Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Phragmén-Lindelöf-type theorems for functions in Homogeneous De Giorgi Classes

Simone Ciani, Ugo Gianazza, Zheng Li

TL;DR

The paper proves Phragmén-Lindelöf-type growth bounds for nonnegative functions in homogeneous De Giorgi classes, showing that the maximal function \\mu_+(r) grows at most like a sublinear power \\alpha \\in (0,1) as \\ r \\to \\infty$, and provides explicit counterexamples that preclude the exponent from reaching 1. It develops a capacity-based framework with fat sets and a new logarithmic estimate, together with a weak Harnack inequality tailored to De Giorgi classes, to derive these growth bounds in both 1<p<N and p=N regimes. The main contributions are (i) a unified approach to obtain PL-type growth via capacity and variational techniques, (ii) sharp counterexamples demonstrating the optimality of the exponent range, and (iii) proofs applicable to unbounded domains and a broad class of DG functions, highlighting limitations of maximum principles in this setting. The results have implications for nonlinear potential theory and the qualitative behavior of solutions governed by degenerate elliptic structures, as they quantify how far DG-class functions can extend into infinity under zero-boundary or boundary-vanishing conditions.

Abstract

We study Phragmén-Lindelöf-type theorems for functions $u$ in homogeneous De Giorgi classes, and we show that the maximum modulus $μ_+(r)$ of $u$ has a power-like growth of order $α\in(0,1)$ when $r\to\infty$. By proper counterexamples, we show that in general we cannot expect $α$ to be $1$.

Phragmén-Lindelöf-type theorems for functions in Homogeneous De Giorgi Classes

TL;DR

The paper proves Phragmén-Lindelöf-type growth bounds for nonnegative functions in homogeneous De Giorgi classes, showing that the maximal function \\mu_+(r) grows at most like a sublinear power \\alpha \\in (0,1) as \\ r \\to \\infty$, and provides explicit counterexamples that preclude the exponent from reaching 1. It develops a capacity-based framework with fat sets and a new logarithmic estimate, together with a weak Harnack inequality tailored to De Giorgi classes, to derive these growth bounds in both 1<p<N and p=N regimes. The main contributions are (i) a unified approach to obtain PL-type growth via capacity and variational techniques, (ii) sharp counterexamples demonstrating the optimality of the exponent range, and (iii) proofs applicable to unbounded domains and a broad class of DG functions, highlighting limitations of maximum principles in this setting. The results have implications for nonlinear potential theory and the qualitative behavior of solutions governed by degenerate elliptic structures, as they quantify how far DG-class functions can extend into infinity under zero-boundary or boundary-vanishing conditions.

Abstract

We study Phragmén-Lindelöf-type theorems for functions in homogeneous De Giorgi classes, and we show that the maximum modulus of has a power-like growth of order when . By proper counterexamples, we show that in general we cannot expect to be .

Paper Structure

This paper contains 24 sections, 16 theorems, 132 equations.

Table of Contents

  1. Introduction and Main Results
  2. Novelty and Significance
  3. A First Counterexample
  4. A Second Counterexample
  5. A Third Counterexample
  6. Structure of the Paper
  7. Ackonwledgements
  8. Notation and Main Tools
  9. Variational Capacity
  10. Fat Sets
  11. A Different Notion of Capacity for 1<p<N
  12. A Different Notion of Capacity for p=N
  13. A Final Comment About Capacities
  14. Preliminary Results
  15. The De Giorgi Lemma
  16. ...and 9 more sections

Key Result

Theorem 1.3

Take $1<p\le N$ and let $u \in [DG]^{+}_p(\mathbb{R}^N_+,\hat{\gamma})$; assume that $u$ is non-negative, $u\not\equiv 0$ in $\mathbb{R}^N_+$, and such that Then, there exists $0<\alpha_1< 1$ depending on $N$, $p$, and $\hat{\gamma}$, such that

Theorems & Definitions (31)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.1
  • Theorem 1.3
  • Remark 1.2
  • Theorem 1.4
  • Remark 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.1
  • ...and 21 more