0000000000466308
AUTHOR
Lorenzo Giacomelli
showing 7 related works from this author
Nonlinear Diffusion in Transparent Media
2021
Abstract We consider a prototypical nonlinear parabolic equation whose flux has three distinguished features: it is nonlinear with respect to both the unknown and its gradient, it is homogeneous, and it depends only on the direction of the gradient. For such equation, we obtain existence and uniqueness of entropy solutions to the Dirichlet problem, the homogeneous Neumann problem, and the Cauchy problem. Qualitative properties of solutions, such as finite speed of propagation and the occurrence of waiting-time phenomena, with sharp bounds, are shown. We also discuss the formation of jump discontinuities both at the boundary of the solutions’ support and in the bulk.
Regular 1-harmonic flow
2017
We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to Lipschitz initial data. We prove uniqueness and, in the case of a convex domain, local existence of solutions to the flow equations. If the target manifold has non-positive sectional curvature or in the case that the datum is small, solutions are shown to exist globally and to become constant in finite time. We also consider the case where the domain is a compact Riemannian manifold without boundary, solving the homotopy problem for 1-harmonic maps under some …
Solutions to the 1-harmonic flow with values into a hyper-octant of the N-sphere
2013
Abstract We announce existence results for the 1-harmonic flow from a domain of R m into the first hyper-octant of the N -dimensional unit sphere, under homogeneous Neumann boundary conditions. The arguments rely on a notion of “geodesic representative” of a BV-vector field on its jump set.
THE 1-HARMONIC FLOW WITH VALUES IN A HYPEROCTANT OF THE N-SPHERE
2014
We prove the existence of solutions to the 1-harmonic flow — that is, the formal gradient flow of the total variation of a vector field with respect to the [math] -distance — from a domain of [math] into a hyperoctant of the [math] -dimensional unit sphere, [math] , under homogeneous Neumann boundary conditions. In particular, we characterize the lower-order term appearing in the Euler–Lagrange formulation in terms of the “geodesic representative” of a BV-director field on its jump set. Such characterization relies on a lower semicontinuity argument which leads to a nontrivial and nonconvex minimization problem: to find a shortest path between two points on [math] with respect to a metric w…
Nonlinear diffusion in transparent media: the resolvent equation
2017
Abstract We consider the partial differential equation u - f = div ( u m ∇ u | ∇ u | ) u-f=\operatornamewithlimits{div}\biggl{(}u^{m}\frac{\nabla u}{|\nabla u|}% \biggr{)} with f nonnegative and bounded and m ∈ ℝ {m\in\mathbb{R}} . We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative boundary datum) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the ℋ N - 1 {{\mathcal{H}}^{N-1}} -Hausdorff measure. Results and proofs extend to more general nonlinearities.
Rotationally symmetric 1-harmonic flows from D2 TO S 2: Local well-posedness and finite time blowup
2010
The 1-harmonic flow from the disk to the sphere with constant Dirichlet boundary conditions is analyzed in the case of rotational symmetry. Sufficient conditions on the initial datum are given, such that a unique classical solution exists for short times. Also, a sharp criterion on the boundary condition is identified, such that any classical solution will blow up in finite time. Finally, nongeneric examples of finite time blowup are exhibited for any boundary condition.
The 1-Harmonic Flow with Values into $\mathbb S^{1}$
2013
We introduce a notion of solution for the $1$-harmonic flow, i.e., the formal gradient flow of the total variation functional with respect to the $L^2$-distance, from a domain of $\mathbb R^m$ into a geodesically convex subset of an $N$-sphere. For such a notion, under homogeneous Neumann boundary conditions, we prove both existence and uniqueness of solutions when the target space is a semicircle and the existence of solutions when the target space is a circle and the initial datum has no jumps of an “angle” larger than $\pi$. Earlier results in [J. W. Barrett, X. Feng, and A. Prohl, SIAM J. Math. Anal., 40 (2008), pp. 1471--1498] and [X. Feng, Calc. Var. Partial Differential Equations, 37…