Volume 12, Issue 2

We maximize the functional ∫Eh⁢(x)⁢𝑑xP⁢(E),\frac{\int_{E}h(x)\,dx}{P(E)}, where E⊂Ω¯{E\subset\overline{\Omega}} is a set of finite perimeter, Ω is an open bounded set with Lipschitz boundary and h is nonnegative. Solutions to this problem are called generalized Cheeger sets in Ω . We show that the Morrey spaces L1,λ⁢(Ω){L^{1,\lambda}(\Omega)}, λ≥n-1{\lambda\geq n-1}, are natural spaces to study this problem. We prove that if h∈L1,λ⁢(Ω){h\in L^{1,\lambda}(\Omega)}, λ>n-1{\lambda>n-1}, then generalized Cheeger sets exist. We also study the embedding of Morrey spaces into Lp{L^{p}} spaces. We show that, for any 0<λ<n{0<\lambda<n}, the Morrey space L1,λ⁢(Ω){L^{1,\lambda}(\Omega)} is not contained in any Lq⁢(Ω){L^{q}(\Omega)}, 1<q<p=nn-λ{1<q<p=\frac{n}{n-\lambda}}. We also show that if h∈L1,λ⁢(Ω){h\in L^{1,\lambda}(\Omega)}, λ>n-1{\lambda>n-1}, then the reduced boundary in Ω of a generalized Cheeger set is C1,α{C^{1,\alpha}} and the singular set has Hausdorff dimension at most n-8{n-8} (empty if n≤7{n\leq 7}). For the critical case h∈L1,n-1⁢(Ω){h\in L^{1,n-1}(\Omega)}, we demonstrate that this strong regularity fails. We prove that a bounded generalized Cheeger set E in ℝn{\mathbb{R}^{n}} with h∈L1⁢(ℝn){h\in L^{1}(\mathbb{R}^{n})} is always pseudoconvex, and any pseudoconvex set is a generalized Cheeger set for some h .

We establish a local monotonicity identity for vector bundle-valued differential k -forms on superlevel sets of appropriate heat kernel-like functions. As a consequence, we obtain new local monotonicity formulæ for the harmonic map and Yang–Mills heat flows on evolving manifolds. We also show how these methods yield local monotonicity formulæ for the Yang–Mills–Higgs flow.

In this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form 1+β⁢θ{1+\beta\,\theta}, where θ denotes the amount of transported mass and β>0{\beta>0} is a fixed parameter (notice that the limit case β=0{\beta=0} corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ({ℱε}ε>0{\{\mathcal{F}_{\varepsilon}\}_{\varepsilon>0}}) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of {ℱε}{\{\mathcal{F}_{\varepsilon}\}} as ε↓0{\varepsilon\downarrow 0}. Our functionals are modeled on the Ambrosio–Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method.

In this article we prove that the first eigenvalue of the ∞{\infty}-Laplacian {min⁡{-Δ∞⁢v,|∇⁡v|-λ1,∞⁢(Ω)⁢v}=0in ⁢Ω,v=0on ⁢∂⁡Ω,\left\{\begin{aligned} \displaystyle\min\{-\Delta_{\infty}v,|\nabla v|-\lambda% _{1,\infty}(\Omega)v\}&\displaystyle=0&&\displaystyle\text{in }\Omega,\\ \displaystyle v&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right. has a unique (up to scalar multiplication) maximal solution. This maximal solution can be obtained as the limit as ℓ↗1{\ell\nearrow 1} of concave problems of the form {min⁡{-Δ∞⁢vℓ,|∇⁡vℓ|-λ1,∞⁢(Ω)⁢vℓℓ}=0in ⁢Ω,vℓ=0on ⁢∂⁡Ω.\left\{\begin{aligned} \displaystyle\min\{-\Delta_{\infty}v_{\ell},|\nabla v_{% \ell}|-\lambda_{1,\infty}(\Omega)v_{\ell}^{\ell}\}&\displaystyle=0&&% \displaystyle\text{in }\Omega,\\ \displaystyle v_{\ell}&\displaystyle=0&&\displaystyle\text{on }\partial\Omega.% \end{aligned}\right. In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the sub-homogeneous problems as happens for the usual eigenvalue problem for the p -Laplacian for a fixed 1<p<∞{1<p<\infty}.

We prove that balls centered at the origin and with small radius are stable local minimizers of the Gaussian perimeter among all symmetric sets. Precisely, using the second variation of the Gaussian perimeter, we show that if the radius is smaller than n+1{\sqrt{n+1}}, then the ball is a local minimizer, while if it is larger, the ball is not a local minimizer.

We prove a compactness principle for the anisotropic formulation of the Plateau problem in codimension one, along the same lines of previous works of the authors [9, 10]. In particular, we perform a new strategy for proving the rectifiability of the minimal set, avoiding Preiss’ Rectifiability Theorem [22].