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Quantitative and exact concavity principles for parabolic and elliptic equations

Marco Gallo, Riccardo Moraschi, Marco Squassina

TL;DR

The paper develops a comprehensive framework to analyze concavity properties of parabolic and elliptic PDEs of the form $u_t-\Delta u = a(x,t) f(u)$ in convex domains. By constructing and applying concavity and harmonic concavity maximum principles, and by employing log- and power-transformations of the solution, it obtains exact and quantitative concavity results under broad nonlinearities (including $f(u)=1$, $u^q$, $u$, $u\log u$, and $\frac{u^2}{1+u}$) and varying weights $a(x,t)$, even when $a$ is not strongly concave. The work provides explicit conditions under which $\log(u(\cdot,t))$ is concave for each $t$, and derives perturbative estimates that measure how far a solution is from exact concavity in terms of the oscillation or concavity/monotonicity of $a$, with applications to weighted Lane-Emden, logistic-type, and torsion-type problems, along with elliptic counterparts. Overall, the results offer a robust, quantitative toolkit for understanding the qualitative structure of solutions and their maxima in heterogeneous media, linking parabolic concavity to classical elliptic results and extending Ishige–Salani-type analyses.

Abstract

Goal of this paper is to study classes of Cauchy-Dirichlet problems which include parabolic equations of the type $$u_t -Δu= a(x,t)f(u)\quad\hbox{in $Ω\times(0,T)$}$$ with $Ω\subset\mathbb{R}^N$ bounded, convex domain and $T\in(0,+\infty]$. Under suitable assumptions on $a$ and $f$, we show logarithmic or power concavity (in space, or in space-time) of the solution $u$; under some relaxed assumptions on $a$, we show moreover that $u$ enjoys concavity properties up to a controlled error. The results include relevant examples like the torsion $f(u)=1$, the Lane-Emden equation $f(u)=u^q$, $q\in(0,1)$, the eigenfunction $f(u)=u$, the logarithmic equation $f(u)=u\log(u^2)$, and the saturable nonlinearity $f(u)=\frac{u^2}{1+u}$. The logistic equation $f(x,u)=a(x)u-u^2$ can be treated as well. Some exact results give a different approach, as well as generalizations, to [Ishige-Salani2013, Ishige-Salani2016]. Moreover, some quantitative results are valid also in the elliptic framework $-Δu=a(x)f(u)$ and refine [Bucur-Squassina2019, Gallo-Squassina2024].

Quantitative and exact concavity principles for parabolic and elliptic equations

TL;DR

The paper develops a comprehensive framework to analyze concavity properties of parabolic and elliptic PDEs of the form in convex domains. By constructing and applying concavity and harmonic concavity maximum principles, and by employing log- and power-transformations of the solution, it obtains exact and quantitative concavity results under broad nonlinearities (including , , , , and ) and varying weights , even when is not strongly concave. The work provides explicit conditions under which is concave for each , and derives perturbative estimates that measure how far a solution is from exact concavity in terms of the oscillation or concavity/monotonicity of , with applications to weighted Lane-Emden, logistic-type, and torsion-type problems, along with elliptic counterparts. Overall, the results offer a robust, quantitative toolkit for understanding the qualitative structure of solutions and their maxima in heterogeneous media, linking parabolic concavity to classical elliptic results and extending Ishige–Salani-type analyses.

Abstract

Goal of this paper is to study classes of Cauchy-Dirichlet problems which include parabolic equations of the type with bounded, convex domain and . Under suitable assumptions on and , we show logarithmic or power concavity (in space, or in space-time) of the solution ; under some relaxed assumptions on , we show moreover that enjoys concavity properties up to a controlled error. The results include relevant examples like the torsion , the Lane-Emden equation , , the eigenfunction , the logarithmic equation , and the saturable nonlinearity . The logistic equation can be treated as well. Some exact results give a different approach, as well as generalizations, to [Ishige-Salani2013, Ishige-Salani2016]. Moreover, some quantitative results are valid also in the elliptic framework and refine [Bucur-Squassina2019, Gallo-Squassina2024].

Paper Structure

This paper contains 13 sections, 39 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On parabolic equations
  4. Power concavity and concavity functions
  5. Concavity and harmonic concavity maximum principles
  6. Maximum principles with boundary conditions
  7. Concavity
  8. Applications
  9. Harmonic concavity
  10. Applications
  11. Further results on the weighted torsion problem
  12. Comparison principles
  13. Properties of concavity functions

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^n$, $n\geq 2$, be a bounded, smooth, strongly convex domain. Let $T>0$, $a\colon\Omega\times(0,T]\to \mathbb{R}$, $u_0\in C^1(\overline{\Omega})$ with $u_0=0$ on $\partial\Omega$ and $u \in C^2_x(\overline{\Omega}) \cap C^1(\overline{\Omega}\times[0,T])$ be a positive where $\nu$ is the interior normal to $\partial \Omega$. Assume moreover that $u_0$ is $\log$-conca

Theorems & Definitions (93)

  • Theorem 1.1: Weighted eigenfunctions and logarithmic equations
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4: Weighted Lane-Emden
  • Remark 1.5
  • Theorem 1.6: Sum of powers
  • Theorem 1.7: Quantitative concavity, I
  • Theorem 1.8: Quantitative concavity, II
  • Remark 1.9
  • Theorem 1.10: $\theta$-concavity estimate
  • ...and 83 more