On the problem of maximal LqL^q-regularity for viscous Hamilton-Jacobi equations

For $q>2, \gamma > 1$, we prove that maximal regularity of $L^q$ type holds for periodic solutions to $-\Delta u + |Du|^\gamma = f$ in $\mathbb{R}^d$, under the (sharp) assumption $q > d \frac{\gamma-1}\gamma$.Comment: 11 page

On the existence and uniqueness of solutions to time-dependent fractional MFG

We establish existence and uniqueness of solutions to evolutive fractional Mean Field Game systems with regularizing coupling, for any order of the fractional Laplacian $s\in(0,1)$. The existence is addressed via the vanishing viscosity method. In particular, we prove that in the subcritical regime $s>1/2$ the solution of the system is classical, while if $s\leq 1/2$ we find a distributional energy solution. To this aim, we develop an appropriate functional setting based on parabolic Bessel potential spaces. We show uniqueness of solutions both under monotonicity conditions and for short time horizons.Comment: 42 page

Lipschitz regularity for viscous Hamilton-Jacobi equations with LpL^p terms

We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (Fokker-Planck) equation. Here, the regularizing effect is due to the non-degenerate diffusion and coercivity of the Hamiltonian in the gradient variable.Comment: 31 page

High-order estimates for fully nonlinear equations under weak concavity assumptions

This paper studies a priori and regularity estimates of Evans-Krylov type in H\"older spaces for fully nonlinear uniformly elliptic and parabolic equations of second order when the operator fails to be concave or convex in the space of symmetric matrices. In particular, it is assumed that either the level sets are convex or the operator is concave, convex or close to a linear function near infinity. As a byproduct, these results imply polynomial Liouville theorems for entire solutions of elliptic equations and for ancient solutions to parabolic problems

On the strong maximum principle for fully nonlinear parabolic equations of second order

We provide a proof of strong maximum and minimum principles for fully nonlinear uniformly parabolic equations of second order. The approach is of parabolic nature, slightly differs from the earlier one proposed by L. Nirenberg and does not exploit the parabolic Harnack inequality

Some new Liouville-type results for fully nonlinear PDEs on the Heisenberg group

We prove new (sharp) Liouville-type properties via degenerate Hadamard three-sphere theorems for fully nonlinear equations structured over Heisenberg vector fields. As model examples, we cover the case of Pucci's extremal operators perturbed by suitable semilinear and gradient terms, extending to the Heisenberg setting known contributions valid in the Euclidean framework

Interior H\"older and Calder\'on-Zygmund estimates for fully nonlinear equations with natural gradient growth

We establish local H\"older estimates for viscosity solutions of fully nonlinear second order equations with quadratic growth in the gradient and unbounded right-hand side in $L^q$ spaces, for an integrability threshold $q$ guaranteeing the validity of the maximum principle. This is done through a nonlinear Harnack inequality for nonhomogeneous equations driven by a uniformly elliptic Isaacs operator and perturbed by a Hamiltonian term with natural growth in the gradient. As a byproduct, we derive a new Liouville property for entire $L^p$ viscosity solutions of fully nonlinear equations as well as a nonlinear Calder\'on-Zygmund estimate for strong solutions of such equations

Some new Liouville-type results for fully nonlinear PDEs on the Heisenberg group

We prove new (sharp) Liouville-type properties via degenerate Hadamard three-sphere theorems for fully nonlinear equations structured over Heisenberg vector fields. As model examples, we cover the case of Pucci's extremal operators perturbed by suitable semilinear and gradient terms, extending to the Heisenberg setting known contributions valid in the Euclidean framework.Comment: 17 page

Back to Eudaimonia as a Social Relation: What Does the Covid Crisis Teach Us about Individualism and its Limits?

The current health crisis that has spread worldwide has raised many questions regarding our relations to the Other and to ourselves. Through isolating people, Covid-19 has demonstrated the need we face, as human beings, to socialize and to get in contact, physically speaking, with others. As Aristotle stated, human beings are political animals, meaning social animals that can flourish only in the polis through the process of interacting with each other in quest of eudaimonia, i.e. happiness. Along with the rise of socio-physical distancing imposed due to the pandemic, people around the world have experienced isolation and the lack of human contact and interaction. In the Western world this isolation has led to an increase in mental health issues, and this fact has to be taken into consideration by the government when making decisions regarding the reinforcement or the slackening of measures in the context of Covid. The pandemic has shed a light on the limits of individualism as it has developed in some places. The quest for happiness has slowly led some societies to create a kind of a solipsistic world in which there would exist no reality, no truth outside individuals’ perceptions. Consequently, each human being is considered as “the measure of all things,” as Protagoras noted. This unique experience could then give us the grounds to question our relations to each other, to investigate our understanding of eudaimonia, and to revisit what it means to live in a society