1,470 research outputs found
Local Estimates for Viscosity Solutions of Neumann-type Boundary Value Problems
In this article, we prove the local regularity and provide
estimates for viscosity solutions of fully nonlinear, possibly
degenerate, elliptic equations associated to linear or nonlinear Neumann type
boundary conditions. The interest of these results comes from the fact that
they are indeed regularity results (and not only a priori estimates), from the
generality of the equations and boundary conditions we are able to handle and
the possible degeneracy of the equations we are able to take in account in the
case of linear boundary conditions
Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations
In this article, we provide existence results for a general class of nonlocal
and nonlinear second-order parabolic equations. The main motivation comes from
front propagation theory in the cases when the normal velocity depends on the
moving front in a nonlocal way. Among applications, we present level-set
equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo
systems
Uniqueness Results for Nonlocal Hamilton-Jacobi Equations
We are interested in nonlocal Eikonal Equations describing the evolution of
interfaces moving with a nonlocal, non monotone velocity. For these equations,
only the existence of global-in-time weak solutions is available in some
particular cases. In this paper, we propose a new approach for proving
uniqueness of the solution when the front is expanding. This approach
simplifies and extends existing results for dislocation dynamics. It also
provides the first uniqueness result for a Fitzhugh-Nagumo system. The key
ingredients are some new perimeter estimates for the evolving fronts as well as
some uniform interior cone property for these fronts
A short proof of the --regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications
Recently I. Capuzzo Dolcetta, F. Leoni and A. Porretta obtain a very
surprising regularity result for fully nonlinear, superquadratic, elliptic
equations by showing that viscosity subsolutions of such equations are locally
H\"older continuous, and even globally if the boundary of the domain is regular
enough. The aim of this paper is to provide a simplified proof of their
results, together with an interpretation of the regularity phenomena, some
extensions and various applications
Continuous dependence results for Non-linear Neumann type boundary value problems
We obtain estimates on the continuous dependence on the coefficient for
second order non-linear degenerate Neumann type boundary value problems. Our
results extend previous work of Cockburn et.al., Jakobsen-Karlsen, and
Gripenberg to problems with more general boundary conditions and domains. A new
feature here is that we account for the dependence on the boundary conditions.
As one application of our continuous dependence results, we derive for the
first time the rate of convergence for the vanishing viscosity method for such
problems. We also derive new explicit continuous dependence on the coefficients
results for problems involving Bellman-Isaacs equations and certain quasilinear
equation
A New Stability Result for Viscosity Solutions of Nonlinear Parabolic Equations with Weak Convergence in Time
We present a new stability result for viscosity solutions of fully nonlinear
parabolic equations which allows to pass to the limit when one has only weak
convergence in time of the nonlinearities
(Almost) Everything You Always Wanted to Know About Deterministic Control Problems in Stratified Domains
We revisit the pioneering work of Bressan \& Hong on deterministic control
problems in stratified domains, i.e. control problems for which the dynamic and
the cost may have discontinuities on submanifolds of R N . By using slightly
different methods, involving more partial differential equations arguments, we
(i) slightly improve the assumptions on the dynamic and the cost; (ii) obtain a
comparison result for general semi-continuous sub and supersolutions (without
any continuity assumptions on the value function nor on the
sub/supersolutions); (iii) provide a general framework in which a stability
result holds
Uniqueness for unbounded solutions to stationary viscous Hamilton--Jacobi equations
We consider a class of stationary viscous Hamilton--Jacobi equations as
\left\{\begin{array}{l} \la u-{\rm div}(A(x) \nabla u)=H(x,\nabla u)\mbox{in
}\Omega, u=0{on}\partial\Omega\end{array} \right. where \la\geq 0,
is a bounded and uniformly elliptic matrix and is convex in
and grows at most like , with and f \in \elle {\frac
N{q'}}. Under such growth conditions solutions are in general unbounded, and
there is not uniqueness of usual weak solutions. We prove that uniqueness holds
in the restricted class of solutions satisfying a suitable energy--type
estimate, i.e. (1+|u|)^{\bar q-1} u\in \acca, for a certain (optimal)
exponent . This completes the recent results in \cite{GMP}, where the
existence of at least one solution in this class has been proved
On the regularizing effect for unbounded solutions of first-order Hamilton-Jacobi equations
We give a simplified proof of regularizing effects for first-order
Hamilton-Jacobi Equations of the form in
in the case where the idea is to first estimate .
As a consequence, we have a Lipschitz regularity in space and time for coercive
Hamiltonians and, for hypo-elliptic Hamiltonians, we also have an H\''older
regularizing effect in space following a result of L. C. Evans and M. R. James
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