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Optimal shapes for positivity preserving

Sascha Eichmann

Abstract

We are looking for an optimal convex domain on which the boundary value problem $$\left\{\begin{array}{cc}(-Δ)^2 u_γ-γΔu_γ= f,& \mbox{ in }Ω\\ u_γ=\partial_νu_γ=0,& \mbox{ on }\partialΩ\end{array}\right.$$ admits a nonnegative solution for the most $γ$, if $f$ is a given nonnegative function.

Optimal shapes for positivity preserving

Abstract

We are looking for an optimal convex domain on which the boundary value problem admits a nonnegative solution for the most , if is a given nonnegative function.
Paper Structure (5 sections, 5 theorems, 62 equations, 2 figures)

This paper contains 5 sections, 5 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hausdorff distance and convexity
  3. Proof of the main Theorem
  4. Open problems
  5. Positivity for large tension

Key Result

Theorem 1.1

There exists an $\Omega_f\in M$, such that for all $\Omega\in M$ we have

Figures (2)

  • Figure 1: Seperating $x_m$ from $\Omega_m$ by Hahn-Banach.
  • Figure 2: Counterexample for Thm. \ref{['2_2']} without convexity.

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.1: Blaschke1916 p.61 & p.62
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • proof
  • Theorem A.1: see Thm. 1.1 in EichmannSchaetzlePosHighTens