An elliptic equation with spike solutions concentrating at local minima of the Laplacian of the potential.
An Elliptic Neumann Problem with Subcritical Nonlinearity
We establish the existence of a solution to the Neumann problem in the half-space with a subcritical nonlinearity on the boundary. Solutions are obtained through the constrained minimization or minimax. The existence of solutions depends on the shape of a boundary coefficient.
An elliptic problem with arbitrarily small positive solutions.
An elliptic problem with critical exponent and positive Hardy potential.
An elliptic semilinear equation with source term involving boundary measures: the subcritical case.
We study the boundary behaviour of the nonnegative solutions of the semilinear elliptic equation in a bounded regular domain Ω of RN (N ≥ 2),⎧ Δu + uq = 0, in Ω⎨⎩ u = μ, on ∂Ωwhere 1 < q < (N + 1)/(N - 1) and μ is a Radon measure on ∂Ω. We give a priori estimates and existence results. The lie on the study of superharmonic functions in some weighted Marcinkiewicz spaces.
An enclosure generating modification of the method of discretization in time
An endpoint Littlewood-Paley inequality for BVP associated with the Laplacian on Lipschitz domains.
We prove a commutator inequality of Littlewood-Paley type between partial derivatives and functions of the Laplacian on a Lipschitz domain which gives interior energy estimates for some BVP. It can be seen as an endpoint inequality for a family of energy estimates.
An energy analysis of degenerate hyperbolic partial differential equations.
An energy analysis is carried out for the usual semidiscrete Galerkin method for the semilinear equation in the region (E) , subject to the initial and boundary conditions, on and . (E) is degenerate at and thus, even in the case , time derivatives of will blow up as . Also, in the case where is locally Lipschitz, solutions of (E) can blow up for in finite time. Stability and convergence of the scheme in is shown in the linear case without assuming (which can blow up as is...
An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment
We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in...
An equality for the curvature function of a simple and closed curve on the plane.
An equation for the limit state of a superconductor with pinning sites.
An equilibrated residual method with a computable error approximation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes
Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in a discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both, the perturbation parameters of the problem and the anisotropy of the mesh. The equilibrated residual method has been shown to provide one...
An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics
This paper gives an error analysis of the multi-configuration time-dependent Hartree (MCTDH) method for the approximation of multi-particle time-dependent Schrödinger equations. The MCTDH method approximates the multivariate wave function by a linear combination of products of univariate functions and replaces the high-dimensional linear Schrödinger equation by a coupled system of ordinary differential equations and low-dimensional nonlinear partial differential equations. The main result of this...
An error estimate for finite volume methods for the Stokes equations.
An error estimate for the parabolic approximation of multidimensional scalar conservation laws with boundary conditions
An error estimate uniform in time for spectral Galerkin approximations for the equations for the motion of a chemical active fluid.
We study error estimates and their convergence rates for approximate solutions of spectral Galerkin type for the equations for the motion of a viscous chemical active fluid in a bounded domain. We find error estimates that are uniform in time and also optimal in the L2-norm and H1-norm. New estimates in the H(-1)-norm are given.
An essay on the Interpolation Theorem of Józef Marcinkiewicz - Polish patriot
An estimate for solutions to the Schrödinger equation.
An estimate of the gap of the first two eigenvalues in the Schrödinger operator
An estimate on convergence for an homogeneisation problem for variational inequalities