Speakers

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.

Lipschitz Regularity of almost minimizers for the p-Laplacian

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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.

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.

Singular analysis of the optimizers of the principal eigenvalue in weighted Neumann problems

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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, 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 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.

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.