arXiv subjects

Carlo Gasparetto

Publications and source records attributed to Carlo Gasparetto.

6 recordsLinked to original sources

arXiv2025

Non-rigidity of the absolutely continuous part of $\mathcal{A}$-free measures

We generalize a result by Alberti, showing that, if a first-order linear differential operator $\mathcal{A}$ belongs to a certain class, then any $L^1$ function is the absolutely continuous part of a measure $μ$ satisfying $\mathcal{A}μ=0$. When $\mathcal{A}$ is scalar valued, we provide a necessary and sufficient condition for the above property to hold true and we prove dimensional estimates on the singular part of $μ$. Finally, we show that operators in the above class satisfy a Lusin-type property.

math.AP↗

arXiv2025

Regularity for free boundary surfaces minimizing degenerate area functionals

We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\mathrm{Per}_{w}(E)=\int_{\partial^*E}w\,\mathrm{d} {\mathscr{H}}^{n-1}$, where $w$ is a weight asymptotic to $d(\cdot,\mathbb{R}^n\setminusΩ)^a$ near $\partialΩ$ and $a>0$. This implies that the boundaries of almost-minimizers are $C^{1,γ_0}$-surfaces that touch $\partial Ω$ orthogonally, up to a Singular Set $\mathrm{Sing}(\partial E)$ whose Hausdorff dimension satisfies the bound $d_{\mathscr{H}}(\mathrm{Sing}(\partial E)) \leq n +a -(5+\sqrt{8})$.

math.AP↗

arXiv2024

Regularity of minimal surfaces with capillary boundary conditions

We prove $\varepsilon$-regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya-Tonegawa \cite{KaTo}. We establish a uniform first variation control for all such varifolds (and free-boundary varifolds generally) satisfying a sharp density bound and prove that if a capillary varifold has bounded mean curvature and is close to a capillary half-plane with angle not equal to $\tfracπ{2}$, then it coincides with a $C^{1,α}$ properly embedded hypersurface. We apply our theorem to deduce regularity at a generic point along the boundary in the region where the density is strictly less than $1$.

math.DG↗

arXiv2024

Epsilon-regularity for the Brakke flow with boundary

We prove that, if a Brakke flow with boundary is close enough to a stationary half-plane with density one, then it is $C^{1,α}$. Our approach is based on viscosity techniques introduced by Savin in the context of elliptic equations. The same techniques can be used to give a proof of Brakke's (interior) regularity theorem which is alternative to the original one.

math.AP↗

arXiv2023

A short proof of Allard's and Brakke's regularity theorems

We give new short proofs of Allard's regularity theorem for varifolds with bounded first variation and Brakke's regularity theorem for integral Brakke flows with bounded forcing. They are based on a decay of flatness, following from weighted versions of the respective monotonicity formulas, together with a characterization of non-homogeneous blow-ups using the viscosity approach introduced by Savin.

math.AP↗

arXiv2017

Resonance tongues for the Hill equation with Duffing coefficients and instabilities in a nonlinear beam equation

We consider a class of Hill equations where the periodic coefficient is the squared solution of some Duffing equation plus a constant. We study the stability of the trivial solution of this Hill equation and we show that a criterion due to Burdina (V.I. Burdina, Boundedness of solutions of a system of differential equations) is very helpful for this analysis. In some cases, we are also able to determine exact solutions in terms of Jacobi elliptic functions. Overall, we obtain a fairly complete picture of the stability and instability regions. These results are then used to study the stability of nonlinear modes in some beam equations.

math.CA↗