Obtain Pointwise Estimates Research Articles

Capacitary estimates for solutions of the Dirichlet problem for second order elliptic equations in divergence form

We consider the Dirichlet problem for A-harmonic functions, i.e. the solutions of the uniformly elliptic equation div(A(x)del u(x)) = 0 in an n-dimensional domain Omega, n greater than or equal to 3. The matrix A is assumed to have bounded measurable entries. We obtain pointwise estimates for the A-harmonic functions near a boundary point. The estimates are in terms of the Wiener capacity and the so called capacitary interior diameter. They imply pointwise estimates for the A-harmonic measure of the domain Omega, which in turn lead to a sufficient condition for the Holder continuity of A-harmonic functions at a boundary point. The behaviour of A-harmonic functions at infinity and near a singular point is also studied and theorems of Phragmen-Lindelof type, in which the geometry of the boundary is taken into account, are proved. We also obtain pointwise estimates for the Green function for the operator -div(A(.)del u(.)) in a domain Omega and for the solutions of the nonhomogeneous equation -div(A(x)del u(x)) = mu with measure on the right-hand side.

Pointwise Green's function approach to stability for scalar conservation laws

We study the pointwise behavior of perturbations from a viscous shock solution to a scalar conservation law, obtaining an estimate independent of shock strength. We find that for a perturbation with initial data decaying algebraically or slower, the perturbation decays in time at the rate of decay of the integrated initial data in any Lp norm, p ≥ 1. Stability in any Lp norm is a direct consequence. The approach taken is that of obtaining pointwise estimates on the perturbation through a Duhamel's principle argument that employs recently developed pointwise estimates on the Green's function for the linearized equation. © 1999 John Wiley & Sons, Inc.

Pointwise Green's function approach to stability for scalar conservation laws

We study the pointwise behavior of perturbations from a viscous shock solution to a scalar conservation law, obtaining an estimate independent of shock strength. We find that for a perturbation with initial data decaying algebraically or slower, the perturbation decays in time at the rate of decay of the integrated initial data in any L p norm, p 1. Stability in any L p norm is a direct consequence. The approach taken is that of obtaining pointwise estimates on the perturbation through a Duhamel’s principle argument that employs recently developed pointwise estimates on the Green’s function for the linearized equation. c 1999 John Wiley & Sons, Inc.

Pointwise convergence to shock waves for viscous conservation laws

We are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws. It is shown that, besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and Burgers equations, a perturbation also gives rise to algebraically decaying terms, which measure the coupling of waves of different characteristic families. Our technique is a combination of time-asymptotic expansion, construction of approximate Green functions, and analysis of nonlinear wave interactions. The pointwise estimates yield optimal Lp convergence of the perturbation to the shock and diffusion waves, 1 ≤ p ≤ ∞. The new approach of obtaining pointwise estimates based on the Green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks. These are being explored elsewhere. © 1997 John Wiley & Sons, Inc.

On the asymptotics of the heat equation and bounds on traces associated with the Dirichlet Laplacian

We use the Feynman-Kac formula and a decomposition of the Brownian bridge to obtain pointwise estimates on the diagonal elements of the heat kernel. We also compute bounds on trace ( e tΔ D ) where Δ D is the Dirichlet Laplacian for (i) a bounded region D in R n with smooth boundary or (ii) an unbounded region D in R 2 with finite area.

Decay estimates for second-order quasilinear partial differential equations

This article is concerned with second-order quasilinear equations (not necessarily elliptic) in two independent variables of the form div[ ϱ(x, u, grad u)grad u] = 0. For Dirichlet (or Neumann) problems on a semi-infinite strip, with non-zero data on one end only, differential inequality techniques are employed to establish exponential decay estimates for quadratic (“energy”) integrals and to obtain pointwise estimates. The results have application to Saint-Venant principles for nonlinear elasticity and compressible fluid flows as well as to theorems of Phragmén-Lindelöf type.

The behavior of the derivatives of the algebraic polynomials of best approximation

In a recent paper M. Hasson [3] has investigated the behavior of the derivatives of the polynomials of best approximation of a function f on (1, 11. These investigations have led to norm estimates on the derivatives of the polynomials and on the distance between these derivatives and the respective derivatives of the function. Separate results were obtained when the norms were taken over the whole interval and when they were taken on a subinterval (where the estimates are significantly better). This calls for pointwise estimates in the spirit of the results of Timan [6] and Trigub [7]. In Section 2 we obtain pointwise estimates on the distance between the derivatives of the polynomials of best approximation off and the respective derivatives of J These results extend and unify those of Hasson [3]. In Section 3 we consider higher-order derivatives and obtain estimates on the growth of the sequence of derivatives of the polynomials of best approximation. Some special cases of these results are due to Hasson [3].

Maximum principles and pointwise error estimates for torsion of shells of revolution

This paper is concerned with the question of assessing the quality of approximate thin shell solutions for the problem of axisymmetric torsion of elastic shells of revolution, an issue previously considered by Ho and Knowles [1]. In both works, the objective is to obtain pointwise estimates, based on the three-dimensional theory of elasticity, for the errors involved in using an approximate shell theory.

A nonlinear generalization of Bihari’s inequality

The aim of the present note is to prove a nonlinear generalization of the well-known integral inequality due to Bihari. This generalization is useful in obtaining pointwise estimates of solutions of nonlinear Volterra integral equations.

Pointwise bounds for eigenfunctions of one-electron systems

In this letter the complementary character of two recently proposed methods to obtain pointwise estimates for eigenfunctions of one-electron systems is pointed out. An illustrative example is discussed and a criterion for the accuracy of approximate wavefunctions is questioned.

Bounds in the Cauchy Problem for a Fourth Order Quasi-Linear Equation

Consider the Cauchy problem for the fourth order quasi-linear equation \[ \Delta ^2 v = h( {x,v,v_{,i} ,\Delta v,\nabla v_{,i} } ) \]where $\Delta $ is the Laplace operator, the comma notation denotes partial differentiation with respect to $x_i $, h satisfies a uniform Lipschitz condition in all but the x-variables, and upper bounds for the error in measurement of the Cauchy data are known. Pointwise bounds for the solution and the square of the gradient are obtained by means of an a priori inequality which is derived under the additional assumption that the solution is uniformly bounded in the domain under consideration. These bounds can be improved by the Ritz method. The technique can be utilized to obtain pointwise estimates in the Cauchy problem for certain coupled systems of elliptic partial differential equations.