Papers by Gianmarco Giovannardi
Journal of Mathematical Analysis and Applications
arXiv (Cornell University), Apr 16, 2021
We establish the Schauder estimates at the boundary away from the characteristic points for the D... more 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 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 G. This is the first instance where a reflection-type argument appears to be useful in the sub-Riemannian setting.
arXiv (Cornell University), Feb 7, 2023
arXiv (Cornell University), Jul 27, 2022
We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body... more We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body K 0 ⊆ R 2n , for t-graphs on a bounded domain Ω in the Heisenberg group H n. When the prescribed datum H is constant and strictly smaller that the Finsler mean curvature of ∂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.
The aim of this PhD thesis is to study the area functional for submanifolds immersed in an equire... more The aim of this PhD thesis is to study the area functional for submanifolds immersed in an equiregular graded manifold. This setting, extends the sub-Riemannian one, removing the bracket generating condition. However, even in the sub-Riemannian setting only sub-manifolds of dimension or codimension one have been extensively studied. We will study the general case and observe that in higher codimension new phenomena arise, which can not show up in the Riemannian case. In particular, we will prove the existence of isolated surfaces, which do not admit degree preserving variation: a phenomena observed by now only for curves, related to the notion of abnormal geodesics.
arXiv: Metric Geometry, 2019
We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped wit... more 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.
arXiv: Differential Geometry, 2020
For a strictly convex set $K\subset \mathbb{R}^2$ of class $C^2$ we consider its associated sub-F... more 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 prove that in the Heisenberg group H with a sub-Finsler structure, a complete, stable, Euclide... more We prove that in the Heisenberg group H with a sub-Finsler structure, a complete, stable, Euclidean Lipschitz and H-regular surface is a vertical plane.
Journal of Differential Equations, 2022
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fra... more 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 ⊂ X with µ X (X \ Ω) > 0, and a function f in the Besov class B θ 2,2 (X) ∩ L 2 (X), we study the problem of finding a function u ∈ B θ 2,2 (X) such that u = f in X \ Ω and E θ (u, u) ≤ E θ (h, h) whenever h ∈ B θ 2,2 (X) with h = f in X \ Ω. 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.
Journal of Differential Equations, 2021
For a strictly convex set K ⊂ R 2 of class C 2 we consider its associated sub-Finsler K-perimeter... more For a strictly convex set K ⊂ R 2 of class C 2 we consider its associated sub-Finsler K-perimeter |∂E| K in H 1 and the prescribed mean curvature functional |∂E| K − 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 .
In the Heisenberg group H 1 equipped with a left invariant and not necessarily symmetric norm in ... more In the Heisenberg group 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 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 out of the characteristic points for the Diri... more We establish the Schauder estimates at the boundary out of the characteristic points for the Dirichlet problem by means of the double layer potential in first Heisenberg H group. 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 H. This is the first instance where a reflection-type argument appears to be useful in the sub-Riemannian setting.
Calculus of Variations and Partial Differential Equations
We consider in this paper an area functional defined on submanifolds of fixed degree immersed int... more 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.
Analysis and Geometry in Metric Spaces
The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be... more 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 to 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 [29], and we provide a characterization for singularities as well as a deformability criterion.
Analysis and Geometry in Metric Spaces
The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be... more 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 to 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 [29], and we provide a characterization for singularities as well as a deformability criterion.
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Papers by Gianmarco Giovannardi