Comparison Results for a class of Neumann Problems of the $p$-Laplace Equation on Riemannian Manifolds

Technical Summary: Comparison Results for a Class of Neumann Problems of the p-Laplace Equation on Riemannian Manifolds

Problem Statement
The paper investigates nonhomogeneous Neumann boundary value problems for the $p$-Laplace equation on Riemannian manifolds with nonnegative Ricci curvature. Specifically, the authors consider a complete noncompact manifold $(M, g)$ with nonnegative Ricci curvature and positive asymptotic volume ratio, or a compact manifold satisfying $\text{Ric} \geq (n-1)$. The primary equation under study is:
$$\begin{cases} -\text{div}(|\nabla u|^{p-2}\nabla u) = f, & \text{in } \Omega, \\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu} + \beta^{p-1} = 0, & \text{on } \partial\Omega, \end{cases}$$
where $\Omega \subset M$ is a bounded domain with Lipschitz boundary. A central challenge addressed is the intrinsic ambiguity of Neumann problems, where solutions are determined only up to additive constants. Unlike Dirichlet or Robin problems, establishing meaningful comparison principles requires specific normalization conditions to fix this ambiguity.

Methodology
The authors employ spherical symmetrization (rearrangement) techniques adapted to Riemannian manifolds. The methodology proceeds as follows:

1. Symmetrization Framework: The solution $u$ on the manifold $\Omega$ is compared with a symmetrized solution $v$ defined on a model space $\Omega^\sharp$.

   • For the noncompact case ($\kappa=0$), the model space is the Euclidean space $\mathbb{R}^n$, and $\Omega^\sharp$ is a Euclidean ball.
   • For the compact case ($\kappa=1$), the model space is the standard sphere $S^n$, and $\Omega^\sharp$ is a geodesic spherical cap.
   • The symmetrization is normalized by a factor $\theta_\kappa$ (related to the asymptotic volume ratio or the total volume of the sphere) to ensure compatibility with the isoperimetric inequalities on the manifold.

2. Normalization Condition: To handle the non-uniqueness of Neumann solutions, the authors impose a specific compatibility condition linking the boundary $L^q$-moments of the original solution $u$ and the symmetrized solution $v$:
   $$\frac{\beta^\sharp}{\beta} \int_{\partial \Omega} u^q d\sigma = \theta_\kappa \int_{\partial \Omega^\sharp} v^q d\sigma$$
   where $q \geq 1$ is an arbitrary parameter. The authors note that this flexibility in choosing $q$ is a distinctive feature of the Neumann setting, contrasting with the stricter constraints found in Robin problems.

3. Analytical Tools:

   • Isoperimetric Inequalities: The proof relies on the isoperimetric inequality on manifolds with nonnegative Ricci curvature (Brendle's inequality for noncompact and Lévy-Gromov for compact), which relates the boundary measure of $\Omega$ to that of the symmetrized ball $\Omega^\sharp$.
   • Differential Inequalities: The authors derive differential inequalities involving the distribution functions $\mu(t) = |\{x \in \Omega : u(x) > t\}|$ and $\phi(t) = |\{x \in \Omega^\sharp : v(x) > t\}|$.
   • Gronwall's Inequality: A crucial step involves applying a generalized Gronwall inequality to the differential inequalities derived from the weak formulation of the $p$-Laplace equation, allowing for the comparison of integral norms.

Key Contributions and Results
The paper establishes Talenti-type comparison results in Lorentz spaces $L^{\lambda, q}$. The main theorems provide pointwise and norm-based comparisons between the solution $u$ on the manifold and the solution $v$ on the model space.

• General Source Term ($f$): For a general non-negative source $f$, the authors prove that under the normalization condition (1.7), the Lorentz norms of $u$ are bounded by those of $v$ (scaled by $\theta_\kappa$). Specifically:
  $$\|u\|_{L^{\lambda, 1}(\Omega)} \leq \theta_\kappa^{1/\lambda} \|v\|_{L^{\lambda, 1}(\Omega^\sharp)}$$
  and similar estimates hold for $L^{\lambda, q}$ norms, provided the parameters $\lambda$ and $p$ satisfy specific constraints depending on the dimension $n$ and the curvature case ($\kappa=0$ or $1$).

• Constant Source Term ($f \equiv 1$): In the special case where $f \equiv 1$, the authors derive a stronger pointwise comparison:
  $$u^\sharp(x) \leq \theta_\kappa v(x), \quad \forall x \in \Omega^\sharp$$
  This holds when $1 \leq p \leq \frac{n}{n-1}$. For $p > \frac{n}{n-1}$, the pointwise result is replaced by sharp Lorentz norm estimates.

• Relaxation of Constraints: A significant finding is that the Neumann setting admits weaker constraints on the normalization parameters compared to the Robin case. The authors demonstrate that the comparison argument is not restricted to specific choices of the exponent $q$, highlighting a characteristic flexibility of Neumann problems due to the invariance under additive constants.

Significance and Claims
The paper claims to extend the classical Talenti comparison theory, originally developed for Dirichlet problems in Euclidean space, to Neumann problems on Riemannian manifolds with nonnegative Ricci curvature.

• Novelty in Normalization: The authors emphasize that their approach resolves the ambiguity of Neumann solutions through a flexible normalization condition involving boundary moments. They argue that this flexibility is a unique feature of the Neumann setting, distinguishing it from the Robin case where boundary conditions impose stricter constraints.
• Optimality of Assumptions: The paper suggests that the derived constraints on the parameters (such as the range of $\lambda$) are likely optimal for the current method, though it notes that identifying the absolute minimal set of assumptions for such comparison results remains an open problem.
• Context: The work builds upon previous extensions of Talenti's ideas to manifolds (Dirichlet and Robin cases) and generalizes them to the $p$-Laplacian with Neumann boundary conditions, providing a unified framework for both compact and noncompact manifolds with nonnegative Ricci curvature.

In summary, the paper provides a rigorous framework for comparing solutions of $p$-Laplace Neumann problems on curved spaces to their symmetric counterparts in model spaces, yielding sharp functional inequalities in Lorentz spaces under appropriate geometric and normalization assumptions.