The paper is a comprehensive study of Lp and Schauder estimates for higher-order divergence type parabolic systems with discontinuous coefficients on a half space and cylindrical domains with the conormal derivative boundary conditions. For the Lp estimates, we assume that the leading coefficients are only bounded and measurable in the t variable and have vanishing mean oscillations (VMOx) with...

Flow of two phases in a heterogeneous porous medium is modeled by a scalar conservation law with a discontinuous coefficient. As solutions of conservation laws with discontinuous coefficients depend explicitly on the underlying small scale effects, we consider a model where the relevant small scale effect is dynamic capillary pressure. We prove that the limit of vanishing dynamic capillary pres...

Elliptic partial differential equations (PDEs) with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electro-magnetic field propagation on heterogeneous media, and diffusion processes on rough surfaces. The standard approach to numerically treating such problems using finite element methods is to assume that the discontinuities lie on th...

We study the asymptotic behavior of the solution of semi-linear PDEs. Neither periodicity nor ergodicity assumptions are assumed. The coefficients admit only a limit in a C̀esaro sense. In such a case, the limit coefficients may have discontinuity. We use probabilistic approach based on weak convergence techniques for the associated backward stochastic differential equation in the S-topology. We...

We study the asymptotic behavior of solution of semi-linear PDEs. Neither periodicity nor ergodicity will be assumed. In return, we assume that the coefficients admit a limit in C̀esaro sense. In such a case, the averaged coefficients could be discontinuous. We use probabilistic approach based on weak convergence for the associated backward stochastic differential equation in the S-topology to d...

This paper is devoted to the investigation of the investigation of the convergence of iterative methods for solving boundary va.!ue problems with discontinuous coeficients. The dependence of the rate" of convergence on the size of the discontinuity of coefficients is ana.!yzed for three popular genera.! iterative methods. A new criterion on the applicability of such methods is proposed and inve...

Abstract. We consider Hamilton–Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an L∞ contractive semigroup. We define a viscosity solution by treating the discontinuities in the coefficients analogously to “interna...

— We introduce small viscosity solutions of hyperbolic systems with discontinuous coefficients accross the fixed noncharacteristic hypersurface {xd = 0}. Under a geometric stability assumption, our first result is obtained, in the multi-D framework, for piecewise smooth coefficients. For our second result, the considered operator is ∂t+a(x)∂x, with sign(xa(x)) > 0 (expansive case not included i...

Ž Ž . Nq 1. q z s x, t g R , where the principal part is uniformly elliptic on R , q F N, and the constant matrix B is upper triangular and such that the operator obtained by freezing the coefficients a at any point z g R Nq 1 is hypoelliptic. We prove i j 0 p Ž . local L -estimates for the derivatives ­ u i, j s 1, . . . , q of a solution to the x x i j equation Lu s f , under the assumption t...

We consider the heat equation with a diffusion coefficient that is discontinuous at an interface. We give global Carleman estimates for solutions of this equation, even if the jump of the coefficient across the interface has not a constant sign. AMS classification scheme numbers: 35K05, 35K55, 35R05, 35R30