Challenges in nonlocal operators: existence and regularity

Nonlocal operators are mathematical tools that intend to describe a variety of physical phenomena characterized by the following fact: what is happening in a specific location may be affected by what is happening "far away" from that location. On the contrary, in the case of local operators, what matters is just a "small" neighbourhood of that specific location. A concrete example that fits with our project is the nonlocal (or anomalous) diffusion of particles. Naively speaking, standard diffusion takes place when particles move randomly around almost exclusively with small (local) displacements. On the other hand, if long jumps play a significant role, the system presents nonlocal features and gives rise to anomalous diffusion. Such type of nonlocal interactions occurs for instance in modelling phenomena arising in economy, biology, statistical mechanics, and epidemiology. Our project focuses on the mathematical properties of the nonlocal operator used to build such models. This theoretical approach is essential since it provides a reliable and coherent theory as a basis for further developments and applications. One of the main issues that we aim to investigate is the well-posedness of the problem associated to a wide class of nonlocal operators. This means that the problem admits a solution and that such solution is unique and stable under small perturbations. In other words, we check that the operator behaves as we expect or, if not, whether some unexpected features or behaviour is hidden inside it. We also address the regularity properties of nonlocal operators. This is quite a technical and delicate topic that tries to understand the relation between the regularity of the data of the problem and the regularity of the solution. Beyond the mathematical interest, such type of analysis provides important information in case of sources that concentrate in extremely small regions (like points or lines) or discontinuities and dishomogeneities of the medium.

This research project mainly addresses some mathematical objects of non-local or fractional nature. The great appeal of fractional tools comes from the need of modeling certain peculiar phenomena that escape the classical local approach. In this framework, the so-called Fractional Laplaciangained a lot of importance in the last decades as a particularly fit candidate to describe nonlocal or anomalous diffusion. Naively speaking, standard diffusion takes place when particles move randomly around almost exclusively with small (local) displacements. On the contrary, if long jumps play a significant role, the system presents nonlocal features and gives rise to anomalous diffusion. Our aim is to better understand the behaviour of such nonlocal tools under a theoretical point of view. This will improve the applicability of such tools in realworld.orientedproblems.