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Boundary estimates for a fully nonlinear Yamabe problem on Riemannian manifolds

Weisong Dong, Yanyan Li, Luc Nguyen

TL;DR

This work addresses the Dirichlet boundary value problem for fully nonlinear Yamabe equations on compact Riemannian manifolds with boundary by leveraging a subsolution to obtain sharp boundary $C^2$-estimates. The authors introduce an interpolating one-parameter family $(f_t,\\Gamma_t)$ that connects the fully nonlinear equation to a semilinear Yamabe form, and establish a priori boundary estimates that are uniform in $t$, enabling existence of smooth solutions for each $t$ via degree theory. The core technical contribution is a detailed analysis of boundary derivatives, including double tangential, mixed tangential-normal, and double normal estimates, built from barrier arguments and careful use of the linearized operator. They also demonstrate the necessity of boundary regularity phenomena by presenting a $C^1$ solution that is not $C^2$ at the boundary, illustrating limitations when subsolutions are not sufficiently controlling boundary behavior. Overall, the results advance boundary regularity theory for Hessian-type conformal equations and yield robust existence results for prescribing Schouten-tensor eigenvalue functions on manifolds with boundary.

Abstract

In this paper, we consider the Dirichlet boundary value problem for fully nonlinear Yamabe equations on Riemannian manifolds with boundary. Assuming the existence of a subsolution, we derive \emph{a priori} boundary second derivative estimates and consequently obtain the existence of a smooth solution. Moreover, with respect to a family of equations interpolating the fully nonlinear Yamabe equation and the classical semi-linear Yamabe equation, our estimates remain uniform. Finally, an example of a $C^1$ solution which is smooth in the interior but not smooth at the boundary is also given.

Boundary estimates for a fully nonlinear Yamabe problem on Riemannian manifolds

TL;DR

This work addresses the Dirichlet boundary value problem for fully nonlinear Yamabe equations on compact Riemannian manifolds with boundary by leveraging a subsolution to obtain sharp boundary -estimates. The authors introduce an interpolating one-parameter family that connects the fully nonlinear equation to a semilinear Yamabe form, and establish a priori boundary estimates that are uniform in , enabling existence of smooth solutions for each via degree theory. The core technical contribution is a detailed analysis of boundary derivatives, including double tangential, mixed tangential-normal, and double normal estimates, built from barrier arguments and careful use of the linearized operator. They also demonstrate the necessity of boundary regularity phenomena by presenting a solution that is not at the boundary, illustrating limitations when subsolutions are not sufficiently controlling boundary behavior. Overall, the results advance boundary regularity theory for Hessian-type conformal equations and yield robust existence results for prescribing Schouten-tensor eigenvalue functions on manifolds with boundary.

Abstract

In this paper, we consider the Dirichlet boundary value problem for fully nonlinear Yamabe equations on Riemannian manifolds with boundary. Assuming the existence of a subsolution, we derive \emph{a priori} boundary second derivative estimates and consequently obtain the existence of a smooth solution. Moreover, with respect to a family of equations interpolating the fully nonlinear Yamabe equation and the classical semi-linear Yamabe equation, our estimates remain uniform. Finally, an example of a solution which is smooth in the interior but not smooth at the boundary is also given.

Paper Structure

This paper contains 8 sections, 19 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminary results
  3. Some facts on $(f,\Gamma)$ and $(f_t, \Gamma_t)$
  4. Some estimates for the linearized operator
  5. A priori boundary second derivative estimates
  6. Double tangential and mixed normal-tangential derivative estimates
  7. Double normal derivative estimates
  8. Existence of non-smooth solutions

Key Result

Theorem 1.1

Let $(M^n, g)$ be a compact smooth Riemannian manifold of dimension $n \geq 3$ with non-empty smooth boundary $\partial M \neq \emptyset$. Let $\Gamma \subset \mathbb{R}^n$ be an open convex symmetric cone with vertex at the origin satisfying $\Gamma_n \subset \Gamma \subset \Gamma_1$ and $f\in C^{\ where $C$ only depends on $\underline{u}$, $(M, g)$, $(f, \Gamma)$, $\psi$ and $\varphi$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Proposition 1.2
  • Example 1.3
  • Definition 2.1: CNS
  • Lemma 2.2
  • proof
  • Definition 2.3: Trudinger
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 26 more