Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An elliptic problem in dimension N with a varying drift term bounded in $L^N$

Juan Casado-Díaz

Abstract

The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in $L^N(Ω)$, with $N$ the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space $H^1_0(Ω)$. However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates in \cite{boc1}, and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems.

An elliptic problem in dimension N with a varying drift term bounded in $L^N$

Abstract

The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in , with the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space . However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates in \cite{boc1}, and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems.
Paper Structure (3 sections, 4 theorems, 68 equations)

This paper contains 3 sections, 4 theorems, 68 equations.

Table of Contents

  1. Introduction.
  2. Some reminders about elliptic problems with a drift or convection term.
  3. Passing to the limit with a varying drift term.

Key Result

Theorem 2.1

For every $f\in H^{-1}(\Omega)$, $E\in L^2(\Omega)^N$, and $a\in L^1(\Omega)$, $a\geq 0$ a.e. in $\Omega$, there exists an entropy solution of in the following sense Moreover, there exists $C>0$, independent of $f$ and $E$ such that If there exists $\gamma>0$ such that and $f$ belongs to $L^1(\Omega)$, such solution is also in $L^1(\Omega)$ and satisfies

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 3.1
  • Remark 3.2