Abstract. The main result includes features of a Hardy-type inequality and an inequality of either Sobolev or Gagliardo-Nirenberg type. It is inspired by the method of proof of a recent improved Sobolev inequality derived by M. Ledoux which brings out the connection between Sobolev embeddings and heat kernel bounds. Here Ledoux’s technique is applied to the operator L := x · ∇ and the analysis ...

ANALYSIS AND PDE ON METRIC MEASURE SPACES: SOBOLEV FUNCTIONS AND VISCOSITY SOLUTIONS Xiaodan Zhou, PhD University of Pittsburgh, 2016 We study analysis and partial differential equations on metric measure spaces by investigating the properties of Sobolev functions or Sobolev mappings and studying the viscosity solutions to some partial differential equations. This manuscript consists of two par...

Least squares methods are effective for solving systems of partial differential equations. In the case of nonlinear systems the equations are usually linearized by a Newton iteration or successive substitution method, and then treated as a linear least squares problem. We show that it is often advantageous to form a sum of squared residuals first, and then compute a zero of the gradient with a ...

s – Lectures Thursday, December 1, 9:00-10:30 Lecture 1: Spline approximation Stefan Takacs RICAM, Computational Methods for PDEs Thomas Takacs JKU Linz, Institute of Applied Geometry In the first part of the lecture we discuss standard approximation properties for B-spline and NURBS spaces. B-splines are piecewise polynomial functions of a given prescribed smoothness. NURBS stands for non-unif...

Action of p ? on distributions is examined in the context of Sobolev spaces, weighted L 2 spaces and weighted Sobolev spaces, respectively. The results obtained are as follows: Let k be a real number. (1) If f is in H

We develop a new method to obtain symmetrization inequalities of Sobolev type. Our approach leads to new inequalities and considerable simplification in the theory of embeddings of Sobolev spaces based on rearrangement invariant spaces.

We investigate asymptotic behaviour of approximation numbers of Sobolev embeddings between weighted function spaces of Sobolev–Hardy–Besov type with polynomials weights. The exact estimates are proved in almost all cases. © 2005 Elsevier Inc. All rights reserved.

We investigate properties of measures in infinite dimensional spaces in terms of Poincaré inequalities. A Poincaré inequality states that the L2 variance of an admissible function is controlled by the homogeneous H1 norm. In the case of Loop spaces, it was observed by L. Gross [17] that the homogeneous H1 norm alone may not control the L2 norm and a potential term involving the end value of the...