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Optimization of the L1 norm of the solution of a Fisher-KPP equations in the small diffusivity regime

Grégoire Nadin

TL;DR

This work analyzes the problem of maximizing the L1 influence of the Fisher KPP steady state under spatially heterogeneous growth in the small diffusivity limit. It proves that all maximizers are BV and, in the regime $c\in(1,3)$, they are bang-bang with a finite number of blocks, yielding a clear BV structure. A central contribution is the construction of a periodic-like quasi-maximizer whose BV norm scales as $\sim 1/\sqrt{\mu}$ and the demonstration that the maximal value converges to a limit $G(\overline{\mu}_{inf})$ as $\mu\to0$, with $\overline{\mu}_{inf}$ defined via the crenel-based function $G$. The paper also establishes an equality case linking symmetry properties to a unique maximizer of $G$ and outlines open questions about uniqueness of the limit maximizer, the potential for uniform BV bounds, and extensions to higher dimensions.

Abstract

We investigate in the present paper the maximization problem for the L1 norm of the unique positive solution of an heterogeneous Fisher-KPP equation with respect to the growth rate. It is already known that the BV norms of maximizers of this functional blow up when the diffusivity tends to zero. Here, we first show that the maximizers are always BV. Next, we completely characterize the limit of the maximas of this functional as the diffusivity tends to zero, and we show that one can construct a quasi-maximizer which is periodic, in a sense, and with a BV norm behaving like the inverse of the square root of the diffusivity. Lastly, we prove that along a subsequence of diffusivities, any maximizer is periodic, in a sense.

Optimization of the L1 norm of the solution of a Fisher-KPP equations in the small diffusivity regime

TL;DR

This work analyzes the problem of maximizing the L1 influence of the Fisher KPP steady state under spatially heterogeneous growth in the small diffusivity limit. It proves that all maximizers are BV and, in the regime , they are bang-bang with a finite number of blocks, yielding a clear BV structure. A central contribution is the construction of a periodic-like quasi-maximizer whose BV norm scales as and the demonstration that the maximal value converges to a limit as , with defined via the crenel-based function . The paper also establishes an equality case linking symmetry properties to a unique maximizer of and outlines open questions about uniqueness of the limit maximizer, the potential for uniform BV bounds, and extensions to higher dimensions.

Abstract

We investigate in the present paper the maximization problem for the L1 norm of the unique positive solution of an heterogeneous Fisher-KPP equation with respect to the growth rate. It is already known that the BV norms of maximizers of this functional blow up when the diffusivity tends to zero. Here, we first show that the maximizers are always BV. Next, we completely characterize the limit of the maximas of this functional as the diffusivity tends to zero, and we show that one can construct a quasi-maximizer which is periodic, in a sense, and with a BV norm behaving like the inverse of the square root of the diffusivity. Lastly, we prove that along a subsequence of diffusivities, any maximizer is periodic, in a sense.
Paper Structure (9 sections, 17 theorems, 79 equations, 2 figures)

This paper contains 9 sections, 17 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Earlier works
  3. Statement of the results
  4. $BV$ regularity of the maximizers
  5. Existence of a quasi-maximizer
  6. Construction of the quasi-maximizer $\hat{m}_{\mu}$
  7. Estimates on $\mu/\delta_{\mu}^{2}$
  8. The equality case
  9. Discussion and open problems

Key Result

Theorem 1.1

Assume that $c\in (1,3)$. Let $\overline{m}_{\mu}$be a maximizer of $F_{\mu}$ and assume that $\overline{m}_{\mu}\not\equiv 0$ and $\overline{m}_{\mu}\not\equiv \kappa$. Then the function $\theta_{\overline{m}_{\mu},\mu}'$ admits a finite number of zeros, and $\overline{m}_{\mu}$ admits exactly one

Figures (2)

  • Figure 1: Construction of $\hat{m}_{\mu}$. First step: select the interval $(a_{i},a_{i+1})$ maximizing $A_{i}$. Second step: repeat periodically this pattern. Third step: stretch it so that it ends at $x=1$.
  • Figure 2: An illustration of the definitions of $\textcolor{black}{\overline{\mu}_{inf}}$ and $\textcolor{black}{\overline{\mu}_{sup}}$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 3.1
  • Lemma 3.2
  • ...and 11 more