INdAM Intensive Period - Geometric Properties of Nonlinear Local and Nonlocal Problems

AIMS OF THIS INTENSIVE PERIOD

The interplay between Analysis and Geometry has always been very fruitful and even a rough summary is not possible here. Recently, a new series of results have underlined a twofold perspective: in a lot of different nonlinear problems, a significant advancement can be achieved only gaining a thorough geometric insight; on the other hand, the geometry of the reference space can allow for the extension of classical Euclidean results to more general metric settings. In the following we point out some analytical problems, mainly stemming from the theory of Partial Differential Equations and Calculus of Variations, where Geometry has played a significant role in the solution, or where a new geometric perspective is felt necessary, in order to make a step forward.

The role played by Geometry in characterizing Free Boundaries is obvious. Recently new results have been proved for a classical free boundary problem, namely the so-called ``boundary obstacle'' problem. Boundary obstacle problems arise in elasticity (the Signorini problem), when an elastic body is at rest, partially laying on a surface, in optimal control of temperature across a surface, in the modelling of semipermeable membranes where some saline concentration can flow through the membrane only in one direction, and in financial mathematics, when the random variation of underlying asset changes in a discontinuous fashion (a Levi process). There is considerable literature on the regularity properties of the solution. In particular, two of the authors proved (Athanasopoulos and Caffarelli, 2004) the optimal regularity of solutions to such a problem. This opened the way to study the properties of the interface by using geometric PDE techniques. It comes out that there is one basic global non-degenerate profile after blow up, and that in a neighborhood of a point that has this profile, the free boundary is a C^1,α curve on the boundary (i.e., an n-2 dimensional graph on the n-1 dimensional boundary). The previous results turn out to be useful in the study of regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian (Caffarelli, Salsa and Silvestre, 2008). Constrained variational problems with fractional diffusion appear in the study of the quasi-geostrophic flow model, anomalous diffusion, and American options with jump processes.

There is no general notion of degeneracy and/or singularity in PDE's. In view of their relevance to physical phenomena, mathematicians have only recently began exploring such a notion, commencing from some specific examples such as the parabolic p-Laplacian equation and its quasi-linear variants. The main recent finding is that a Harnack estimate holds for a waiting time intrinsically determined by the solution itself and it is false otherwise. Thus, within its own intrinsic time-geometry, such degenerate equations behave as though they were not degenerate. Moreover solutions of these equations behave, roughly speaking, as potentials (DiBenedetto, Gianazza and Vespri, 2008). This is surprising on two accounts; first there is no notion of potential for quasi-linear equations, and second, the classical proofs of Moser and Krylov-Safonov were possible precisely by avoiding potentials. Thus it appears as though this potential is the driving energy of the underlying physical process, and not the commonly-looked-at pointwise laws. What is the reason for this lingering of fundamental solutions into quasi-linear structures? What are the implications to interior and boundary regularity theory? These findings suggest a change in perspective in at least two ways. First, degeneracy in physical models surfaces only if one insists in describing them in the flat Euclidean geometry, and it would disappear if described in its own intrinsic phase space. Second, within such an intrinsic phase space the system admits some sort of potential. This point of view is further supported by corresponding results for singular equations: there seems to be a link between Harnack inequalities and existence of potentials. A better understanding of this phenomenon will have also a bearing in physical modeling: basic conservation laws (mass and/or momentum), which are at the basis of most models, might have to be reinterpreted in the intrinsic geometry dictated by the underlying physics; in retrospect this idea is rather natural, as it links back to the notion of conservation along Lagrangian paths in Classical Mechanics for discrete, or rigid systems.

A similar situations occurs for those PDEs where the degeneracy and/or singularity is not intrinsic but rather determined by coefficients that may vanish, such as Hörmander vector fields: they might have to be interpreted in a geometric framework where the basic first order operators (vector fields) are adapted to the degeneracy of the equation. While the linear theory has been developed in great detail, non-linear developments are much recent and we are still far from anything close to a satisfactory theory. Parallel to the historical development in the Euclidean case, we understand the subelliptic p-Laplacian best when p is close to 2. This is the Cordes case studied in (Domokos and Manfredi, 2005; Chanillo and Manfredi, 2007). In the simpler cases (Heisenberg group, Grušin plane, and strictly pseudoconvex domains) it is clear that the obstacle to regularity is the integrability of the commutator derivative. This is a second differentiability result that involves delicate fractional difference quotients along vector fields. In the one-dimensional Heisenberg group and in the Grušin plane this is the case when 2≤p<4. In (Manfredi and Mingione, 2008) the Lipschitz continuity is established in the one-dimensional Heisenberg group for p<4. The role of the critical value p=4 remains to be understood. Are p-harmonic functions Lipschitz continuous when p≥4? More recently Mingione, Zatorska-Goldstein, and Zhong made a step forward: they give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in (Manfredi and Mingione, 2008), where only dimension dependent bounds for the growth exponent are given. More precisely, now the authors allow 2<p<4 in any dimension. These last results extend to the sub-elliptic setting a few classical nonlinear Euclidean results due to Iwaniec and DiBenedetto-Manfredi, and to the nonlinear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations.

The Harnack inequality holds also for quasi-minima in the Calculus of Variations (DiBenedetto and Trudinger, 1984) and for functions in the DeGiorgi classes. Since these are only indirectly related to Partial Differential Equations, it might be that the Harnack inequality is a structural property of some classes rather that PDE's, and it raises the question of identifying such classes (see Caffarelli, 1999). For example a change in perspective would result if functions in the DeGiorgi classes in a open set E⊆R^N, would satisfy a Wiener-type criterion, at points of ∂E. This would identify the Wiener test as an energy-(quasi)minimization phenomenon, more than a partial differential equation property. A closely related topic is the study of nonlinear potential theory related to quasiminimizers on a metric measure space equipped with a doubling measure and supporting a Poincar\'e inequality. In (Kinnunen and Martio, 2003) the authors show that quasiminimizers create an interesting potential theory with new features although from the potential theoretic point of view they have several drawbacks: They do not provide a unique solution to the Dirichlet problem, they do not obey the comparison principle and they do not form a sheaf. However, many potential theoretic concepts such as harmonic functions, superharmonic functions and the Poisson modification have their counterparts in the theory of quasiminimizers. These results are part of a general working project: verify how much of the classical potential theory can be extended on metric spaces; otherwise stated, show how potential theory depends on the geometric properties of the underlying space; see (Kinnunen and Martio, 2000; Kinnunen and Martio, 2002; Kinnunen and Shanmugalingam, 2001; Shanmugalingam, 2001) just to have a taste of what is going on under this point of view).

In a series of recent papers, Acerbi and Mingione (Acerbi and Mingione, 2007; Mingione, 2007) develops a suitable Calderon-Zygmund theory for nonlinear elliptic and parabolic problems. The methods have a large impact and promise to have a wide applicability. In the first paper the authors establish the local Calderon-Zygmund theory for a class of parabolic systems, including the parabolic p-laplacian, thereby extending previous elliptic work due to Iwaniec, and DiBenedetto-Manfredi. The results extend to general degenerate parabolic equations in divergence form of the type considered by DiBenedetto. Acerbi and Mingione also treat parabolic equations and systems with discontinuous coefficients of suitable VMO/BMO type. The peculiarity of the work is that no use of Harmonic Analysis tools is made here, and the proofs are purely PDE ones. This approach is somehow unescapable, as the systems under consideration scale differently in space and time, and this rules out the use of maximal operators, previously employed by Iwaniec and DiBenedetto-Manfredi. As for the discontinuous coefficients case, the usual singular integrals techniques are ruled out at the very beginning, by the non-linearity of the systems.

There is a deep interplay among Riemannian manifolds satisfying a scale-invariant parabolic Harnack inequality, their characterization in terms of Poincar\'e inequality and volume growth, ultracontractivity of the heat diffusion semigroup, Gaussian heat kernel bounds and analogouos results obtained with probabilistic methods; if the general framework is well understood thanks to recent results, obviously not all the relative links have been fully explored. How much of this can be extended to the nonlinear setting (just consider the p-laplacian case), is still a matter of intensive research. An interesting survey, although now a bit old, is represented by (Saloff-Coste, 2002), whereas more recent results in the same spirit are in (Grigorýan and Saloff-Coste, 2005).