Fausto Ferrari, Nicolò Forcillo
The aim of this paper is to study the existence of an Alt-Caffarelli-Friedman monotonicity type formula in the Heisenberg group.
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In this paper we revisit the proof of the Alt-Caffarelli-Friedman monotonicity formula. Then, in the framework of the Heisenberg group, we discuss the existence of an analogous monotonicity formula introducing a necessary condition for its existence, recently proved in \cite{FeFo}.
A self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula
本文提供了一个自包含的Alt-Caffarelli-Friedman单调性公式证明，利用了一个有用的凸性性质来证明核心的Friedland-Hayman不等式。
Alt-Caffarelli-Friedman monotonicity formula and mean value properties in Carnot groups with applications
本文提出一种基于梯度范数函数均值单调递增性质的新方法来处理Alt-Caffarelli-Friedman单调性公式，并证明该论证在Carnot群的一般框架下成立。
A counterexample to the monotone increasing behavior of an Alt-Caffarelli-Friedman formula in the Heisenberg group
本文在海森堡群的非交换框架下，针对文献中引入的 Alt-Caffarelli-Friedman 单调性公式伴生函数，构造了一个关于其单调递增行为的反例。
On a fractional Alt-Caffarelli-Friedman-type monotonicity formula
本文利用 s-调和函数的均值性质，在非局部框架下引入了模仿 Alt-Caffarelli-Friedman 泛函的单调性公式，方法纯粹非局部，不依赖延拓技术，并建立了内部非局部梯度估计和 Bochner 恒等式的非局部类比。
Some counterexamples to Alt-Caffarelli-Friedman monotonicity formulas in Carnot groups
本文在阶数大于1的Carnot群中研究Alt-Caffarelli-Friedman单调性公式，利用调和多项式缺乏正交性给出存在单调递增行为反例的充分条件，推广了已有结果。
We give an example of a pair of nonnegative subharmonic functions with disjoint support for which the Alt-Caffarelli-Friedman monotonicity formula has strictly positive limit at the origin, and yet the interface between their supports lacks a (unique) tangent there. This clarifies a remark appearing in the literature (see \cite{cs05}) that the positivity of the limit of the ACF formula implies unique tangents; this is true under some additional assumptions, but false in general. In our example, blow-ups converge to the expected piecewise linear two-plane function along subsequences, but the li…
We study the regularity of the interface between the disjoint supports of a pair of nonnegative subharmonic functions. The portion of the interface where the Alt-Caffarelli-Friedman (ACF) monotonicity formula is asymptotically positive forms an $\mathcal{H}^{n-1}$-rectifiable set. Moreover, for $\mathcal{H}^{n-1}$-a.e. such point, the two functions have unique blowups, i.e. their Lipschitz rescalings converge in $W^{1,2}$ to a pair of nondegenerate truncated linear functions whose supports meet at the approximate tangent plane. The main tools used include the Naber-Valtorta framework and our r…
The objective of this paper is two-fold. First, we establish new sharp quantitative estimates for Faber-Krahn inequalities on simply connected space forms. We prove that the gap between the first eigenvalue of a given set $Ω$ and that of the ball quantitatively controls both the $L^1$ distance of this set from a ball {\it and} the $L^2$ distance between the corresponding eigenfunctions: \[ λ_1(Ω) - λ_1(B) \gtrsim |ΩΔB|^2 + \int |u_Ω - u_B|^2, \] where $B$ denotes the nearest geodesic ball to $Ω$ with $|B|=|Ω|$ and $u_Ω$ denotes the first eigenfunction with suitable normalization. On Euclidean …
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…
We prove that for minimizers of the vectorial Alt-Caffarelli functional the two-phase singular set of the free boundary is rectifiable and the blow-up is unique almost everywhere on it. While the first conclusion is an application of the recent techniques developed by Naber and Valtorta, the uniqueness part follows from the rectifiability and a new application of the Alt-Caffarelli-Friedman monotonicity formula.
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.
On the supremal version of the Alt-Caffarelli minimization problem
本文作为近期研究无穷拉普拉斯算子内部Bernoulli自由边界工作的伴随论文，探讨了对应的变分问题，即Alt-Caffarelli极小化问题的上确界版本。
We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper "The regularity of free boundaries in higher dimensions."
Global minimizers for a two-sided biharmonic Alt-Caffarelli problem
本文研究双侧双调和Alt-Caffarelli问题的全局极小解，发现半空间解及具有常数拉普拉斯变换的解为全局极小解，并指出单侧问题与双侧问题在偏微分方程满足性上的差异。
We eliminate the existence of cusps in a class of \textit{degenerate} free-boundary problems for the Alt-Caffarelli functional $J_{Q}(v, Ω):= \int_Ω|\nabla v|^2 + Q^2(x)χ_{\{v>0\}}dx,$ so-called because $Q(x) = \text{dist}(x, Γ)^γ$ for $Γ$ an affine $k$-plane and $0< γ$. This problem is inspired by a generalization of the variational formulation of the Stokes Wave by Arama and Leoni. The elimination of cusps implies that the results of [Mccurdy20] in fact describe the entire free-boundary as it intersects $Γ$.
A biharmonic analogue of the Alt-Caffarelli problem
该研究针对Dirichlet和Navier边界条件下的Alt-Caffarelli问题双调和类似问题，证明了极小元的存在性、基本性质与$C^{1,α}$正则性，得到Navier问题径向边界数据下的对称性结果，并验证了该正则性的最优性。
Linearized equation and generic regularity in the Alt-Caffarelli problem
本文研究 Alt-Caffarelli 问题中能量极小元的自由边界正则性，通过分析齐次极小元周围线性化方程的正解、证明 Harnack 不等式及主特征值维数下界，得到六维下一般边界数据的解析性并改进奇异集维数。
An anisotropic Alt-Caffarelli problem of higher order
本文研究二维高阶 Alt-Caffarelli 问题，用各向异性弯曲能替代 Dirichlet 能，证明光滑各向异性不影响极小化子的最优 $C^{2,1}$ 正则性，并推广了 Frehse 估计，为高阶自由边界问题研究奠定基础。
Regularity of Lipschitz free boundaries for weak solutions of Alt-Caffarelli type problems
本文研究了Alt-Caffarelli型问题弱解的Lipschitz自由边界正则性，证明在Lipschitz领域下弱解实际上具有无限次可微性，并提供了Serrin问题的另一种解法。
A minimizing cone from Cartan polynomials in the Alt-Caffarelli problem
本文研究 Cartan 等参三次多项式导出的一相问题 1-齐次解，证明其在维度 5、8、14 不稳定，在维度 26 极小，首次给出非轴对称且非 Lawson 型的奇异极小锥。
This paper considers the Alt-Caffarelli free boundary problem in a periodic medium. This is a convenient model for several interesting phenomena appearing in the study of contact lines on rough surfaces, pinning, hysteresis and the formation of facets. We show the existence of an interval of effective pinned slopes at each direction $e \in S^{d-1}$. In $d=2$ we characterize the regularity properties of the pinning interval in terms of the normal direction, including possible discontinuities at rational directions. These results require a careful study of the families of plane-like solutions av…
In this paper we prove a $C^{1,α}$ regularity result in dimension two for almost-minimizers of the constrained one-phase Alt-Caffarelli and the two-phase Alt-Caffarelli-Friedman functionals for an energy with variable coefficients. As a consequence, we deduce the complete regularity of solutions of a multiphase shape optimization problem for the first eigenvalue of the Dirichlet-Laplacian up to the fixed boundary. One of the main ingredient is a new application of the epiperimetric-inequality of Spolaor-Velichkov [CPAM, 2018] up to the boundary. While the framework that leads to this applicati…
We prove an Alt-Caffarelli-Friedman montonicity formula for pairs of functions solving elliptic equations driven by different ellipticity matrices in their positivity sets. As application, we derive Liouville-type theorems for subsolutions of some elliptic systems, and we analyze segregation phenomena for systems of equations where the diffusion of each density is described by a different operator.
Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and Carleson's $\varepsilon^2$ conjecture in higher dimensions
本文证明了高维Dirichlet拉普拉斯算子Faber-Krahn不等式的定量稳定性，推广并部分证明了Carleson的$ε^2$猜想，还量化了Hayman-Friedland不等式并否定回答了相关公开问题。
In this paper we prove uniqueness of blow-ups and $C^{1,\log}$-regularity for the free-boundary of minimizers of the Alt-Caffarelli functional at points where one blow-up has an isolated singularity. We do this by establishing a (log-)epiperimetric inequality for the Weiss energy for traces close to that of a cone with isolated singularity, whose free-boundary is graphical and smooth over that of the cone in the sphere. With additional assumptions on the cone, we can prove a classical epiperimetric inequality which can be applied to deduce a $C^{1,α}$ regularity result. We also show that these…
For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace-Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds.
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 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。
Boundary regularity of a fourth order Alt-Caffarelli problem and applications to the minimization of the critical buckling load
本文研究了四阶Alt-Caffarelli问题的边界正则性，通过改进的单调性公式和epiperimetric不等式，得出了最优形状在临界屈曲载荷最小化中的正则性结果。
Generic properties in free boundary problems
该研究针对自由边界问题，证明了含Alt-Caffarelli、Alt-Phillips泛函在内的一大类能量泛函极小元的通有唯一性，以及对应单相泛函极小元自由边界的通有正则性，明确了不同维度下的光滑性结果与奇异集维数特征。
In this note we prove two monotonicity formulas for solutions of $Δ_H f = c$ and $Δ_H f - \p_t f = c$ in Carnot groups. Such formulas involve the right-invariant \emph{carré du champ} of a function and they are false for the left-invariant one. The main results, Theorems 1.1 and 1.2, display a resemblance with two deep monotonicity formulas respectively due to Alt-Caffarelli-Friedman for the standard Laplacian, and to Caffarelli for the heat equation. In connection with this aspect we ask the question whether an "almost monotoniccity" formula be possible. In the last section we discuss the fai…
Monotonicity formulas in positively curved settings with applications to two-phase free boundary problems
本文针对二维正曲率环境构造ACF泛函的变体并证明其几乎单调性公式，借此得到两相自由边界问题极小元的新利普希茨界，常数仅依赖总积分曲率，还恢复了特定情形下的已有利普希茨界。
In this article we study functionals of the type considered in \cite{HS21}, i.e. $$ J(v):=\int_{B_1} A(x,u)|\nabla u|^2 +f(x,u)u+ Q(x)λ(u)\,dx $$ here $A(x,u)= A_+(x)χ_{\{u>0\}}+A_-(x) χ_{\{u<0\}}$, $f(x,u)= f_+(x)χ_{\{u>0\}}+f_-(x) χ_{\{u<0\}}$ and $λ(x,u) = λ_+(x) χ_{\{u>0\}} + λ_-(x) χ_{\{u\le 0\}}$. We prove the optimal $C^{0,1^-}$ regularity of minimizers of the functional indicated above (with precise Hölder estimates) when the coefficients $A_{\pm}$ are continuous functions and $μ\le A_{\pm}\le \frac{1}μ$ for some $0<μ<1$, with $f \in L^N(B_1)$ and $Q$ bounded. We do this by presenting …
Optimal Hölder regularity for solutions to Signorini-type obstacle problems
本文研究一类障碍问题弱解的存在性、唯一性和正则性，建立了Signorini型问题（障碍条件施加在余维1子集上）的最优Hölder正则性，并应用于不规则障碍的经典障碍问题。
ACF-Monotonicity Formula on RCD(0,N) Metric Measure Cones
本文将研究两相自由边界问题的ACF单调性公式推广至RCD(0,N)度量测度锥，并以此为应用给出了该锥的刚性结果。
Liouville theorems and optimal regularity in elliptic equations
该研究建立了含可测系数椭圆偏微分方程解的最优正则性与无穷远刘维尔性质的关联，通过单调性公式、填洞技巧等方法分别处理二维与高维情形，探讨了增长性与正则性指数的相互作用。
Serrin's overdetermined theorem and weak Bernoulli laws without Alt--Caffarelli regularity
本文研究几何不规则领域中的分布Bernoulli型条件，证明了在无Alt-Caffarelli正则性假设下，存在非球形有限周长领域满足弱Bernoulli定律，但表面密度界失效，从而反驳了Serrin型过定问题。
In this paper we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions. We show that solutions with non-flat profile can be found variationally as global minimizers of the classical Alt-Caffarelli energy functional.
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.
An extension problem of higher order operators and operators of logarithmic type via renormalization
本文借助重整化思想构造 Caffarelli-Silvestre 型高阶延拓问题，并为已有对数 Laplacian 延拓理论提供新的解释视角。
We show the existence of a Lipschitz viscosity solution $u$ in $Ω$ to a system of fully nonlinear equations involving Pucci-type operators. We study the regularity of the interface $\partial \{ u> 0 \}\cap\Om$ and we show that the viscosity inequalities of the system imply, in the weak sense, the free boundary condition $u^{+}_{ν_{+}} = u^{-}_{ν_{-}}$, and hence $u$ is a solution to a two-phase free boundary problem. We show that we can apply the classical method of sup-convolutions developed by the first author in \cite{caffarelli_harnack_1987,caffarelli_harnack_1989}, and generalized by Wang…
In this paper we give an unified approach to some questions arising in different fields of nonlinear analysis, namely: (a) the study of the structure of the Fuč\'ık spectrum and (b) possible variants and extensions of the monotonicity formula by Alt--Caffarelli--Friedman \cite{acf}. In the first part of the paper we present a class of optimal partition problems involving the first eigenvalue of the Laplace operator. Beside establishing the existence of the optimal partition, we develop a theory for the extremality conditions and the regularity of minimizers. As a first application of this appr…
Consider the following coupled elliptic system of equations \begin{equation*} \label{} (-Δ)^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} \ \ \mathbb{R}^n , \end{equation*} where $0<s\le 2$, $p>1$, $m\ge1$, $u=(u_i)_{i=1}^m$ and $u_i:\mathbb R^n\to \mathbb R$. The qualitative behavior of solutions of the above system has been studied from various perspectives in the literature including the free boundary problems and the classification of solutions. For the case of local scalar equation, that is when $m=1$ and $s=1$, Gidas and Spruck in \cite{gs} and later Caffarelli, Gidas …
Let $Ω\subset\mathbb{R}^{n+1}$, $n\geq2$, be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^*$ is the operator in divergence form associated with the transpose matrix of $A$, then $\partialΩ$ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded…
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