In this paper we prove a comparison result between semicontinuous viscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form in where the Hamiltonian H may be noncoercive in the gradient Du. As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.
The nonlocal Fisher equation has been proposed as a simple model exhibiting Turing instability and the interpretation refers to adaptive evolution. By analogy with other formalisms used in adaptive dynamics, it is expected that concentration phenomena (like convergence to a sum of Dirac masses) will happen in the limit of small mutations. In the present work we study this asymptotics by using a change of variables that leads to a constrained Hamilton-Jacobi equation. We prove the convergence analytically...
This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators. We establish such an estimate for the parabolic Cauchy problem in the whole space [0, +∞) × ℝn and, under some periodicity and either ellipticity or controllability assumptions, we deduce a similar estimate for the ergodic constant associated to the operator. An interesting byproduct of the latter result will be the local uniform convergence for some classes of singular perturbation problems.
A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.
A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.
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.
Many inverse problems for differential equations can be formulated as optimal control problems. It is well known that inverse problems often need to be regularized to obtain good approximations. This work presents a systematic method to regularize and to establish error estimates for approximations to some control problems in high dimension, based on symplectic approximation of the Hamiltonian system for the control problem. In particular the work derives error estimates and constructs regularizations...