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Mirco Piccinini

Publications and source records attributed to Mirco Piccinini.

16 recordsLinked to original sources
arXiv2025

Heteroclinic connections for fractional Allen-Cahn equations with degenerate potentials

We investigate existence, uniqueness and asymptotic behavior of minimizers of a family of non-local energy functionals of the type $$ \frac{1}{4}\iint_{\mathbb{R}^{2n}\setminus (\mathbb{R}^n \setminus \Omega)^2}|u(x)-u(y)|^2 K(x-y) \,dx dy + \int_\Omega W(u(x)) \,dx. $$ Here, $W$ is a possibly degenerate double well potential with a polynomial control on its second derivative near the wells. Also, ${K}$ belongs to a wide class of measurable kernels and is modeled on that of the fractional Laplacian.

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arXiv2024

De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusions

We extend the De Giorgi-Nash-Moser theory to a class of nonlocal hypoelliptic equations arising naturally in kinetic theory, in which a first-order transport operator is coupled with an elliptic nonlocal operator involving fractional derivatives only in part of the variables. Under the sole assumption that the nonlocal tail in velocity of weak solutions is $p$-summable along the drift variables, we prove a local $L^2$-$L^\infty$ estimate for kinetic integral equations and a corresponding strong Harnack inequality. The tail condition is satisfied in standard kinetic regimes considered in the literature, for instance under the usual boundedness of the mass density in the Boltzmann equation without cut-off, and it is consistent with the recent counterexample by Kassmann and Weidner (Adv. Math. 2024). These estimates further lead to a geometric characterization of the Harnack inequality, in the spirit of the seminal work of Aronson and Serrin (Arch. Ration. Mech. Anal. 1967) for the local parabolic counterpart.

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arXiv2024

Regularity for multi-phase problems at nearly linear growth

Minima of the log-multiphase variational integral $$ w \mapsto \int_{\Omega} \left[|Dw|\log(1+|Dw|) + a(x)|Dw|^q + b(x)|Dw|^s\right] \, {\rm d}x\,, $$ have locally H\"older continuous gradient under sharp quantitative bounds linking the growth powers $(q,s)$ to the H\"older exponents of the modulating coefficients $a(\cdot)$ and $b(\cdot)$ respectively.

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arXiv2023

A limiting case in partial regularity for quasiconvex functionals

Local minimizers of nonhomogeneous quasiconvex variational integrals with standard $p$-growth of the type $$ w \mapsto \int \left[F(Dw)-f\cdot w\right]dx $$ feature almost everywhere $\mbox{BMO}$-regular gradient provided that $f$ belongs to the borderline Marcinkiewicz space $L(n,\infty)$.

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arXiv2023

Struwe's Global Compactness and energy approximation of the critical Sobolev embedding in the Heisenberg group

We investigate some of the effects of the lack of compactness in the critical Folland-Stein-Sobolev embedding in very general (possible non-smooth) domains, by proving via De Giorgi's $\Gamma$-convergence techniques that optimal functions for a natural subcritical approximations of the Sobolev quotient concentrate energy at one point. In the second part of the paper, we try to restore the compactness by extending the celebrated Global Compactness result to the Heisenberg group via a completely different approach with respect to the original one by Struwe [37].

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arXiv2023

Nonlinear fractional equations in the Heisenberg group

We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order $(s,p)$, with summability exponent $p \in (1,\infty)$ and differentiability exponent $s\in (0,1)$, whose prototype is the fractional subLaplacian in the Heisenberg group. We present very recent boundedness and regularity estimates (up to the boundary) for the involved weak solutions, and we introduce the nonlocal counterpart of the Perron Method in the Heisenberg group, by recalling some results on the fractional obstacle problem. Throughout the paper we also list various related open problems.

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arXiv2023

Asymptotic approach to singular solutions for the CR Yamabe equation

We study the loss of compactness for optimal functions of a subcritical approximations of the sharp Folland-Stein-Sobolev embedding in the Heisenberg group. When blow-up is prevent at the boundary we show that centered-symmetric maximizers in a Kor\'any ball concentrate at its center. Moreover, we obtain the exact blow-up rate and the pointwise asymptotic profile away from the pole, in turn establishing a natural counterpart of the classical Brezis and Peletier asymptotics for nearly critical Sobolev problems.

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arXiv2023

New perspectives on recent trends for Kolmogorov operators

After carrying out an overview on the non Euclidean geometrical setting suitable for the study of Kolmogorov operators with rough coefficients, we list some properties of the functional space $\mathcal{W}$, mirroring the classical $H^1$ theory for uniformly elliptic operators. Then we provide the reader with the proof of a new Sobolev embedding for functions in $\mathcal{W}$. Additionally, after reviewing recent results regarding weak regularity theory, we discuss some of their recent applications to real life problems arising both in Physics and in Economics. Finally, we conclude our analysis stating some recent results regarding the study of nonlinear nonlocal kinetic Kolmogorov-Fokker-Planck operators.

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arXiv2023

Hölder continuity and boundedness estimates for nonlinear fractional equations in the Heisenberg group

We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator on the Heisenberg-Weyl group $\mathbb{H}^n$. Amongst other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates.

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arXiv2023

The obstacle problem and the Perron Method for nonlinear fractional equations in the Heisenberg group

We study the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional subLaplacian in the Heisenberg group. We prove both the existence and uniqueness of the solution, and that solutions inherit regularity properties of the obstacle such as boundedness, continuity and Hölder continuity up to the boundary. We also prove some independent properties of weak supersolutions to the class of problems we are dealing with. Armed with the aforementioned results, we finally investigate the Perron-Wiener-Brelot generalized solution by proving its existence for very general boundary data.

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arXiv2023

Nonlocal Harnack inequalities in the Heisenberg group

We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group $\mathbb{H}^n$, whose prototype is the Dirichlet problem for the $p$-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is $p=2$, we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent $s$ goes to $1$.

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arXiv2022

Borderline global regularity for nonuniformly elliptic systems

We establish sharp global regularity results for solutions to nonhomogeneous, nonunifomrly elliptic systems with zero boundary conditions. In particular, we obtain everywhere Lipschitz continuity under borderline Lorentz assumptions on the forcing term, thus positively settling the optimality issue raised in \cite{bdms}.

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arXiv2021

The Dirichlet problem for a family of totally degenerate differential operators

In the framework of Potential Theory we prove existence or the Perron-Weiner-Brelot-Bauer solution to the Dirichlet problem related to a family of totally degenerate, in the sense of Bony, differential operators. We also state and prove a Wiener-type criterium and an exterior cone condition for the regularity of a boundary point. Our results apply to a wide family of strongly degenerate operators that includes the following example $\mathcal{L} = t^2\Delta_x + \langle x, \nabla_y \rangle -\partial_t$, for $(x,y,t) \in \mathbb{R}^N \times \mathbb{R}^{N} \times \mathbb{R}$.

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