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On the nonexistence of Green's function and failure of the strong maximum principle

Luigi Orsina, Augusto C. Ponce

TL;DR

This work analyzes the Schrödinger operator $-\Delta + V$ with nonnegative Borel potential $V$ on a smooth bounded domain $\Omega$. It identifies a universal zero-set $Z$ where a Green's function cannot exist and shows that $\Omega\setminus Z$ splits into Sobolev-connected components on which the strong maximum principle holds; solvability of $-\Delta u + Vu = \mu$ with nonnegative measure datum $\mu$ occurs if and only if $\mu(Z)=0$. The authors develop a duality-solution framework, establish a comparison principle, and prove a detailed Green's representation off $Z$, along with an orthogonality principle that links measure data to the geometry of $Z$. They further introduce a Green's-function-based decomposition of $\Omega\setminus S$ (where $S$ is the torsion-zero set), define Sobolev-open/connected structures for the level sets $U_x$, and prove sharp maximum-principle results within each resulting component, including a strong maximum principle for measure data. Collectively, the results illuminate how singular potentials shape the existence of Green's functions and the validity of maximum principles through a fine decomposition of the domain into Sobolev-structured regions.

Abstract

Given any Borel function $V : Ω\to [0, +\infty]$ on a smooth bounded domain $Ω\subset \mathbb{R}^{N}$, we establish that the strong maximum principle for the Schrödinger operator $-Δ+ V$ in $Ω$ holds in each Sobolev-connected component of $Ω\setminus Z$, where $Z \subset Ω$ is the set of points which cannot carry a Green's function for $- Δ+ V$. More generally, we show that the equation $- Δu + V u = μ$ has a distributional solution in $W_{0}^{1, 1}(Ω)$ for a nonnegative finite Borel measure $μ$ if and only if $μ(Z) = 0$.

On the nonexistence of Green's function and failure of the strong maximum principle

TL;DR

This work analyzes the Schrödinger operator with nonnegative Borel potential on a smooth bounded domain . It identifies a universal zero-set where a Green's function cannot exist and shows that splits into Sobolev-connected components on which the strong maximum principle holds; solvability of with nonnegative measure datum occurs if and only if . The authors develop a duality-solution framework, establish a comparison principle, and prove a detailed Green's representation off , along with an orthogonality principle that links measure data to the geometry of . They further introduce a Green's-function-based decomposition of (where is the torsion-zero set), define Sobolev-open/connected structures for the level sets , and prove sharp maximum-principle results within each resulting component, including a strong maximum principle for measure data. Collectively, the results illuminate how singular potentials shape the existence of Green's functions and the validity of maximum principles through a fine decomposition of the domain into Sobolev-structured regions.

Abstract

Given any Borel function on a smooth bounded domain , we establish that the strong maximum principle for the Schrödinger operator in holds in each Sobolev-connected component of , where is the set of points which cannot carry a Green's function for . More generally, we show that the equation has a distributional solution in for a nonnegative finite Borel measure if and only if .

Paper Structure

This paper contains 13 sections, 38 theorems, 325 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Method of sub- and supersolutions
  3. Distributional solutions are duality solutions
  4. Duality solutions as distributional solutions
  5. Existence of a distributional solution with datum $\chi_{\Omega \setminus Z}$
  6. Orthogonality principle
  7. Comparison principle
  8. Proofs of \ref{['theoremGreen', 'theoremGoodMeasuresCharacterization']}
  9. Green's functions and decomposition of $\Omega \setminus S$
  10. Sobolev-openness of $U_{x}$
  11. Sobolev-connectedness of $U_{x}$
  12. Proofs of \ref{['theoremMaximumPrincipleFull']} and \ref{['theoremMaximumPrinciplePrecise']}
  13. Strong maximum principle for distributional solutions involving measures

Key Result

Theorem 1.1

For every Borel function $V : \Omega \to [0, +\infty]$, the Sobolev-open set $\Omega \setminus Z$ can be uniquely decomposed as a finite or countably infinite union of disjoint Sobolev-connected-open sets $(D_{j})_{j \in J}$ and any solution $w \in W_{0}^{1, 2}(\Omega) \cap L^{\infty}(\Omega)$ of th

Figures (3)

  • Figure 1: Sobolev-open set which is not open.
  • Figure 2: Relation among the classes of quasi-, fine- and Sobolev-open sets.
  • Figure 3: (a) Path-connected set which is not Sobolev-connected; (b) Sobolev-connected set.

Theorems & Definitions (101)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Definition 1.1
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Example 1.5
  • Example 1.6
  • ...and 91 more