On the regularity of a free boundary for a nonlinear obstacle problem arising in superconductor modelling
Convexity estimates for nonlinear elliptic equations and application to free boundary problems
On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two
In this paper we establish a Liouville type theorem for fully nonlinear elliptic equations related to a conjecture of De Giorgi in . We prove that if the level lines of a solution have bounded curvature, then these level lines are straight lines. As a consequence, the solution is one-dimensional. The method also provides a result on free boundary problems of Serrin type.
A Hölder infinity Laplacian
In this paper we study the limit as → ∞ of minimizers of the fractional -norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results....
A Hamilton-Jacobi approach to junction problems and application to traffic flows
This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They...
A Hölder infinity Laplacian
In this paper we study the limit as → ∞ of minimizers of the fractional -norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results....
Stability of travelling waves in a model for conical flames in two space dimensions
Travelling graphs for the forced mean curvature motion in an arbitrary space dimension
We construct travelling wave graphs of the form , , , solutions to the -dimensional forced mean curvature motion () with prescribed asymptotics. For any -homogeneous function , viscosity solution to the eikonal equation , we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by . We also describe in terms of a probability measure on .
Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics
We prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. The equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, which arises in the theory of dislocation dynamics. We show that if an anisotropic mean curvature motion is approximated by equations of this type then it is always of variational type, whereas the converse is true only in dimension two.
Junction of elastic plates and beams
We consider the linearized elasticity system in a multidomain of . This multidomain is the union of a horizontal plate with fixed cross section and small thickness , and of a vertical beam with fixed height and small cross section of radius . The lateral boundary of the plate and the top of the beam are assumed to be clamped. When and tend to zero simultaneously, with , we identify the limit problem. This limit problem involves six junction conditions.
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