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Simone Ciani

Publications and source records attributed to Simone Ciani.

28 records · Page 2Linked to original sources
arXiv2023

Liouville rigidity and time-extrinsic Harnack estimates for an anisotropic slow diffusion

We prove that ancient non-negative solutions to a fully anisotropic prototype evolution equation are constant if they satisfy a condition of finite speed of propagation and if they are both one-sided bounded, and bounded in space at a single time level. A similar statement is valid when the bound is given at a single space point. As a general paradigm, Hölder estimates provide the basics for rigidity. Finally, we show that recent intrinsic Harnack estimates can be improved to a Harnack inequality valid for non-intrinsic times. Locally, they are equivalent.

math.AP↗
arXiv2022

A note on the point-wise behaviour of bounded solutions for a non-standard elliptic operator

In this brief note we discuss local Hölder continuity for solutions to anisotropic elliptic equations of the type $ \sum_{i=1}^s \partial_{ii} u+ \sum_{i=s+1}^N \partial_i \bigg(A_i(x,u,\nabla u) \bigg) =0,$ for $x \in Ω\subset \subset \mathbb{R}^N$ and $1\leq s \leq N-1$, where each operator $A_i$ behaves directionally as the singular $p$-Laplacian, $1< p < 2$ and the supercritical condition $p+(N-s)(p-2)>0$ holds true. We show that the Harnack inequality can be proved without the continuity of solutions and that in turn this implies Hölder continuity of solutions.

math.AP↗
arXiv2022

On a particular scaling for the prototype anisotropic p-Laplacian

In this brief note we show that under a volume non-preserving scaling it is possible to recover the basics for a regularity theory regarding local weak solutions to a parabolic fully anisotropic equation. We characterize self-similar solutions regarding this particular scaling and we show that semi-continuity for solutions to this equation is a consequence of a simple property that is itself invariant under scaling.

math.AP↗
arXiv2022

Evolution equations with nonlocal initial conditions and superlinear growth

We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as particular cases, the Cauchy multipoint problem, the weighted mean value problem and the periodic problem. The dynamic is transformed into an abstract setting and by combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results. By the compactness of the semigroup generated by the linear operator, we do not assume any Lipschitzianity, nor compactness on the nonlinear term or on the nonlocal initial condition. In addition, the exploited approximation technique coupled to a Hartman-type inequality argument, allows to treat nonlinearities with superlinear growth. Moreover, regarding the periodic case we are able to prove the existence of at least one periodic solution on the half line.

math.AP↗
arXiv2021

Dimensional lower bounds for contact surfaces of Cheeger sets

We carry on an analysis of the size of the contact surface of a Cheeger set $E$ with the boundary of its ambient space $Ω$. We show that this size is strongly related to the regularity of $\partial Ω$ by providing bounds on the Hausdorff dimension of $\partial E\cap \partialΩ$. In particular we show that, if $\partial Ω$ has $C^{1,α}$ regularity then $\mathcal{H}^{d-2+α}(\partial E\cap \partialΩ)>0$. This shows that a sufficient condition to ensure that $\mathcal{H}^{d-1}(\partial E\cap \partial Ω)>0$ is that $\partial Ω$ has $C^{1,1}$ regularity. Since the Hausdorff bounds can be inferred in dependence of the regularity of $\partial E$ as well, we obtain that $Ω$ convex, which yields $\partial E\in C^{1,1}$, is also a sufficient condition. Finally, we construct examples showing that such bounds are optimal in dimension $d=2$.

math.AP↗
arXiv2021

Parabolic Harnack estimates for anisotropic slow diffusion

We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker-Planck equation and construct a self-similar Barenblatt solution. We exploit translation invariance to obtain positivity near the origin via a self-iteration method and deduce a sharp anisotropic expansion of positivity. This eventually yields a scale invariant Harnack inequality in an anisotropic geometry dictated by the speed of the diffusion coefficients. As a corollary, we infer Hölder continuity, an elliptic Harnack inequality and a Liouville theorem.

math.AP↗
arXiv2020

A new short proof of regularity for local weak solutions for a certain class of singular parabolic equations

We shall establish the interior Hölder continuity for locally bounded weak solutions to a class of parabolic singular equations whose prototypes are \begin{equation} u_t= \nabla \cdot \bigg( |\nabla u|^{p-2} \nabla u \bigg), \quad \text{ for } \quad 1 3-\frac{p}{N}, \end{equation} via a new and simplified proof using recent techniques on expansion of positivity and $L^{1}$-Harnack estimates.

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