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.
In this paper, we consider the weighted fourth order equation $$Δ(|x|^{-α}Δu)+λ\text{div}(|x|^{-α-2}\nabla u)+μ|x|^{-α-4}u=|x|^βu^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\},$$ where $n\geq 5$, $-n<α<n-4$, $p>1$ and $(p,α,β,n)$ belongs to the critical hyperbola $$\frac{n+α}{2}+\frac{n+β}{p+1}=n-2.$$ We prove the existence of radial solutions to the equation for some $λ$ and $μ$. On the other hand, let $v(t):=|x|^{\frac{n-4-α}{2}}u(|x|)$, $t=-\ln |x|$, then for the radial solution $u$ with non-removable singularity at origin, $v(t)$ is a periodic function if $α\in (-2,n-4)$ and $λ$, $μ…
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 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 $Γ$.
We prove the (optimal) $W^{1,\infty}$-regularity of weak solutions to the equation $-Δu = Q \; \mathcal{H}^{n-1} \llcorner Γ$ in a domain $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions, where $Γ\subset \subset Ω$ is a compact (Lipschitz) manifold and $Q \in L^\infty(Γ)$. We also discuss optimality and necessity of the assumptions on $Q$ and $Γ$. Our findings can be applied to study the regularity of solutions for several free boundary problems, in particular the biharmonic Alt-Caffarelli Problem.
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…
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 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…
We prove full boundary regularity for minimizing biharmonic maps with smooth Dirichlet boundary conditions. Our result, similarly as in the case of harmonic maps, is based on the nonexistence of nonconstant boundary tangent maps. With the help of recently derivated boundary monotonicity formula for minimizing biharmonic maps by Altuntas we prove compactness at the boundary following Scheven's interior argument. Then we combine those results with the conditional partial boundary regularity result for stationary biharmonic maps by Gong--Lamm--Wang.
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 …
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.
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 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…
The study of fourth order partial differential equations has flourished in the last years, however, a $p(\cdot)$-biharmonic problem with no-flux boundary condition has never been considered before, not even for constant $p$. This is an important step further, since surfaces that are impermeable to some contaminants are appearing quite often in nature, hence the significance of such boundary condition. By relying on several variational arguments, we obtain the existence and the multiplicity of weak solutions to our problem. We point out that, although we use a mountain pass type theorem in orde…
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.
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.
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.
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.
We obtain density estimates for the free boundaries of minimizers $u \ge 0$ of the Alt-Phillips functional involving negative power potentials $$\int_Ω\left(|\nabla u|^2 + u^{-γ} χ_{\{u>0\}}\right) \, dx, \quad \quad γ\in (0,2).$$ These estimates remain uniform as the parameter $γ\to 2$. As a consequence we establish the uniform convergence of the corresponding free boundaries to a minimal surface as $γ\to 2$. The results are based on the $Γ$-convergence of these energies (properly rescaled) to the Dirichlet-perimeter functional $$\int_Ω|\nabla u|^2 dx + Per_Ω(\{ u=0\}),$$ considered by Athana…
We study a Bernoulli type free boundary problem with two phases $$J[u]=\int_Ω|\nabla u(x)|^2\,dx+Φ\big({\mathcal M}_-(u), {\mathcal{M}}_+(u)\big), \quad u-\bar u\in W^{1,2}_0(Ω), $$ where $\bar u\in W^{1,2}(Ω)$ is a given boundary datum. Here, ${\mathcal M}_1$ and ${\mathcal M}_2$ are weighted volumes of $\{u\le 0\}\cap Ω$ and $\{u>0\}\cap Ω$, respectively, and $Φ$ is a nonnegative function of two real variables. We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the wei…
In this paper we prove the rectifiability of and measure bounds on the singular set of the free boundary for minimizers of a functional first considered by Alt-Caffarelli. Our main tools are the Quantitative Stratification and Rectifiable-Reifenberg framework of Naber-Valtorta, which allow us to do a type of "effective dimension-reduction." The arguments are sufficiently robust that they apply to a broad class of related free boundary problems as well.
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 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 …
We prove a strong maximum principle for minimizers of the one-phase Alt-Caffarelli functional. We use this to construct a Hardt-Simon-type foliation associated to any 1-homogenous global minimizer.
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.
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…
