Obtain Pointwise Estimates Research Articles

Sharp estimates of semistable radial solutions of k-Hessian equations

AbstractWe consider semistable, radially symmetric and increasing solutions of Sk(D2u) = g(u) in the unit ball of ℝn, where Sk(D2u) is the k-Hessian operator of u and g ∈ C1 is a general positive nonlinearity. We establish sharp pointwise estimates for such solutions in a proper weighted Sobolev space, which are optimal and do not depend on the specific nonlinearity g. As an application of these results, we obtain pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation Sk(D2u) = λg(u), posed in B1, with Dirichlet data $u\arrowvert _{B_1}=0$, where g is a continuous, positive, nonincreasing function such that lim t→−∞g(t)/|t|k = +∞.

Interpolatory estimates in monotone piecewise polynomial approximation

Given a monotone function f∈Cr[−1,1], r≥1, we obtain pointwise estimates for its monotone approximation by piecewise polynomials involving the second order modulus of smoothness of f(r). These estimates are interpolatory estimates, namely, the piecewise polynomials interpolate the function at the endpoints of the interval. However, they are valid only for n≥N(f,r). We also show that such estimates are in general invalid with N independent of f.

Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.

Pointwise estimates for solutions of semilinear parabolic inequalities with a potential

We obtain pointwise estimates for solutions of semilinear parabolic equations with a potential on connected domains both of \mathbb R^n and of general Riemannian manifolds. Our results generalize to the parabolic case those shown in [10] for elliptic equations.

From integral estimates of functions to uniform and locally averaged ones

Obtaining pointwise estimates from above for a function or its averages, under known integral restrictions on the growth of the function, often arise in the function theory. We offer an approach to such problems based on integral Jensen’s inequality with a convex function.

Interpolatory estimates in monotone piecewise polynomial approximation

Given a monotone function f∈Cr[−1,1], r≥1, we obtain pointwise estimates for its monotone approximation by piecewise polynomials involving the second order modulus of smoothness of f(r). These estimates are interpolatory estimates, namely, the piecewise polynomials interpolate the function at the endpoints of the interval. However, they are valid only for n≥N(f,r). We also show that such estimates are in general invalid with N independent of f.

Pointwise Estimates and Regularity in Geometric Optics and Other Generated Jacobian Equations

AbstractThe study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge‐Ampère type known as generated Jacobian equations. This class of equations, whose general existence theory has been recently developed by Trudinger, goes beyond the framework of optimal transport. We obtain pointwise estimates for weak solutions of such equations under minimal structural and regularity assumptions, covering situations analogous to those of costs satisfying the A3‐weak condition introduced by Ma, Trudinger, and Wang in optimal transport. These estimates are used to develop a C1,α regularity theory for weak solutions of Aleksandrov type. The results are new even for all known near‐field reflector/refractor models, including the point source and parallel beam reflectors, and are applicable to problems in other areas of geometry, such as the generalized Minkowski problem.© 2017 Wiley Periodicals, Inc.

Resolvent bounds for jump generators

The paper deals with jump generators with a convolution kernel. Assuming that the kernel decays either exponentially or polynomially, we prove a number of lower and upper bounds for the resolvent of such operators. In particular we focus on sharp estimates of the resolvent kernel for small values of the spectral parameter. We consider two applications of these results. First we obtain pointwise estimates for principal eigenfunction of jump generators perturbed by a compactly supported potential (so-called nonlocal Schrödinger operators). Then we consider the Cauchy problem for the corresponding inhomogeneous evolution equations and study the behaviour of its solutions.

Subdyadic square functions and applications to weighted harmonic analysis

Through the study of novel variants of the classical Littlewood–Paley–Stein g-functions, we obtain pointwise estimates for broad classes of highly-singular Fourier multipliers on Rd satisfying regularity hypotheses adapted to fine (subdyadic) scales. In particular, this allows us to efficiently bound such multipliers by geometrically-defined maximal operators via general weighted L2 inequalities, in the spirit of a well-known conjecture of Stein. Our framework applies to solution operators for dispersive PDE, such as the time-dependent free Schrödinger equation, and other highly oscillatory convolution operators that fall well beyond the scope of the Calderón–Zygmund theory.

Pointwise Inequalities for Elliptic Boundary Value Problems

We introduce a new approach for obtaining pointwise estimates for solutions of elliptic boundary value problems when the operator satisfies a certain type of weighted integral inequalities. The method is illustrated on several examples, including a scalar secondorder elliptic equation, the 3D Lamé system, and a scalar higher-order elliptic equation. The techniques can be extended to other elliptic boundary value problems provided that the corresponding weighted integral inequalities are satisfied. Bibliography: 36 titles.

Higher order Glaeser inequalities and optimal regularity of roots of real functions

We prove a higher order generalization of the Glaeser inequality, according to which one can estimate the first derivative of a function in terms of the function itself and the H¨older constant of its k-th derivative. We apply these inequalities in order to obtain pointwise estimates on the derivative of the (k + α)-th root of a function of class Ck whose derivative of order k is α-Holder continuous. Thanks to such estimates, we prove that the root is not just absolutely continuous, but its derivative has a higher summability exponent. Some examples show that our results are optimal

Solvability of boundary singular integral operators of elasticity on surfaces with conic points

A system of boundary singular integral equations of the linear isotropic elasticity is studied in the case where the double layer potential is generated by the stress operator. These equations are considered on a bounded two-dimensional surface without boundary which is smooth outside a finite number of conical points. The solvability of the system is proved in various weighted spaces of differentiable functions. We obtain a representation of the inverse operator of the system in terms of the inverse operators of some boundary value problems. We also obtain pointwise estimates for the kernel of this operator and “quasilocal” estimates for solutions of the integral equations in question. Bibliography: 12 titles.

The penalized Lebesgue constant for surface spline interpolation

Problems involving approximation from scattered data where data is arranged quasi-uniformly have been treated by RBF methods for decades. Treating data with spatially varying density has not been investigated with the same intensity and is far less well understood. In this article we consider the stability of surface spline interpolation (a popular type of RBF interpolation) for data with nonuniform arrangements. Using techniques similar to Hangelbroek, Narcowich and Ward, who discuss interpolation from quasi-uniform data on manifolds, we show that surface spline interpolation on R d \mathbb {R}^d is stable, but in a stronger, local sense. We also obtain pointwise estimates showing that the Lagrange function decays very rapidly, and at a rate determined by the local spacing of datasites. These results, in conjunction with a Lebesgue lemma, show that surface spline interpolation enjoys the same rates of convergence as those of the local approximation schemes recently developed by DeVore and Ron.

Comparison inequalities for heat semigroups and heat kernels on metric measure spaces

We prove a certain inequality for a subsolution of the heat equation associated with a regular Dirichlet form. As a consequence of this inequality, we obtain various interesting comparison inequalities for heat semigroups and heat kernels, which can be used for obtaining pointwise estimates of heat kernels. As an example of application, we present a new method of deducing sub-Gaussian upper bounds of the heat kernel from on-diagonal bounds and tail estimates.

Pointwise error estimates for a stabilized Galerkin method: non-selfadjoint problems

Highly localized pointwise error estimates for a stabilized Galerkin method are provided for second-order non-selfadjoint elliptic partial differential equations. The estimates show a local dependence of the error on the derivative of the solution u and weak dependence on the global norm. The results in this paper are an extension of the previous pointwise error estimates for the self-adjoint problems. In order to provide pointwise error estimates in the presence of the first-order term in the differential equations, we prove that the stabilized Galerkin solution is higher order perturbation to the Ritz projection of the true solutions. Then, we proceed to obtain pointwise estimates using the so-called discrete Green’s function. Application to error expansion inequalities and a posteriori error estimators are briefly discussed.

Blow-Up Estimates at Horizontal Points and Applications

Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Caratheodory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure at all horizontal points of C1,1 smooth submanifolds. As an application, we establish an integral formula to compute the spherical Hausdorff measure of any C1,1 submanifold. Our technique also shows that C2 smooth submanifolds everywhere admit an intrinsic blow-up and that the limit set is an intrinsically homogeneous algebraic variety.

Maximal L 2 and Pointwise Hölder Estimates for □ b on CR Manifolds of Class C 2

In this paper we prove the optimal pointwise Hölder estimates for the Kohn's Laplacian □ b for CR manifolds of class C 2. We first derive optimal L 2 theory for solutions of □ b directly without using pseudo-differential operators. Then we use the scaling method to obtain pointwise estimates in non-istotropic Campanato spaces based on the maximal L 2 theory.

Conditional Stability and Convergence of a Fully Discrete Scheme for Three-Dimensional Navier–Stokes Equations with Mass Diffusion

We construct a fully discrete numerical scheme for three-dimensional incompressible fluids with mass diffusion (in density-velocity-pressure formulation), also called the Kazhikhov-Smagulov model. We will prove conditional stability and convergence, by using at most $C^0$-finite elements, although the density of the limit problem will have $H^2$-regularity. The key idea of our argument is first to obtain pointwise estimates for the discrete density by imposing the constraint $\lim_{(h,k)\to 0}h/k=0$ on the time and space parameters $(k,h)$. Afterwards, under the same constraint on the parameters, strong estimates for the discrete density in $l^\infty(H^1)$ and for the discrete Laplacian of the density in $l^2(L^2)$ are obtained. From here, the compactness and convergence of the scheme can be concluded with similar arguments as we used in [Math. Comp., to appear], where a different scheme is studied for two-dimensional domains which is unconditionally stable and convergent. Moreover, we study the asymptotic behavior of the numerical scheme as the diffusion parameter $\lambda$ goes to zero, obtaining convergence as $(k,h,\lambda)\to 0$ towards a weak solution of the density-dependent Navier-Stokes system provided that the constraint $\lim_{(\lambda,h,k)\to 0}h/(\lambda^2 k)=0$ on $(h,k,\lambda)$ is satisfied.

Fine continuity on metric spaces

We obtain pointwise estimates for solutions of obstacle problems on metric measure spaces and prove that p-superharmonic functions are p-finely continuous. Consequently, we show that p-quasicontinuous functions are p-finely continuous at p-quasievery point. As a byproduct, we obtain the sufficiency part of the Wiener criterion in metric spaces without the assumption of linear local connectedness.

Pointwise Estimates for the Coconvex Approximation of Differentiable Functions

We obtain pointwise estimates for the coconvex approximation of functions of the class Wr, r > 3.