arXiv subjects

M. Cristina Caputo

Publications and source records attributed to M. Cristina Caputo.

7 recordsLinked to original sources

arXiv2014

Duality for the left and right fractional derivatives

We prove duality between the left and right fractional derivatives, independently on the type of fractional operator. Main result asserts that the right derivative of a function is the dual of the left derivative of the dual function or, equivalently, the left derivative of a function is the dual of the right derivative of the dual function. Such duality between left and right fractional operators is useful to obtain results for the left operators from analogous results on the right operators and vice versa. We illustrate the usefulness of our duality theory by proving a fractional integration by parts formula for the right Caputo derivative and by proving a Tonelli-type theorem that ensures the existence of minimizer for fractional variational problems with right fractional operators.

math.OC↗

arXiv2011

Regularity for non-local almost minimal boundaries and applications

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-De Giorgi-Tamanini regularity theory. The main result has several applications, among these $C^{1,α}$ regularity for sets with prescribed non-local mean curvature in $L^p$ and regularity of solutions to non-local obstacle problems.

math.AP↗

arXiv2010

Time scales: from Nabla calculus to Delta calculus and vice versa via duality

In this note we show how one can obtain results from the nabla calculus from results on the delta calculus and vice versa via a duality argument. We provide applications of the main results to the calculus of variations on time scales.

math.OC↗

arXiv2009

Minimally invasive surgery for Ricci flow singularities

In this paper, we construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on S^(n+1), without performing an intervening surgery. In the restrictive context of rotational symmetry, this construction gives evidence in favor of Perelman's hope for a "canonically defined Ricci flow through singularities".

math.DG↗

arXiv2009

On the evolution of convex hypersurfaces by the $Q_k$ flow

We prove the existence and uniqueness of a $C^{1,1}$ solution of the $Q_k$ flow in the viscosity sense for compact convex hypersurfaces $Σ_t$ embedded in $R^{n+1}$ ($n \geq 2$) . In particular, for compact convex hypersurfaces with flat sides we show that, under a certain non-degeneracy initial condition, the interface separating the flat from the strictly convex side, becomes smooth, and it moves by the $Q_{k-1}$ flow at least for a short time.

math.AP↗

arXiv2009

Global regularity of solutions to systems of reaction-diffusion with Sub-Quadratic Growth in any dimension

This paper is devoted to the study of the regularity of solutions to some systems of reaction--diffusion equations, with reaction terms having a subquadratic growth. We show the global boundedness and regularity of solutions, without smallness assumptions, in any dimension $N$. The proof is based on blow-up techniques. The natural entropy of the system plays a crucial role in the analysis. It allows us to use of De Giorgi type methods introduced for elliptic regularity with rough coefficients. In spite these systems are entropy supercritical, it is possible to control the hypothetical blow-ups, in the critical scaling, via a very weak norm. Analogies with the Navier-Stokes equation are briefly discussed in the introduction.

math.AP↗

arXiv2008

Highly Degenerate Harmonic Mean Curvature Flow

We study the evolution of a weakly convex surface $Σ_0$ in $\R^3$ with flat sides by the Harmonic Mean Curvature flow. We establish the short time existence as well as the optimal regularity of the surface and we show that the boundaries of the flat sides evolve by the curve shortening flow. It follows from our results that a weakly convex surface with flat sides of class $C^{k,γ}$, for some $k\in \mathbb{N}$ and $0 < γ\leq 1$, remains in the same class under the flow. This distinguishes this flow from other, previously studied, degenerate parabolic equations, including the porous medium equation and the Gauss curvature flow with flat sides, where the regularity of the solution for $t >0$ does not depend on the regularity of the initial data.

math.AP↗