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We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter ε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution...
The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described...
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...
In this paper we study the limit as p → ∞ of minimizers of the
fractional Ws,p-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.
In this paper we study the limit as p → ∞ of minimizers of the
fractional Ws,p-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.
We prove Hölder regularity of the gradient, up to the boundary for solutions of some fully-nonlinear, degenerate elliptic equations, with degeneracy coming from the gradient.
We consider an evolution equation similar to that introduced by Vese in [Comm.
Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution
converges in large time to the convex envelope of the initial datum. We give a stochastic
control representation for the solution from which we deduce, under quite general
assumptions that the convergence in the Lipschitz norm is in fact exponential in time.
We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.
We consider an evolution equation similar to that introduced by Vese in [Comm.
Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution
converges in large time to the convex envelope of the initial datum. We give a stochastic
control representation for the solution from which we deduce, under quite general
assumptions that the convergence in the Lipschitz norm is in fact exponential in time.
This paper is concerned with the Hölder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth...
In this paper, a singular semi-linear parabolic PDE with locally periodic
coefficients is homogenized. We substantially weaken previous assumptions on
the coefficients. In particular, we prove new ergodic theorems. We show that
in such a weak setting on the coefficients, the proper statement of the
homogenization property concerns viscosity solutions, though we need a
bounded Lipschitz terminal condition.
In this paper we furnish mean value characterizations for subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution . These characterizations are based on suitable average operators on the level sets of . Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach, Reade and Saks. We analyze as well the notion of subharmonic function...
We prove the existence of a principal eigenvalue associated to the ∞-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.
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