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Michele Gatti

Publications and source records attributed to Michele Gatti.

7 recordsLinked to original sources
arXiv2026

The Liouville equation on Riemannian surfaces: the role of volume growth in classification and rigidity results

We study the Liouville equation $-Δu = e^u$ on a complete, connected, non-compact, boundaryless Riemannian surface $(M, g)$ with non-negative Ricci curvature. Assuming only some asymptotic lower bound on the solution, we establish classification results for both the solutions and the ambient manifold, discussing also their optimality. Our results reveal a close connection between the volume growth of the manifold and the classification of both the solutions and the underlying manifold.

math.AP↗
arXiv2026

On the anisotropic critical $p$-Laplace equation: classification, decomposition, and stability results

We investigate both qualitative and quantitative issues related to the classification of non-negative energy solutions to the anisotropic critical $p$-Laplace equation in $\mathbb{R}^n$, for $1<p<n$. Specifically, we establish an anisotropic version of Struwe's decomposition, along with the interaction estimate for the family of bubbles in this decomposition. Moreover, we provide a short proof of the classification result as well as a quantitative stability result, proving that every energy solution to a perturbation of the anisotropic critical equation must be closed to a bubble, in the absence of bubbling.

math.AP↗
arXiv2026

Fractional Dirichlet problems with an overdetermined nonlocal Neumann condition

We investigate symmetry and quantitative approximate symmetry for an overdetermined problem related to the fractional torsion equation in a regular open, bounded set $Ω\subseteq \mathbb{R}^n$. Specifically, we show that if $\overlineΩ$ has positive reach and the nonlocal normal derivative introduced in (Dipierro, Ros-Oton, Valdinoci, Rev. Mat. Iberoam. 33 (2017), no. 2, 377-416) is constant on an external surface parallel and sufficiently close to $\partial Ω$, then $Ω$ must be a ball. Remarkably, this conclusion remains valid under the sole assumption that $Ω$ is convex. Moreover, we analyze the quantitative stability of this result under two distinct sets of assumptions on $Ω$. Finally, we extend our analysis to a broader class of overdetermined Dirichlet problems involving the fractional Laplacian.

math.AP↗
arXiv2025

On the stability of the critical $p$-Laplace equation

For $1<p<n$, it is well-known that non-negative, energy weak solutions to $Δ_p u + u^{p^{\ast}-1} =0$ in $\mathbb{R}^n$ are completely classified. Moreover, due to a fundamental result by Struwe and its extensions, this classification is stable up to bubbling. In the present work, we investigate the stability of perturbations of the critical $p$-Laplace equation for any $1<p<n$, under a condition that prevents bubbling. In particular, we show that any solution $u \in \mathcal{D}^{1,p}(\mathbb{R}^n)$ to such a perturbed equation must be quantitatively close to a bubble. This result generalizes a recent work by the first author, together with Figalli and Maggi (Int. Math. Res. Not. IMRN 2018 (2018), no. 21, 6780-6797), in which a sharp quantitative estimate was established for $p=2$. However, our analysis differs completely from theirs and is based on a quantitative $P$-function approach.

math.AP↗
arXiv2025

Approximate radial symmetry for $p$-Laplace equations via the moving planes method

We investigate quasi-symmetry for small perturbations of the Gidas-Ni-Nirenberg problem involving the $p$-Laplacian and for small perturbations the critical $p$-Laplace equation for $p>2$. To achieve these results, we provide a quantitative review of the work by Damascelli & Sciunzi (Calc. Var. Partial Differential Equations 25 (2006), no. 2, 139-159) concerning the weak Harnack comparison inequality and the local boundedness comparison inequality. Moreover, we prove a comparison principle for small domains.

math.AP↗
arXiv2025

A quantitative study of radial symmetry for solutions to semilinear equations in $\mathbb{R}^n$

A celebrated result by Gidas, Ni & Nirenberg asserts that positive classical solutions, decaying at infinity, to semilinear equations $Δu +f(u)=0$ in $\mathbb{R}^n$ must be radial and radially decreasing. In this paper, we consider both energy solutions in $\mathcal{D}^{1,2}(\mathbb{R}^n)$ and non-energy local weak solutions to small perturbations of these equations, and study its quantitative stability counterpart. To the best of our knowledge, the present work provides the first quantitative stability result for non-energy solutions to semilinear equations involving the Laplacian, even for the critical nonlinearity.

math.AP↗