Archivio 2022 270 seminari

Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprise, among the others, equations involving both standard and Caputo time-derivative, and diffusion operators as the classic and fractional Laplacian, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators. Both local and fractional space diffusion are taken into account, possibly in a nonlinear setting. The different quantitative behaviours, which distinguish polynomial decays from exponential ones, depend heavily on the structure of the time-derivative involved in the equation. This work was done in collaboration with Enrico Valdinoci.

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We present some properties of a nonlocal version of the Neumann boundary conditions associated to problems involving the fractional p-Laplacian. For this problems, we show some regularity results for the general case and some existence results for particular types of problems. When p=2, we give a generalization of the boundary conditions in which both the nonlocal and the classic Neumann conditions are present, and we consider problems involving both nonlocal and local interactions.

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I will consider a class of degenerate nonlinear operators that are extremal among operators with one dimensional fractional diffusion and that approximate the so-called truncated Laplacians. I will show some properties of these operators, emphasizing the differences both with the local equivalent operators and with more standard nonlocal operators such as the fractional Laplacian. In particular, continuity properties, validity of comparison and maximum principles, and their relation with principal eigenvalues, will be presented. Joint work with Isabeau Birindelli and Giulio Galise.

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When one writes the fractional heat equation in self-similar variables a drift term appears. We study the associated eigenvalue problem for this equation, which has a fractional Laplacian and a first order term under competition. Our main contribution is to give explicit Euclidean formulae of the fractional analogue of Hermite polynomials. A crucial tool is the Mellin transform, which is essentially the Fourier transform in logarithmic variable and which turns the gradient into multiplication. This is joint work with Hardy Chan, Marco Fontelos and Juncheng Wei.

In the limit of vanishing but moderate external magnetic field, we derived a few years ago together with S. Conti, F. Otto and S. Serfaty a branched transport problem from the full Ginzburg-Landau model. In this regime, the irrigated measure is the Lebesgue measure and, at least in a simplified 2d setting, it is possible to prove that the minimizer is a self-similar branching tree. In the regime of even smaller magnetic fields, a similar limit problem is expected but this time the irrigation of the Lebesgue measure is not imposed as a hard constraint but rather as a penalization. While an explicit computation of the minimizers seems here out of reach, I will present some ongoing project with G. De Philippis and B. Ruffini relating local energy bounds to dimensional estimates for the irrigated measure.

We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain \Omega\subset R^N, within a suitable class of sign-changing weights. This problem naturally arises in population dynamics. Denoting with u the optimal eigenfunction and with D its super-level set associated to the optimal weight, we perform the analysis of the singular limit of the optimal eigenvalue as the measure of D tends to zero. We show that, when the measure of D is sufficiently small, u has a unique local maximum point lying on the boundary of \Omega and D is connected. Furthermore, the boundary of D intersects the boundary of the box \Omega, and more precisely, ${\mathcal H}^{N-1}(\partial D \cap \partial \Omega)\ge C|D|^{(N-1)/N} $ for some universal constant C>0. Though widely expected, these properties are still unknown if the measure of D is arbitrary. This is a joint project with B. Pellacci and G. Verzini.

Elliptic operators of fractional order were popularized, mainly thanks to Luis Caffarelli, during the early 2000's. Suddenly, we learnt that every classical PDE had a fractional counterpart (or even more than one in some cases!). Also, fractional versions of most important techniques and results in PDE were developed. In this context, the invention in the late 2000's of fractional minimal surfaces may not seem a very striking milestone. Over the years, however, the interest and depth of these new surfaces is becoming unquestionable, to the point that they may be a fundamental tool in order to better understand certain (famously delicate) questions on classical minimal surfaces, such as Yau's conjecture. In the talk I will describe some very recent works that, I hope, may help to convince a fraction of the remaining skeptics about the beauty and usefulness of nonlocal minimal surfaces.

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We describe how different regularity assumptions on the x-dependence of the energy impact the regularity of minimizers of some non-autonomous functionals having nonuniform ellipticity of moderate size. We put particular emphasis on double phase functionals with logarithmic phase transition, including some new results.

In this talk, I will consider a two-phase obstacle type problem driven by the fractional Laplacian and I will present some results concerning the local behavior of solutions and the regularity of their nodal set. Some time will be devoted to the description of the main tools, namely Almgren and Monneau type monotonicity formulas. This is a joint work with D. Danielli.

In this talk we will analyze the existence and the structure of different sign-changing solutions to the Yamabe equation in the whole space and we will use them to find positive solutions to critical competitive systems in dimension 4.

In this talk we will discuss the Cheeger problem in a weighted domain. In particular, we are interested in the distribution of mass which maximizes the Cheeger constant in a ball (the minimization is always trivial). We will give some results, and notice how they depend on the bounds that we impose on the distribution. Joint work with Leonardi and Saracco.

We consider a class of degenerate equations satisfying a parabolic Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal Hölder regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin-type diffusions.