The project focuses on a selection of nonlinear Partial Differential Equations (PDEs) which have at the same time a novel and challenging structure and a robust connection to models in Science and Engineering:
A - the 1-harmonic flow, a well-grounded tool in Image Processing which is also interesting per se, as a prototype of manifold-constrained gradient flows in BV spaces;
B - variants of the total variation flow, such as the (nonlinear) relativistic heat equation, which find applications whenever a-priori bounds on speed or flux of a diffusion process are modeling-wise relevant, thus sharing interesting qualitative features with nonlinear conservation laws;
C - fractional laplacian equations, an extremely active field in current mathematical research which arises in the modeling of nonlocal interactions, such as for the fractional Schroedinger equation;
D - free boundary problems for the thin-film equations, a fourth-order degenerate parabolic equation modelling wetting phenomena at "small" scales.
Our ambition is to give more insight on specific issues of relevance to applications while at the same time innovating mathematics, possibly achieving advances in the general theory of PDEs. Depending on the problem under consideration, our focus will be on well-posedness, regularity, and/or qualitative properties of solutions. In most cases, however, these three issues are strictly related to each other.
Our approach is strongly collaborative and inclined towards training. The project's goals will be pursued together with both young, talented researchers and more senior, internationally visible scientists. In addition, most topics are appropriate for post-doc research programs. The budget is distributed accordingly: besides overhead and a basic support to the Team members' mobility, funds are only requested for co-working with external collaborators and co-financing post-doc positions (and related equipment).
*** TASK A
The long-term goal is to develop an existence (and, more ambitiously, uniqueness) theory for the 1-harmonic flow on a generic target manifold.
Together with S Moll, we began studying the basic case of a one-dimensional domain, where additional difficulties related to codimension-two singularities are not present. So far, we developed a well-posedness result for Lipschitz 1-harmonic flows [A4], which will be used as basis of a generic existence result in the BV-setting: we plan to identify the right-hand side of the Euler-Lagrange equation by combining variational and energy methods (for characterizing the jump set of solutions, see [A5]) with Fermi-type coordinate formulations of the flow (for characterizing solutions' behavior on the jump set).
In parallel, we plan to start investigating multidimensional domains. In local coordinates, 1-harmonic flows turn into evolutionary, unconstrained 1-laplacian systems with natural growth in the gradient: here, we plan to combine the above-mentioned Team's expertise on gradient flows in BV spaces [PI4-5,PI9-11,PI14,NP2-3] with those on p-laplacian equations with natural growth [TP3,TP5,TP23-24] in order to develop, starting from prototype systems, a global-in-time existence theory in BV under suitable smallness assumptions on the initial datum.
[A4] L Giacomelli, M Lasica, S Moll. arXiv:1711.07460, submitted
[A5] L Giacomelli, M Lasica. arXiv:1806.02456, submitted
*** TASK B1
Our final goal is to develop a general theory on the mechanism and the dynamics of shock formation in singular/degenerate flux-saturating diffusion equations. To this aim, we are currently looking at a prototypical equation of the form
(3) u_t = div ( u^m \frac{\nabla u}{|\nabla u|} ), m \in R.
We expect that its simple, purely hyperbolic scaling will make the study of shock formation easier to describe qualitatively. In contrast, however, a well-posedness theory for (3) is still missing. We have recently obtained existence and uniqueness results for the associated resolvent equation [PI1], which will be used to obtain analogous results for entropy solutions to (3). We then plan to use this theory to obtain both explicit examples and (in one space dimension, through comparison arguments based on the entropy condition) general results on the dynamics of shock formation.
*** TASK B2
As mentioned above, our main goal is to develop a comprehensive theory for the Dirichlet problem with singular reaction in the case p=1, including the case of "strong" singularities. Two issues have potential to yield improvements to the general theory. The first one is uniqueness, which requires to identify the boundary datum: here, use will be made of new results on the Anzellotti-Chen-Frid pairing and trace theory [B12-13], recently obtained in [B14-15]. The second one is regularity, in the sense of global BV estimates on the solution: indeed, explicit examples and scaling arguments suggest that, in contrast to p>1, such bounds hold true regardless of the strength of the singularity.
[B12] G Anzellotti. Ann. Mat. Pura Appl. (4) 135 (1983) 293-318
[B13] G-Q Chen, H Frid. ARMA 147 (1999) 89-118
[B14] G Crasta, V De Cicco. arXiv:1708.00792, submitted
[B15] G Crasta, V De Cicco. arXiv:1804.06249, to appear in J. Functional Analysis
*** TASK C
Our ambition is to extend the solid knowledge on existence and multiplicity of solutions to (2) from the local (s=1) to the fractional case (0
[C9] Rey, O.: Comm. PDE 22 (1997) 1055-1139
[C10] C Gui, J Wei. Canad. J. Math. 52 (2000), 522-538
*** TASK D
The Ren-E class of free boundary conditions relates the traces of first and third derivatives of the unknown function in a nonlinear and possibly degenerate way. For this reason, extending the aforementioned well-posedness theory to this class is an extremely challenging goal which would improve the general theory on (1).
Within the span of this project we plan to look at a special case in such class, which preserves the equation's self-similarity. Such structure will allow to develop a regularity theory for source-type solutions via ODE methods, in the spirit of [PI12]. As in the case of zero-contact angle conditions (cf. [PI8]), we expect that these results will guide toward the study of generic initial data.