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Applications of P-functions to Fully Nonlinear Elliptic equations: Gradient Estimates and Rigidity Results

Dimitrios Gazoulis

TL;DR

The paper develops a unifying $P$-function framework for fully nonlinear elliptic equations, establishing general criteria to construct $P$-functions with $\mu L P \le 0$ and deriving consequences such as pointwise gradient bounds, Harnack-type inequalities, and rigidity results. It then applies the method to Pucci’s equations, Monge–Ampère-type equations, Allen–Cahn-type settings, and higher-order nonlinear PDEs, yielding sharp gradient bounds (including a Pucci-modica-type bound), Liouville-type theorems, and De Giorgi-type properties. Key contributions include a comprehensive abstract gradient bound theorem, concrete $P$-function constructions, and higher-order extensions that provide both local and global estimates for derivatives. Collectively, the work offers a versatile toolkit for sharp gradient control and rigidity across second- and higher-order nonlinear elliptic PDEs, with broad implications for nonlinear PDE theory and its applications.

Abstract

We introduce the notion of $ P -$functions for fully nonlinear equations and establish a general criterion for obtaining such quantities for this class of equations. Some applications are gradient bounds, De Giorgi-type properties of entire solutions and rigidity results. In particular, we establish a pointwise gradient bound and a rigidity result for Pucci's equations. This pointwise gradient bound generalizes the Modica inequality in the case of fully nonlinear elliptic equations. Furthermore, we prove Harnack-type inequalities and local estimates for the gradient of solutions. In addition, we consider such quantities for higher order nonlinear equations and for equations of order greater than two we obtain Liouville-type theorems and pointwise estimates for the Laplacian.

Applications of P-functions to Fully Nonlinear Elliptic equations: Gradient Estimates and Rigidity Results

TL;DR

The paper develops a unifying -function framework for fully nonlinear elliptic equations, establishing general criteria to construct -functions with and deriving consequences such as pointwise gradient bounds, Harnack-type inequalities, and rigidity results. It then applies the method to Pucci’s equations, Monge–Ampère-type equations, Allen–Cahn-type settings, and higher-order nonlinear PDEs, yielding sharp gradient bounds (including a Pucci-modica-type bound), Liouville-type theorems, and De Giorgi-type properties. Key contributions include a comprehensive abstract gradient bound theorem, concrete -function constructions, and higher-order extensions that provide both local and global estimates for derivatives. Collectively, the work offers a versatile toolkit for sharp gradient control and rigidity across second- and higher-order nonlinear elliptic PDEs, with broad implications for nonlinear PDE theory and its applications.

Abstract

We introduce the notion of functions for fully nonlinear equations and establish a general criterion for obtaining such quantities for this class of equations. Some applications are gradient bounds, De Giorgi-type properties of entire solutions and rigidity results. In particular, we establish a pointwise gradient bound and a rigidity result for Pucci's equations. This pointwise gradient bound generalizes the Modica inequality in the case of fully nonlinear elliptic equations. Furthermore, we prove Harnack-type inequalities and local estimates for the gradient of solutions. In addition, we consider such quantities for higher order nonlinear equations and for equations of order greater than two we obtain Liouville-type theorems and pointwise estimates for the Laplacian.
Paper Structure (11 sections, 26 theorems, 127 equations)

This paper contains 11 sections, 26 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. $P-$functions for Fully Nonlinear Elliptic equations
  3. Examples of $P -$functions
  4. Gradient Bounds for entire solutions of Fully Nonlinear equations
  5. A Pointwise Gradient Bound for entire solutions of Pucci's equations
  6. Gradient Bounds for entire solutions by the Examples of subsection 2.1
  7. Rigidity results and properties of entire solutions of fully nonlinear equations
  8. A Harnack-type inequality and Local Estimates for the gradient
  9. Higher order nonlinear equations
  10. Local and Global Pointwise estimates
  11. A Liouville theorem and a De Giorgi-type property

Key Result

Theorem 2.2

Let $u$ be a smooth solution or subsolution of such that $\inf_{\overline{\Omega}} g( \nabla^k u )>0$ for some $g :\mathbb{R}^{n^k \times d} \rightarrow [0, + \infty) \;,\; k \in \lbrace 1,..., m-1 \rbrace$ and suppose that $P = P(x,u, \nabla u,..., \nabla^{m-1} u)$ is a $P-$function of GeneralFullyNonlinearPDE with $\mu = \mu ( g( \nabla^k u )) then $P(x,u, \nabla u,..., \nabla^{m-1} u)$ is cons

Theorems & Definitions (56)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Corollary 2.7
  • ...and 46 more