13 research outputs found
Shape Theorems for Poisson Hail on a Bivariate Ground
We consider the extension of the Euclidean stochastic geometry Poisson Hail
model to the case where the service speed is zero in some subset of the
Euclidean space and infinity in the complement. We use and develop tools
pertaining to sub-additive ergodic theory in order to establish shape theorems
for the growth of the ice-heap under light tail assumptions on the hailstone
characteristics. The asymptotic shape depends on the statistics of the
hailstones, the intensity of the underlying Poisson point process and on the
geometrical properties of the zero speed set.Comment: Final version accepted in Advances in Applied Probabilit
Well-posedness for a transmission problem connecting first and second-order operators
We establish the existence and uniqueness of viscosity solutions within a
domain for a class of equations governed by
elliptic and eikonal type equations in disjoint regions. Our primary motivation
stems from the Hamilton-Jacobi equation that arises in the context of a
stochastic optimal control problem
Regularity for solutions of nonlocal, nonsymmetric equations
We study the regularity for solutions of fully nonlinear integro differential
equations with respect to nonsymmetric kernels. More precisely, we assume that
our operator is elliptic with respect to a family of integro differential
linear operators where the symmetric part of the kernels have a fixed
homogeneity and the skew symmetric part have strictly smaller
homogeneity . We prove a weak ABP estimate and regularity.
Our estimates remain uniform as we take and so that
this extends the regularity theory for elliptic differential equations with
dependence on the gradient.Comment: New sections added with preliminaries and qualitative behavior of
solutions. To appear in ANIHP
Classical solutions to integral equations with zero order kernels
We show global and interior higher-order log-H\"older regularity estimates
for solutions of Dirichlet integral equations where the operator has a
nonintegrable kernel with a singularity at the origin that is weaker than that
of any fractional Laplacian. As a consequence, under mild regularity
assumptions on the right hand side, we show the existence of classical
solutions of Dirichlet problems involving the logarithmic Laplacian and the
logarithmic Schr\"odinger operator.Comment: We added a lower-order regularity estimate for the logarithmic
Laplacian (Corollary 5.8
Fractional-order operators: Boundary problems, heat equations
The first half of this work gives a survey of the fractional Laplacian (and
related operators), its restricted Dirichlet realization on a bounded domain,
and its nonhomogeneous local boundary conditions, as treated by
pseudodifferential methods. The second half takes up the associated heat
equation with homogeneous Dirichlet condition. Here we recall recently shown
sharp results on interior regularity and on -estimates up to the boundary,
as well as recent H\"older estimates. This is supplied with new higher
regularity estimates in -spaces using a technique of Lions and Magenes,
and higher -regularity estimates (with arbitrarily high H\"older estimates
in the time-parameter) based on a general result of Amann. Moreover, it is
shown that an improvement to spatial -regularity at the boundary is
not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in
Mathematics and Statistics: "New Perspectives in Mathematical Analysis -
Plenary Lectures, ISAAC 2017, Vaxjo Sweden