The Rayleigh quotient and dynamic programming.
The RCWA method - A case study with open questions and perspectives of algebraic computations.
The regularity of the positive part of functions in with applications to parabolic equations
Let with be given. Then we show by means of a counter-example that the positive part of has less regularity, in particular it holds in general. Nevertheless, satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.
The regularity of weak and very weak solutions of the Poisson equation on polygonal domains with mixed boundary conditions (part I)
We examine the regularity of weak and very weak solutions of the Poisson equation on polygonal domains with data in L². We consider mixed Dirichlet, Neumann and Robin boundary conditions. We also describe the singular part of weak and very weak solutions.
The regularity of weak and very weak solutions of the Poisson equation on polygonal domains with mixed boundary conditions (part II)
We examine the regularity of weak and very weak solutions of the Poisson equation on polygonal domains with data in L². We consider mixed Dirichlet, Neumann and Robin boundary conditions. We also describe the singular part of weak and very weak solutions.
The Regularization of the Second Order Lagrangians in Example
This paper is devoted to geometric formulation of the regular (resp. strongly regular) Hamiltonian system. The notion of the regularization of the second order Lagrangians is presented. The regularization procedure is applied to concrete example.
The regularized asymptotics in the singularly perturbed parabolic problem with angular boundary layers.
The relation between the porous medium and the eikonal equations in several space dimensions.
We study the relation between the porous medium equation ut = Δ(um), m > 1, and the eikonal equation vt = |Dv|2. Under quite general assumtions, we prove that the pressure and the interface of the solution of the Cauchy problem for the porous medium equation converge as m↓1 to the viscosity solution and the interface of the Cauchy problem for the eikonal equation. We also address the same questions for the case of the Dirichlet boundary value problem.
The relationship between the local temperature and the local heat flux within a one-dimensional semi-infinite domain of heat wave propagation.
The relativistic Enskog equation near the vacuum.
The representation of smooth functions in terms of the fundamental solution of a linear parabolic equation
Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip . We prove a representation theorem for an arbitrary function, in terms of the fundamental solution of the equation Lu=0. Such a theorem was proved in an earlier paper for a parabolic operator in divergence form with coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of...
The Residue of the Local Eta Function at the Origin.
The resolution of the bounded curvature conjecture in general relativity
This paper reports on the recent proof of the bounded curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the -norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.
The resolution of the Navier-Stokes equations in anisotropic spaces.
In this paper we prove global existence and uniqueness for solutions of the 3-dimensional Navier-Stokes equations with small initial data in spaces which are Hδi in the i-th direction, δ1 + δ2 + δ3 = 1/2, -1/2 < δi < 1/2 and in a space which is L2 in the first two directions and B2,11/2 in the third direction, where H and B denote the usual homogeneous Sobolev and Besov spaces.
The resolvent for Laplace-type operators on asymptotically conic spaces
Let be a compact manifold with boundary, and a scattering metric on , which may be either of short range or “gravitational” long range type. Thus, gives the geometric structure of a complete manifold with an asymptotically conic end. Let be an operator of the form , where is the Laplacian with respect to and is a self-adjoint first order scattering differential operator with coefficients vanishing at and satisfying a “gravitational” condition. We define a symbol calculus for...
The resonance phenomenon in the reaction-diffusion systems.
The restriction of the Fourier transform to some curves and surfaces
The restriction theorem for fully nonlinear subequations
Let be a submanifold of a manifold . We address the question: When do viscosity subsolutions of a fully nonlinear PDE on , restrict to be viscosity subsolutions of the restricted subequation on ? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be transformed...
The reverse Hölder inequality for the solution to -harmonic type system.
The Riccati differential equation and a diffusion-type equation.