Families of whiskered tori for a-priori stable/unstable Hamiltonian systems and construction of unstable orbits.
Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations
We consider a mesoscopic model for phase transitions in a periodic medium and we construct multibump solutions. The rational perturbative case is dealt with by explicit asymptotics.
Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde’s
We prove pointwise gradient bounds for entire solutions of pde’s of the form , where ℒ is an elliptic operator (possibly singular or degenerate). Thus, we obtain some Liouville type rigidity results. Some classical results of J. Serrin are also recovered as particular cases of our approach.
Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations
We consider a mesoscopic model for phase transitions in a periodic medium and we construct multibump solutions. The rational perturbative case is dealt with by explicit asymptotics.
Mean curvature properties for -Laplace phase transitions
This paper deals with phase transitions corresponding to an energy which is the sum of a kinetic part of -Laplacian type and a double well potential with suitable growth conditions. We prove that level sets of solutions of possessing a certain decay property satisfy a mean curvature equation in a suitable weak viscosity sense. From this, we show that, if the above level sets approach uniformly a hypersurface, the latter has zero mean curvature.
Alessio Figalli's contributionsto nonlocal minimal surfaces
Alessio has produced in his very intense career an extraordinary number of outstanding results in an impressive variety of topics. Among the multifold research lines in which he acted as a trailblazer, the one focused on nonlocal minimal surfaces offered an excellent opportunity for Alessio to pioneer some of the first settlements in a brand new subject of investigation and pave the way to a broad spectrum of future research.
A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations
Periodic orbits close to elliptic tori and applications to the three-body problem
We prove, under suitable non-resonance and non-degeneracy “twist” conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses...
Bernstein and De Giorgi type problems: new results via a geometric approach
We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in and and of the Bernstein problem on the flatness of minimal area graphs in . A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our...
Liouville results for -Laplace equations of Lane-Emden-Fowler type
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