In this paper we provide a counterexample about the existence of an increasing monotonicity behavior of a function introduced in \cite{FeFo}, companion of the celebrated Alt-Caffarelli-Friedman monotonicity formula, in the noncommutative framework.
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}.
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.
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."
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 $Γ$.
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.
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.
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…
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.
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…
