arXiv subjects
Publications and source records attributed to Gianmarco Giovannardi.
In every sub-Riemannian Heisenberg group $\mathbb H^n$, with $n\geq 2$, we prove the existence of a global perimeter minimizer whose boundary is a smooth, non-characteristic, entire intrinsic graph but it is not a vertical hyperplane.
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.
This paper examines minimal hypersurfaces in sub-Riemannian Heisenberg groups. We extend the celebrated Simons formula and Kato inequality to the sub-Riemannian setting, and we apply them to obtain integral curvature estimates for stable hypersurfaces. These results lead to structural conditions that imply a Bernstein-type rigidity theorem for smooth, non-characteristic hypersurfaces in the second Heisenberg group.
In this paper we study Monge solutions to stationary Hamilton-Jacobi equations associated to discontinuous Hamiltonians in the framework of Carnot groups. After showing the equivalence between Monge and viscosity solutions in the continuous setting, we prove existence and uniqueness for the Dirichlet problem, together with a comparison principle and a stability result.
We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body $K_0 \subseteq \mathbb{R}^{2n}$, for $t$-graphs on a bounded domain $Ω$ in the Heisenberg group $\mathbb{H}^n$. When the prescribed datum $H$ is constant and strictly smaller that the Finsler mean curvature of $\partial Ω$, we prove the existence of a Lipschitz solution to the Dirichlet problem for the sub-Finsler CMC equation by means of a Finsler approximation scheme.
We introduce a class of nonlinear partial differential equations in a product space which are at the interface of Finsler and sub-Riemannian geometry. To such equations we associate a non-isotropic Minkowski gauge $Θ$ for which we introduce a suitable notion of Legendre transform $Θ^0$. We compute the action of the relevant nonlinear PDEs on ``radial" functions, i.e., functions of $Θ^0$, and by exploiting it we are able to compute explicit fundamental solutions of such PDEs.
We prove the existence and multiplicity of weak solutions for a mixed local-nonlocal problem at resonance. In particular, we consider a not necessarily positive operator which appears in models describing the propagation of flames. A careful adaptation of well known variational methods is required to deal with the possible existence of negative eigenvalues.
In the Heisenberg group $\mathbb{H}^1$, equipped with a left-invariant and not necessarily symmetric norm in the horizontal distribution, we provide examples of entire area-minimizing horizontal graphs which are locally Lipschitz in Euclidean sense. A large number of them fail to have further regularity properties. The examples are obtained by prescribing as singular set a horizontal line or a finite union of horizontal half-lines extending from a given point. We also provide examples of families of area-minimizing cones.
We establish the Schauder estimates at the boundary away from the characteristic points for the Dirichlet problem by means of the double layer potential in a Heisenberg-type group $\mathbb{G}$. Despite its singularity we manage to invert the double layer potential restricted to the boundary thanks to a reflection technique for an approximate operator in $\mathbb{G}$. This is the first instance where a reflection-type argument appears to be useful in the sub-Riemannian setting.
In this paper we study the asymptotic behavior of solutions to the subelliptic $p$-Poisson equation as $p\to +\infty$ in Carnot Carathéodory spaces. In particular, introducing a suitable notion of differentiability, we extend the celebrated result of Bhattacharya, DiBenedetto and Manfredi [Rend. Sem. Mat. Univ. Politec. Torino, 1989, Special Issue, 15-68] and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the $\infty-$Laplacian and the Eikonal equation.
We prove that in the Heisenberg group $\mathbb{H}^1$ with a sub-Finsler structure, an $(X,Y)$-Lipschitz surface which is complete, oriented, connected and stable must be a vertical plane. In particular, the result holds for entire intrinsic graphs of Euclidean Lipschitz functions.
We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and this condition is expressed via a differential equation along the curve. In the classical differential geometry setting, the analogous condition was considered by Bryant and Hsu in [Invent. Math., 114(2):435-461, 1993, J. Differential Geom., 36(3):551-589, 1992], who proved that it is equivalent to the surjectivity of a holonomy map. The purpose of this paper is to extend this deformation theory to curves of fixed degree providing several examples and applications. In particular, we give a useful sufficient condition to guarantee the possibility of deforming a curve.
We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns out that the associated variational vector fields must satisfy a system of partial differential equations of first order on the submanifold. Moreover, given a vector field solution of this system, we provide a sufficient condition that guarantees the possibility of deforming the original submanifold by variations preserving its degree. As in the case of singular curves in sub-Riemannian geometry, there are examples of isolated surfaces that cannot be deformed in any direction. When the deformability condition holds we compute the Euler-Lagrange equations. The resulting mean curvature operator can be of third order.
For a strictly convex set $K\subset \mathbb{R}^2$ of class $C^2$ we consider its associated sub-Finsler $K$-perimeter $|\partial E|_K$ in $\mathbb{H}^1$ and the prescribed mean curvature functional $|\partial E|_K-\int_E f$ associated to a function $f$. Given a critical set for this functional with Euclidean Lipschitz and intrinsic regular boundary, we prove that their characteristic curves are of class $C^2$ and that this regularity is optimal. The result holds in particular when the boundary of $E$ is of class $C^1$.
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,μ_X)$ satisfying a $2$-Poincaré inequality. Given a bounded domain $Ω\subset X$ with $μ_X(X\setminusΩ)>0$, and a function $f$ in the Besov class $B^θ_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^θ_{2,2}(X)$ such that $u=f$ in $X\setminusΩ$ and $\mathcal{E}_θ(u,u)\le \mathcal{E}_θ(h,h)$ whenever $h\in B^θ_{2,2}(X)$ with $h=f$ in $X\setminusΩ$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on $Ω$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.
The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [J. Differential Geom., 36(3):551-589, 1992], and we provide a characterization for singularities as well as a deformability criterion.