Francesca Da Lio

Uniqueness Results for Second-Order Bellman--Isaacs Equations under Quadratic Growth Assumptions and Applications

In this paper, we prove a comparison result between semicontinuous viscosity sub- and supersolutions growing at most quadratically of second-order degenerate parabolic Hamilton--Jacobi--Bellman and Isaacs equations. As an application, we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation.

Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations

We study uniformly elliptic fully nonlinear equations

On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations

F(D^2u, Du, u, x)=0,

On the Boundary Ergodic Problem for Fully Nonlinear Equations in Bounded Domains with General Nonlinear Neumann Boundary Conditions

and prove results of Gidas--Ni--Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.

n/p-harmonic maps: regularity for the sphere case

ABSTRACT We study the “generalized” Dirichlet problem (in the sense of viscosity solutions) for quasilinear elliptic and parabolic equations in the case when losses of boundary conditions can actually occur. We prove for such problems comparison results between semicontinuous viscosity sub- and supersolutions (Strong Comparison Principle) in annular domains. As a consequence of the Strong Comparison Principle and the Perrons method we obtain the existence and the uniqueness of a continuous solution. Our approach allow us to handle also the case of “singular” equations, in particular the geometric equations arising in the level-sets approach for defining the motions of hypersurfaces with different types of normal velocities. We are able to provide a level-sets approach for equations set in bounded domains with generalized Dirichlet boundary conditions. *This work was partially supported by the TMR Programme “Viscosity Solutions and their Applications.”

Local C 0,α estimates for viscosity solutions of Neumann-type boundary value problems

Abstract We study the Dirichlet problem for viscous Hamilton–Jacobi equations. Despite this type of equations seems to be uniformly elliptic, loss of boundary conditions may occur because of the strong nonlinearity of the first-order part and therefore the Dirichlet boundary condition has to be understood in the sense of viscosity solutions theory. Under natural assumptions on the initial and boundary data, we prove a Strong Comparison Result which allows us to obtain the existence and the uniqueness of a continuous solution which is defined globally in time.

Blow-up analysis of a nonlocal Liouville-type equation

In this paper we prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. This first order equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, that arises in the theory of dislocations dynamics. We show that if an anisotropic mean curvature motion is approximated by this type of equations then it is always of variational type, whereas the converse is true only in dimension two

A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions

We study nonlinear Neumann type boundary value problems related to ergodic phenomenas. The particularity of these problems is that the ergodic constant appears in the (possibly nonlinear) Neumann boundary conditions. We provide, for bounded domains, several results on the existence, uniqueness and properties of this ergodic constant.