Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Revisiting the Lavrentiev Phenomenon in One Dimension

Wiktor Wichrowski

TL;DR

This work revisits Lavrentiev's one-dimensional results on the Lavrentiev phenomenon, clarifying his original example, exposing flaws in the historic proof, and providing a concise, complete proof of the Lavrentiev Approximation Lemma. It shows that for functionals with appropriate $f$, any $u\in AC_*([0,1])$ can be uniformly approximated by smooth functions with nearly the same energy, yielding $\inf_{AC_*} \mathcal{F} = \inf_{C^\infty_*} \mathcal{F}$ in one dimension. The authors also supply an appendix correcting the original proof by detailing a robust construction of the intermediary sets and a finite interval partition, addressing gaps in Lavrentiev's argument. These results enhance the regularity theory for variational problems and bolster the reliability of numerical schemes by eliminating potential Lavrentiev gaps in 1D.

Abstract

We clarify and extend insights from Lavrentiev's seminal paper. We examine the original theorem on the absence of the Lavrentiev's phenomenon and a counterexample offering a detailed analysis of its proof and providing a new, concise, and complete reasoning. In the appendix, we also provide additional details to supplement the original proof.

Revisiting the Lavrentiev Phenomenon in One Dimension

TL;DR

This work revisits Lavrentiev's one-dimensional results on the Lavrentiev phenomenon, clarifying his original example, exposing flaws in the historic proof, and providing a concise, complete proof of the Lavrentiev Approximation Lemma. It shows that for functionals with appropriate , any can be uniformly approximated by smooth functions with nearly the same energy, yielding in one dimension. The authors also supply an appendix correcting the original proof by detailing a robust construction of the intermediary sets and a finite interval partition, addressing gaps in Lavrentiev's argument. These results enhance the regularity theory for variational problems and bolster the reliability of numerical schemes by eliminating potential Lavrentiev gaps in 1D.

Abstract

We clarify and extend insights from Lavrentiev's seminal paper. We examine the original theorem on the absence of the Lavrentiev's phenomenon and a counterexample offering a detailed analysis of its proof and providing a new, concise, and complete reasoning. In the appendix, we also provide additional details to supplement the original proof.

Paper Structure

This paper contains 8 sections, 9 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Lavrentiev Example
  3. Absence of the Lavrentiev Phenomenon in one-dimensional problems
  4. Sketch of original proof
  5. Simple counterexample
  6. A Simplified Proof of the Lavrentiev Approximation Lemma
  7. Consequences of the Approximation Lemma
  8. Supplementing Lavrentiev's proof

Key Result

Lemma 2.1

The minimum of for $u \in \text{Lip}_*([a,b])$ and any convex function $f\colon \mathbb{R} \to \mathbb{R}$ is attained when $u(x)$ is a straight line. Therefore, the following inequality holds:

Figures (2)

  • Figure 1: In blue:Plot of $y=\frac{1}{4}\sqrt{x}$.In red:Plot of $y=\frac{1}{2}\sqrt{x}$.In green:Tangent line to plot of $y=\frac{1}{4}\sqrt{x}$ in point $x_0$.
  • Figure 3: In blue, there is graph of $u'$, while on the x-axis in green the set $P_\varepsilon$ is marked.

Theorems & Definitions (18)

  • Lemma 2.1
  • Example 2.2: Lavrentiev 1926
  • proof
  • Lemma 3.1: Approximation Lemma, Mascon
  • Lemma 3.2
  • proof
  • Remark 4.1
  • proof
  • Theorem 5.1: Lavrentiev 1927, Mascon
  • proof
  • ...and 8 more