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Beyond the classical strong maximum principle: forcing changing sign near the boundary and flat solutions

Jesús Ildefonso Díaz, Jesús Hernández

Abstract

We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain $Ω$ can be extended, under suitable conditions, to the case in which the forcing term $f(x)$ is changing sign. In addition, in the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solutions). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign is also given.

Beyond the classical strong maximum principle: forcing changing sign near the boundary and flat solutions

Abstract

We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain can be extended, under suitable conditions, to the case in which the forcing term is changing sign. In addition, in the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solutions). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign is also given.
Paper Structure (5 sections, 7 theorems, 131 equations, 2 figures)

This paper contains 5 sections, 7 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The symmetric one-dimensional linear problem
  3. The N-dimensional case and a general bounded domain $\Omega$
  4. Application to some sublinear indefinite semilinear equations
  5. An application to the linear parabolic problem

Key Result

Theorem 2.1

Weassume that $f(x)$ becomes negative near the boundary in the following sense: there exists such that (A) Assume the “ balance condition” and the “ decay condition” Then any symmetric supersolution $u$ satisfies Moreover, if for instance $u\in C^{1}[0,R]$, then we have In addition, assumed (Hypo positivity r0), if $u$ is a solution then $u>0$if and only if the decay condition (Hypo r0 doble)

Figures (2)

  • Figure 1: Representation of the exact solution of (\ref{['probExam1']}), and the value of its normal derivative at the boundary, when the forcing is given by (\ref{['ForcingExamp']}), for different values of $a$.
  • Figure 2: Representation of the solution of problem \ref{['probExam1']} when the forcing is like (\ref{['ForcingExamp']}) but with the oposite sign on $f(x)$.

Theorems & Definitions (28)

  • Theorem 2.1
  • Remark 2.2
  • Example 2.3
  • Corollary 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 18 more