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A variational inequality for discontinuous solutions of degenerate parabolic equations.

Lorina Dascal, Shoshana Kamin, Nir A. Sochen (2005)

RACSAM

The Beltrami framework for image processing and analysis introduces a non-linear parabolic problem, called in this context the Beltrami flow. We study in the framework for functions of bounded variation, the well-posedness of the Beltrami flow in the one-dimensional case. We prove existence and uniqueness of the weak solution using lower semi-continuity results for convex functions of measures. The solution is defined via a variational inequality, following Temam?s technique for the evolution problem...

Absolute continuity for elliptic-caloric measures

Caroline Sweezy (1996)

Studia Mathematica

A Carleson condition on the difference function for the coefficients of two elliptic-caloric operators is shown to give absolute continuity of one measure with respect to the other on the lateral boundary. The elliptic operators can have time dependent coefficients and only one of them is assumed to have a measure which is doubling. This theorem is an extension of a result of B. Dahlberg [4] on absolute continuity for elliptic measures to the case of the heat equation. The method of proof is an...

An extension of Rothe's method to non-cylindrical domains

Komil Kuliev, Lars-Erik Persson (2007)

Applications of Mathematics

In this paper Rothe’s classical method is extended so that it can be used to solve some linear parabolic boundary value problems in non-cylindrical domains. The corresponding existence and uniqueness theorems are proved and some further results and generalizations are discussed and applied.

An observability estimate for parabolic equations from a measurable set in time and its applications

Kim Dang Phung, Gengsheng Wang (2013)

Journal of the European Mathematical Society

This paper presents a new observability estimate for parabolic equations in Ω × ( 0 , T ) , where Ω is a convex domain. The observation region is restricted over a product set of an open nonempty subset of Ω and a subset of positive measure in ( 0 , T ) . This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.

An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence

Sashikumaar Ganesan (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation...

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