Volume 15, Issue 1

Emmanuele DiBenedetto passed away in May 2021, after battling cancer for fifteen months. I have had the unique privilege to collaborate and discuss Mathematics with him, almost up to his final days. Here I briefly present his life and those mathematical results of his, which I consider most familiar with.

We study the local regularity of the solution u of the following nonlinear boundary value problem: { 𝒜 ⁢ u = - div ⁡ [ E ⁢ ( x ) ⁢ u + F ⁢ ( x ) ] in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle\mathcal{A}u&\displaystyle=-\operatorname{% div}{[E(x)u+F(x)]}&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a bounded open subset of ℝ N {\mathbb{R}^{N}} , with N > 2 {N>2} , 𝒜 {\mathcal{A}} is a nonlinear Leray–Lions operator in divergence form, and E ⁢ ( x ) {E(x)} and F ⁢ ( x ) {F(x)} are vector fields satisfying suitable local summability properties.

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.

In this paper, we investigate the fine properties of functions under suitable geometric conditions on the jump set. Precisely, given an open set Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} and given p > 1 {p>1} , we study the blow-up of functions u ∈ GSBV ⁢ ( Ω ) {u\in\mathrm{GSBV}(\Omega)} , whose jump sets belong to an appropriate class 𝒥 p {\mathcal{J}_{p}} and whose approximate gradients are p -th power summable. In analogy with the theory of p -capacity in the context of Sobolev spaces, we prove that the blow-up of u converges up to a set of Hausdorff dimension less than or equal to n - p {n-p} . Moreover, we are able to prove the following result which in the case of W 1 , p ⁢ ( Ω ) {W^{1,p}(\Omega)} functions can be stated as follows: whenever u k {u_{k}} strongly converges to u , then, up to subsequences, u k {u_{k}} pointwise converges to u except on a set whose Hausdorff dimension is at most n - p {n-p} .

We introduce and study certain variants of Gamow’s liquid drop model in which an anisotropic surface energy replaces the perimeter. After existence and nonexistence results are established, the shape of minimizers is analyzed. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. In sharp contrast, Wulff shapes are the unique minimizers for certain crystalline surface tensions. We also introduce and study several related liquid drop models with anisotropic repulsion for which the Wulff shape is the minimizer in the small mass regime.

We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n -space, where φ ⁢ ( t ) {\varphi(t)} is a non-negative convex function vanishing only at t = 0 {t=0} . We show that this property is always satisfied in dimension n = 2 {n=2} , while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when φ ⁢ ( t ) = c ⁢ t 2 {\varphi(t)=ct^{2}} ) in dimension n ≥ 4 {n\geq 4} . The validity of the quadratic rigidity, which we prove in dimension n = 2 {n=2} , implies the existence of the trace of a divergence-measure vector field ξ on an ℋ 1 {\mathcal{H}^{1}} -rectifiable set S , as soon as its weak normal trace [ ξ ⋅ ν S ] {[\xi\cdot\nu_{S}]} is maximal on S . As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.