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Alessandro Goffi

Publications and source records attributed to Alessandro Goffi.

At least 19 recordsLinked to original sources
arXiv2026

Regularizing estimates for positive solutions of the heat equation under geometric flows

We study higher-order global estimates for the heat equation on Riemannian manifolds, both for static metrics and for metrics evolving under the Ricci flow. Under minimal geometric assumptions, we derive first-order regularizing estimates for log-solutions of the heat equation, together with upper second-order bounds with explicit constants. Our quantitative approach is based on integral duality methods proposed by L.\ C.\ Evans, J.-M.\ Lasry and P.-L.\ Lions in different settings.

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arXiv2026

On the rate of the vanishing viscosity approximation for Mean Field Games with nonlocal coupling

We study quantitative convergence rates of the vanishing viscosity approximation of first-order time-dependent Mean Field Games with regularizing coupling acting in the Hamilton-Jacobi equation. Under standard structural assumptions on the Hamiltonian ensuring convergence of the vanishing viscosity approximation, previous results provide either qualitative convergence for the two unknown of the system or quantitative estimates merely for solutions of the Hamilton-Jacobi equation. In this work, we improve these convergence rates under the same assumptions and establish, in addition, quantitative estimates for the convergence of the associated forward Fokker-Planck equation. As a consequence, we obtain quantitative convergence rates for the full Mean Field Game system, thus extending and strengthening the existing theory for the first-order limit.

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arXiv2026

Quantitative maximal $L^2$-regularity for viscous Hamilton-Jacobi PDEs in 2D and Mean Field Games

We discuss quantitative Calder\'on-Zygmund estimates in $W^{2,2}$ for 2D viscous Hamilton-Jacobi equations with natural growth in the gradient. We apply the result to obtain the existence of classical solutions for stationary second order Mean Field Games systems in 2D with (defocusing) coupling behaving like $m^\alpha$ for any $\alpha>0$. We also survey on the known results for the regularity of viscous Hamilton-Jacobi equations and second order Mean Field Games and list several open problems.

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arXiv2026

Transport-diffusion equations with irregular data and applications to stability estimates for second-order Hamilton-Jacobi PDEs

This paper studies quantitative uniqueness properties in $L^p$ spaces for Fokker-Planck and transport-diffusion equations under two new assumptions on their velocity field $b=b(x,t)$. We first prove $L^p$-stability estimates for advection-diffusion PDEs when $\mathrm{div}(b)\in L^r_t(L^q_x)$ with $r\in[2,\infty]$ and $q\in[n/2,\infty)$ satisfying the compatibility condition $n/(2q)+1/r\leq 1$. We then prove a stability result in $L^\infty$ for solutions of viscous transport equations when $\mathrm{div}(b(t))$ fails to be integrable in time. We apply these properties to obtain new continuous dependence estimates for viscous Hamilton-Jacobi equations via integral methods. One of the main novelties in this latter setting is that the constants of the estimates are all explicit with respect to the data of the problem. These imply new uniqueness properties for diffusive Hamilton-Jacobi equations without relying on the theory of viscosity solutions.

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arXiv2025

On the smoothness of solutions of fully nonlinear second order equations in the plane

We study interior $C^{2,\alpha}$ regularity estimates for solutions of fully nonlinear uniformly elliptic equations of the general form $F(D^2u)=0$ in two independent variables and without any geometric condition on $F$. By means of the theory of divergence form equations we prove that $C^2$ solutions of the previous equation are $C^{2,\bar\alpha(\lambda/\Lambda)}$ in the interior of the domain, where $0<\lambda\leq\Lambda$ are the ellipticity constants. We finally exploit the theory of nondivergence equations in the plane to obtain $C^{2,\tilde\alpha}$ regularity for an explicit exponent $\tilde\alpha=\tilde\alpha(\lambda/\Lambda)>\lambda/\Lambda$.

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arXiv2025

Convergence rates for the vanishing viscosity approximation of fully nonlinear, non-convex, second-order Hamilton-Jacobi equations

We obtain new quantitative estimates of the vanishing viscosity approximation for time-dependent, degenerate, Hamilton-Jacobi equations that are neither concave nor convex in the gradient and Hessian entries of the form $\partial_t u+H(x,t,Du,D^2u)=0$ in the whole space. We approximate the PDE with a fully nonlinear, possibly degenerate, elliptic operator $\varepsilon F(x,t,D^2u)$. Assuming that $u\in C^\alpha_x$, $u_0\in C^\eta$, $H\in C^\beta_x$ and having power growth $\gamma$ in the gradient entry, we establish a convergence rate of order $\varepsilon^{\min\left\{\frac{\eta}{2},\frac{\beta+\gamma(\alpha-1)}{\beta+\gamma(\alpha-1)+2-\alpha}\right\}}$. Our novel approach exploits the regularizing properties of sup/inf-convolutions for viscosity solutions and the comparison principle. We also obtain explicit constants and do not assume differentiability properties neither on solutions nor on $H$. The same method provides new convergence rates for the vanishing viscosity approximation of the stationary counterpart of the equation and for transport equations with H\"older coefficients.

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arXiv2025

Rate of convergence of the vanishing viscosity method for Hamilton-Jacobi equations with Neumann boundary conditions

We study the quantitative small noise limit in the $L^\infty$ norm of certain time-dependent Hamilton-Jacobi equations equipped with Neumann boundary conditions, depending on the regularity of the data and the geometric properties of the domain. We first provide a $\mathcal{O}(\sqrt{\varepsilon})$ rate of convergence for Hamilton-Jacobi equations with locally Lipschitz Hamiltonians posed on convex domains of the Euclidean space. We then enhance this speed of convergence in the case of quadratic Hamiltonians proving one-side rates of order $\mathcal{O}(\varepsilon)$ and $\mathcal{O}(\varepsilon^\beta)$, $\beta\in(1/2,1)$. The results exploit recent $L^1$ contraction estimates for Fokker-Planck equations with bounded velocity fields on unbounded domains used to derive differential Harnack estimates for the corresponding Neumann heat flow.

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arXiv2025

Convergence rates for the vanishing viscosity approximation of Hamilton-Jacobi equations: the convex case

We study the speed of convergence in $L^\infty$ norm of the vanishing viscosity process for Hamilton-Jacobi equations with uniformly or strictly convex Hamiltonian terms with superquadratic behavior. Our analysis boosts previous findings on the rate of convergence for this procedure in $L^p$ norms, showing rates in sup-norm of order $\mathcal{O}(\epsilon^\beta)$, $\beta\in(1/2,1)$, or $\mathcal{O}(\epsilon|\log\epsilon|)$ with respect to the vanishing viscosity parameter $\epsilon$, depending on the regularity of the initial datum of the problem and convexity properties of the Hamiltonian. Our proofs are based on integral methods and avoid the use of techniques based on stochastic control or the maximum principle.

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arXiv2024

Remarks on the rate of convergence of the vanishing viscosity process of Hamilton-Jacobi equations

We establish a linear $L^p$ rate of convergence, $1<p<\infty$, with respect to the viscosity $\varepsilon$ for the vanishing viscosity process of semiconcave solutions of Hamilton-Jacobi equations by regularizing the PDE with the half-Laplacian $-\varepsilon(-\Delta)^{1/2}$. Our result reveals a nonlocal phenomenon, since it improves the known estimates obtained via the classical second order vanishing viscosity regularization $\varepsilon\Delta u$. It also highlights a faster rate of convergence than the available $\mathcal{O}(\varepsilon|\log\varepsilon|)$ rate in sup-norm obtained by the doubling of variable technique for this nonlocal approximation. The result is based on integral methods and does not use the maximum principle.

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arXiv2024

Global geometric estimates for the heat equation via duality methods

We discuss first-order and second-order regularization effects for solutions to the classical heat equation. In particular we propose a global approach to study smoothing effects of Hamilton-Li-Yau type: such approach is nonlinear in spirit and it is based on the Bernstein method and duality techniques \`a la Evans. In a similar way, we also deal with the conservation of geometric properties for the heat flow as initiated by Brascamp-Lieb. In contrast to maximum principle methods based on sup-norm procedures, the integral method we adopt relies on contractivity properties for advection-diffusion equations and it applies to problems with homogeneous Neumann conditions posed equally on bounded and unbounded convex domains under suitable assumptions on their geometry.

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arXiv2024

Interpolated time-H\"older regularity of solutions of fully nonlinear parabolic equations

We show interior Schauder estimates for a special class of fully nonlinear parabolic Isaacs equations by the maximum principle, providing an Evans-Krylov result for the model equation $\min\{\inf_{\beta}L_\beta u,\sup_\gamma L_\gamma u\}-\partial_t u=0$, where $L_\beta,L_\gamma$ are linear operators with possibly variable H\"older coefficients. We also give a proof of the Evans-Krylov theorem for fully nonlinear uniformly parabolic equations for which a regularity theory of the stationary non-homogeneous equation is available.

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arXiv2023

Interior H\"older and Calder\'on-Zygmund estimates for fully nonlinear equations with natural gradient growth

We establish local H\"older estimates for viscosity solutions of fully nonlinear second order equations with quadratic growth in the gradient and unbounded right-hand side in $L^q$ spaces, for an integrability threshold $q$ guaranteeing the validity of the maximum principle. This is done through a nonlinear Harnack inequality for nonhomogeneous equations driven by a uniformly elliptic Isaacs operator and perturbed by a Hamiltonian term with natural growth in the gradient. As a byproduct, we derive a new Liouville property for entire $L^p$ viscosity solutions of fully nonlinear equations as well as a nonlinear Calder\'on-Zygmund estimate for strong solutions of such equations.

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arXiv2023

On maximal regularity estimates for quasilinear evolution equations via the integral Bernstein method

This work addresses the problem of (global) maximal regularity for quasilinear evolution equations with sublinear gradient growth and right-hand side in Lebesgue spaces, complemented with Neumann boundary conditions. The proof relies on a suitable variation of the Bernstein technique and the Bochner identity, and provides new results even for the simpler parabolic $p$-Laplacian equation with unbounded source term. As a byproduct we also obtain a second-order estimate that can be of independent interest when the right-side of the equation belongs to $L^m$, $m\neq 2$. This approach leads to new results even for stationary problems.

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arXiv2023

High-order estimates for fully nonlinear equations under weak concavity assumptions

This paper studies a priori and regularity estimates of Evans-Krylov type in Hölder spaces for fully nonlinear uniformly elliptic and parabolic equations of second order when the operator fails to be concave or convex in the space of symmetric matrices. In particular, it is assumed that either the level sets are convex or the operator is concave, convex or close to a linear function near infinity. As a byproduct, these results imply polynomial Liouville theorems for entire solutions of elliptic equations and for ancient solutions to parabolic problems.

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arXiv2023

Quantitative and qualitative properties for Hamilton-Jacobi PDEs via the nonlinear adjoint method

We provide some new integral estimates for solutions to Hamilton-Jacobi equations and we discuss several consequences, ranging from $L^p$-rates of convergence for the vanishing viscosity approximation to regularizing effects for the Cauchy problem in the whole Euclidean space and Liouville-type theorems. Our approach is based on duality techniques \`a la Evans and a careful study of advection-diffusion equations. The optimality of the results is discussed by several examples.

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arXiv2023

Interior a priori estimates for supersolutions of fully nonlinear subelliptic equations under geometric conditions

In this note, we prove interior a priori first- and second-order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that $u$ is semiconvex along the fields. These estimates for supersolutions are new even for linear subelliptic inequalities in nondivergence form, whereas in the nonlinear setting they do not require neither convexity nor concavity on the second derivatives. We complement the analysis exhibiting an explicit example showing that horizontal $W^{2,q}$ regularity of Calder\'on-Zygmund type for fully nonlinear subelliptic equations posed on the Heisenberg group cannot be in general expected in the range $q<Q$, $Q$ being the homogeneous dimension of the group.

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arXiv2022

Gradient estimates for quasilinear elliptic Neumann problems with unbounded first-order terms

This paper studies global a priori gradient estimates for divergence-type equations patterned over the $p$-Laplacian with first-order terms having polynomial growth with respect to the gradient, under suitable integrability assumptions on the source term of the equation. The results apply to elliptic problems with unbounded data in Lebesgue spaces complemented with Neumann boundary conditions posed on convex domains of the Euclidean space.

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