minimizers for Alt-Caffarelli functionals with non-standard growth
本文研究了具有非标准增长的Alt-Caffarelli泛函弱耦合向量近极小解的边界正则性，证明其最优Lipschitz连续性并扩展了相关正则性结果。
Lipschitz regularity for almost minimizers of a one-phase $p$-Bernoulli-type functional in Carnot Groups of step two
本文在二阶Carnot群中研究单相p-Bernoulli型泛函的几乎极小元，证明当p大于临界指数2Q/(Q+2)时，几乎极小元关于Carnot-Carathéodory距离局部Lipschitz连续，从而在欧氏意义下具有Hölder连续正则性。
We perform the classical Ginzburg-Landau analysis originated from the celebrated paper by Bethuel, Brezis, Hélein for optimal boundary data. More precisely, we give optimal regularity assumptions on the boundary curve of planar domains and Dirichlet boundary data on them. When the Dirichlet boundary data is the tangent vector field of the boundary curve, our framework allows us to define a natural energy minimizing frame for simply connected domains enclosing Weil-Petersson curves.
We prove the partial Hölder continuity on boundary points for minimizers of quasiconvex non-degenerate functionals \begin{equation*} \mathcal{F}({\bf u}) \colon =\int_Ω f(x,{\bf u},D{\bf u})\,\mathrm{d}x, \end{equation*} where $f$ satisfies a uniform VMO condition with respect to the $x$-variable, is continuous with respect to ${\bf u}$ and has a general growth with respect to the gradient variable.
Lipschitz regularity of weakly coupled vectorial almost-minimizers for Alt-Caffarelli functionals in Orlicz spaces
本文研究了弱耦合向量近极小解在Orlicz空间中Alt-Caffarelli泛函的Lipschitz正则性，提出了一种新的分析方法并提供了最佳局部Lipschitz正则性的证明。
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…
In [David-Toro 15] and [David-Engelstein-Toro 19], (some of) the authors studied almost minimizers for functionals of the type first studied by Alt and Caffarelli in [Alt-Caffarelli 81] and Alt, Caffarelli and Friedman in [Alt-Caffarelli-Friedman 84]. In this paper we study the regularity of almost minimizers to energy functionals with variable coefficients (as opposed to [DT15, DET19. AC 81] and [ACF84] which deal only with the "Laplacian" setting). We prove Lipschitz regularity up to, and across, the free boundary, generalizing the results of [David-Toro 15] to the variable coefficient setti…
In this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning $C^1$-extremals at which the second variation is strictly positive. The results are presented in the quasiconvex setting, in accordance with the original statement by Grabovsky and Mengesha (2009). The strategy that we follow relies on a decomposition theorem that allows to split a sequence of variations into its oscillating and its concentrating parts, as well as on a sufficiency result according to which smooth extremals are spatially-local minimizers. Furthermore, we prove partial regulari…
Lipschitz regularity of almost minimizers in a Bernoulli problem with non-standard growth
本文研究了具有非标准增长的单相 Bernoulli 型泛函非负几乎极小子的最优 Lipschitz 正则性，并在 Orlicz 模背景下通过 Campanato 型估计提出了新的 Höder 正则性刻画。
We consider $\mathbb{S}^2$-valued maps on a domain $Ω\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $Ω$ and horizontal penalization on $\partialΩ$. We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincaré-type inequality. Then, we prove that any energy minimizer takes its values into a fixed meridian of the sphere $\mathbb{S}^2$, and deduce uniqueness of minimizers up to the action of the appropriate symmetry group. We also prove a comparison principle for minim…
We analyze Ginzburg--Landau minimization problems in two dimensions with either a strong or weak" tangential boundary condition. These problems are motivated by experiments in liquid crystal with boundary defects. In the singular limit when the correlation length tends to zero, we show that boundary defects will be observed for weak anchoring, while both boundary and interior vortices are possible for strong anchoring in the first order limit.
We consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb R^{n+1}_+$ plus the area of the positivity set of that function in $\mathbb R^n$. We establish full regularity of the free boundary for dimensions $n \leq 2$, prove almost everywhere regularity of the free boundary in arbitrary dimension and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for the calculus of varia…
Given a bounded $C^2$ domain $Ω\subset{\mathbb R}^d$ with $d\geq3$, we prove a sharp inequality which relates the perimeter of ${\partialΩ}$ to the endpoint Gagliardo seminorm in $W^{r,2}({\partialΩ})$, corresponding to $r=0$, of the normal vector field on ${\partialΩ}$. The proof of the inequality relies on the use of Bessel potentials and a monotonicity formula; we also show that balls are the unique minimizers. For $1/2<r<1$, the Gagliardo seminorm of the normal vector field on ${\partialΩ}$ is related to a fractional second fundamental form which arises in the study of nonlocal perimeters …
Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the $p$-Laplacian
本文证明了 $p$-Laplacian 弱耦合向量几乎极小子在紧致集上的最优内部 Lipschitz 连续性，并在 $p \geq 2$ 时采用新方法解决了边界 Lipschitz 正则性问题。
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.
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.
Boundary regularity for Oseen-Frank minimizers with arbitrary positive splay, twist, and bend constants
针对Oseen-Frank能量的全局极小化子，在无小各向异性假设下证明其边界正则性，解决了Lin和Liu问题(c)的边界正则性部分。
We continue the analysis of some modifications of the total variation image inpainting method formulated on the space $BV(Ω)^M$ in the sense that we generalize the main results of [32] to the case that a more general data fitting term is involved. As in [32] we deal with vector-valued images, we do not impose any structure condition on our density $F$ and the dimension of the domain $Ω$ is arbitrary. Precisely we discuss existence of generalized solutions of the corresponding variational problem and we will also pass to the associated dual variational problem for which we show unique solvabili…
A priori $L^\infty-$bound for Ginzburg-Landau energy minimizers with divergence penalization
研究带散度惩罚的Ginzburg-Landau能量极小元，证明在$C^{2,1}$边界上一致$L^\infty$界，在$C^{3,1}$边界上Lipschitz常数以$\varepsilon^{-1}$爆破，并推广至混合边界条件的$W^{2,p}$正则性。
We examine a variational free boundary problem of Alt-Caffarelli type for the biharmonic operator with Navier boundary conditions in two dimensions. We show interior C2-regularity of minimizers and that the free boundary consists of finitely many C2-hypersurfaces. With the aid of these results, we can prove that minimizers are in general not unique. We investigate radial symmetry of minimizers and compute radial solutions explicitly.
Optimal regularity up to the boundary for Plateau-quasi-minimizers
本文研究带边界条件的Plateau拟极小化子的最优正则性，证明其可由具有Ahlfors正则边界的双John域刻画，并建立了到边界的Ahlfors正则性与一致可整流性。
We study existence of minimizers of the least gradient problem \[\inf_{v \in BV_g} \int_Ωφ(x, Dv),\] where $BV_g=\{v \in BV(Ω): \int_{\partial Ω}gv=1\}$, $φ(x,p): Ω\times \R^n \rightarrow \R$ is a convex, continuous, and homogeneous function of degree $1$ with respect to the $p$ variable, and $g$ satisfies the comparability condition $\int_{\partial Ω} g dS=0$. We prove that for every $0\not \equiv g \in L^{\infty}(\partial Ω)$ there are infinitely many minimizers in $BV(Ω)$. Moreover there exists a divergence free vector field $T\in (L^{\infty}(Ω))^n$ that determines the structure of level se…
Minimizers in the least gradient problem with discontinuous boundary data need not be unique. However, all of them have a similar structure of level sets. Here, we give a full characterization of the set of minimizers in terms of any one of them and discuss stability properties of an approximate problem.
Borderline regularity in singular free boundary problems
本文研究势项$σ$满足最小假设时能量泛函局部极小值的临界正则性，得出不同$σ$正则性下极小值的正则性结果，包括Log-Lipschitz连续及$C^1$类等，确定了可微性的严格阈值。
In this paper, we extend the related notions of Dirichlet quasiminimizer, $ω-$minimizer and almost minimizer to the framework of multiple-valued functions in the sense of Almgren and prove Holder regularity results. We also give examples of those minimizers with various branch sets.
We study a class of semi-linear problems involving the fractional Laplacian under subcritical or critical growth assumptions. We prove that, for the corresponding functional, local minimizers with respect to a C^0-topology weighted with a suitable power of the distance from the boundary are actually local minimizers in the natural H^s-topology.
Suppose that $ω\subsetΩ\subset R^2$. In the annular domain $A=Ω\setminus\barω$ we consider the class $J$ of complex valued maps having degree 1 on $\partial Ω$ and on $\partialω$. It was conjectured by Berlyand and Mironescu ('04), that he existence of minimizers of the Ginzburg-Landau energy $E_κ$ in $J$ is completely determined by the value of the $H^1$-capacity $cap(A)$ of the domain and the value of the Ginzburg-Landau parameter $κ$. The existence of minimizers of $E_κ$ for all $κ$ when $cap(A)\geqπ$ (domain $A$ is ``thin'') and for small $κ$ when $cap(A)<π$ (domain $A$ is ``thick'') was e…
We examine lower order perturbations of the harmonic map prob- lem from $\mathbb{R}^2$ to $\mathbb{S}^2$ including chiral interaction in form of a helicity term that prefers modulation, and a potential term that enables decay to a uniform background state. Energy functionals of this type arise in the context of magnetic systems without inversion symmetry. In the almost conformal regime, where these perturbations are weighted with a small parameter, we examine the existence of relative minimizers in a non-trivial homotopy class, so-called chiral skyrmions, strong compactness of almost minimizer…
We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.
We study the regularity of minimizers to the functional \[ J(w)=\int_Ω a^{ij}w_iw_j + Qχ_{\{w>0\}}, \] over a bounded domain $Ω$ and among the class of nonnegative functions in $W^{1,2}(Ω)$ with prescribed boundary data. We assume that the coefficients $a^{ij}$ are only bounded and measurable and satisfy an ellipticity in condition. In two dimensions we prove that minimizers are Hölder continuous on subdomains. We also prove that in two dimensions a minimizer $u$ satisfies a linear growth condition from above and below near the free boundary $\partial \{u>0\}$.
Fluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier-Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) en…
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