Free Boundary Regularity of Some Non-Homogeneous Problems

In this work, we deal with the study of some free boundary problems governed by non-homogeneous equations. In particular, we are interested in the regularity of the free boundaries for solutions of one-phase problems associated with non-divergence elliptic operators with variable coefficients

A new glance to the Alt-Caffarelli-Friedman monotonicity formula

In this paper we revisit the proof of the Alt-Caffarelli-Friedman monotonicity formula. Then, in the framework of the Heisenberg group, we discuss the existence of an analogous monotonicity formula introducing a necessary condition for its existence, recently proved in \cite{FeFo}.Comment: arXiv admin note: text overlap with arXiv:2001.0439

A sub-Riemannian maximum modulus theorem

In this note we prove a sub-Riemannian maximum modulus theorem in a Carnot group. Using a nontrivial counterexample, we also show that such result is best possible, in the sense that in its statement one cannot replace the right-invariant horizontal gradient with the left-invariant one

Regularity in degenerate elliptic and parabolic free boundary problems

In this thesis, the main topic is the study of some free boundary problems, more precisely the investigation of regularity issues in degenerate elliptic and parabolic ones. Specifically, three different problems are treated. The first one is the one-phase Stefan problem, for which the regularity of flat free boundaries is dealt with by relying on perturbation arguments leading to a linearization of the problem. This approach is inspired by the elliptic counterpart. The second problem concerns the question of the existence of an Alt-Caffarelli-Friedman monotonicity formula in the Heisenberg group. Following the ideas exploited in the Euclidean setting, a necessary condition about the existence of such tool in that noncommutative setting is found. The last problem faced is related to almost minimizers of the p-Laplacian. In particular, the optimal Lipschitz continuity of almost minimizers, for p greater or equal than 2, is proved as well as the regularity of the free boundary is studied

Enhanced boundary regularity of planar nonlocal minimal graphs and a butterfly effect

In this note, we showcase some recent results obtained in [DSV19] concerning the stickiness properties of nonlocal minimal graphs in the plane. To start with, the nonlocal minimal graphs in the planeenjoy an enhanced boundary regularity, since boundary continuity with respect to the external datum is sufficient to ensure differentiability across the boundary of the domain. As a matter of fact, the Hoelder exponent of the derivative is in this situation sufficiently high to provide the validity of the Euler-Lagrange equation at boundary points as well. From this, using a sliding method, one also deduces that the stickiness phenomenon is generic for nonlocal minimal graphs in the plane, since an arbitrarily small perturbation of continuous nonlocal minimal graphs can produce boundary discontinuities (making the continuous case somehow ``exceptional'' in this framework.In questa nota, presentiamo alcuni risultati recenti ottenuti in [DSV19] relativi alla proprietà di ``appiccicosità'' dei grafici minimi nonlocali nel piano. I grafici minimi non locali nel piano godono di una regolarità ``accresciuta'' al bordo, in quanto la continuità al bordo rispetto al dato esterno è sufficiente a garantire la differenziabilità attraverso il bordo del dominio. Inoltre, l'esponente di Hoelder della derivata è sufficientemente grande da garantire la validità dell'equazione di Eulero-Lagrange anche ai punti di bordo del dominio. Da ciò, usando un metodo di scivolamento, si ottiene anche cheil fenomeno di appiccicosità è generico per grafici minimi non locali nel piano, nel senso che una perturbazione arbitrariamente piccola di i grafici minimi nonlocali continui produce discontinuità al bordo (rendendo quindi il caso continuo in qualche modo ``eccezionale'')

Regolarità della frontiera libera nel problema di Stefan a una fase: un approccio recente

In this note, we discuss about the regularity of the free boundary for the solutions of the one-phase Stefan problem. We start by recalling the classical results achieved by I. Athanasopoulos, L. Caffarelli, and S. Salsa in the more general setting of the two-phase Stefan problem. Next, we focus on some recent achievements on the subject, obtained with Daniela De Silva and Ovidiu Savin starting from the techniques already known for one-phase problems governed by elliptic operators.In questa nota, discutiamo della regolarità della frontiera libera per le soluzioni del problema di Stefan a una fase. Incominciamo richiamando i risultati classici ottenuti da I. Athanasopoulos, L. Caffarelli, e S. Salsa nel setting più generale del problema di Stefan a due fasi, giungendo successivamente ad alcuni più recenti sull'argomento, trovati insieme a Daniela De Silva e Ovidiu Savin partendo dalle tecniche già note per problemi di frontiera libera a una fase governati da operatori ellittici

ON THE ∞-LAPLACIAN ON CARNOT GROUPS

We prove Lipschitz estimates for viscosity solutions to Poisson problem for the infinity Laplacian in general Carnot groups

Lipschitz regularity of almost minimizers in one-phase problems driven by the pp-Laplace operator

We prove that, given~$p>\max\left\{\frac{2n}{n+2},1\right\}$, the nonnegative almost minimizers of the nonlinear free boundary functional $$J_p(u,\Omega):=\int_{\Omega}\Big( |\nabla u(x)|^p+\chi_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz continuous