Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities

1. Problem Statement and Context

The paper addresses two interconnected problems regarding the Gaussian $p$-Laplacian in $\mathbb{R}^n$ equipped with the Gaussian probability measure $\gamma$. The operator is defined as:
$$-\Delta_{p,\gamma} u := -\text{div}(|\nabla u|^{p-2}\nabla u) + \langle x, \nabla u \rangle |\nabla u|^{p-2}$$
where $p > 1$.

The authors investigate the first (nontrivial) Dirichlet eigenvalue $\lambda_{p,\gamma}(\Omega)$ associated with the eigenvalue problem:
$$\begin{cases} -\Delta_{p,\gamma} u = \lambda_{p,\gamma} |u|^{p-2}u & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega. \end{cases}$$
The eigenvalue is characterized variationally via the Rayleigh quotient:
$$\lambda_{p,\gamma}(\Omega) = \inf_{u \in W^{1,p}_0(\Omega, \gamma), u \not\equiv 0} \frac{\int_\Omega |\nabla u|^p d\gamma}{\int_\Omega |u|^p d\gamma}.$$

The primary goals are:

1. To establish a Brunn-Minkowski type inequality for $\lambda_{p,\gamma}$ in the class of bounded Lipschitz domains.
2. To prove the log-concavity of the corresponding positive eigenfunctions when the domain $\Omega$ is convex.

2. Methodology

The authors employ a combination of variational methods, viscosity solution theory, and geometric analysis. The core methodology involves the following steps:

A. Viscosity vs. Weak Solutions Equivalence

A crucial technical hurdle is bridging the gap between viscosity solutions and weak (distributional) solutions for the Gaussian $p$-Laplacian.

• Challenge: Standard regularity theory for the $p$-Laplacian is well-established, but the specific Gaussian weight and the $p$-Laplacian structure (degenerate for $p>2$, singular for $1<p<2$) require careful handling.
• Approach: The authors prove that any bounded, non-negative, upper semi-continuous viscosity subsolution is also a local weak subsolution (Theorem 3.6).
• Technique: They utilize sup-convolutions ($u_\varepsilon$) to approximate the solution. They show that the sup-convolution of a viscosity subsolution satisfies a perturbed equation almost everywhere. By establishing Caccioppoli-type estimates for the gradients of these approximations and using Fatou's Lemma, they pass to the limit to recover the weak formulation.

B. The Prekopa-Leindler / Brunn-Minkowski Argument

To prove the Brunn-Minkowski inequality and log-concavity, the authors construct a specific test function based on the geometric mean of solutions in different domains.

• Construction: For domains $\Omega_0, \Omega_1$ and a parameter $t \in [0,1]$, let $\Omega_t = (1-t)\Omega_0 + t\Omega_1$. They define a function $u_t$ on $\Omega_t$ via the sup-convolution of the eigenfunctions $u_0$ and $u_1$:
  $$u_t(x) = \sup \{ u_0(x_0)^{1-t} u_1(x_1)^t : x = (1-t)x_0 + tx_1 \}.$$
• Viscosity Verification: They demonstrate that $u_t$ is a viscosity subsolution to the eigenvalue problem with eigenvalue $\lambda_t = (1-t)\lambda_0 + t\lambda_1$. This involves a detailed calculation of the Hessian and gradient of the geometric mean, utilizing the Lagrange multiplier conditions at the maximum points.
• Comparison Principle: By proving $u_t$ is a viscosity subsolution, they invoke the equivalence theorem (viscosity $\implies$ weak) to treat $u_t$ as a weak subsolution. Comparing the Rayleigh quotient of $u_t$ with the definition of $\lambda_{p,\gamma}(\Omega_t)$ yields the desired inequality.

3. Key Contributions and Results

Result 1: Brunn-Minkowski Inequality for $\lambda_{p,\gamma}$

Theorem 1.1: For $p > 1$ and $t \in [0,1]$, if $\Omega_0, \Omega_1$ are open bounded Lipschitz domains, then:
$$\lambda_{p,\gamma}(\Omega_t) \leq (1-t)\lambda_{p,\gamma}(\Omega_0) + t\lambda_{p,\gamma}(\Omega_1).$$

• Significance: This extends previous results known for the Laplacian ($p=2$) and the Gaussian Laplacian to the general $p$-Laplacian case in Gauss space. It confirms that the first eigenvalue is concave with respect to Minkowski addition of domains.

Result 2: Urysohn-type Inequality (Corollary 1.4)

Using the Brunn-Minkowski inequality, the authors derive a lower bound for the eigenvalue in terms of the mean width $w(\Omega)$.

• Statement: $\lambda_{p,\gamma}(\Omega) \geq \lambda_{p,\gamma}(\Omega^\sharp)$, where $\Omega^\sharp$ is a centered ball with the same mean width as $\Omega$.
• Insight: Unlike the classical Faber-Krahn inequality in Gauss space (where half-spaces are minimizers), this result relates to the mean width, a metric quantity independent of the measure's tail behavior.

Result 3: Log-Concavity of Eigenfunctions

Theorem 1.5: If $\Omega$ is an open, bounded, and convex domain, and $u$ is a positive solution to the eigenvalue problem, then $u$ is log-concave (i.e., $\ln u$ is concave).

• Methodology: The proof sets $\Omega_0 = \Omega_1 = \Omega$ and $u_0 = u_1 = u$. The constructed function $u_t$ becomes the convex envelope of $\ln u$. The Brunn-Minkowski argument shows that $u_t$ is an eigenfunction with the same eigenvalue. By uniqueness of the positive eigenfunction (up to scaling), $u_t$ must coincide with $u$, implying $u$ is log-concave.
• Significance: This generalizes the classic Brascamp-Lieb result (for the Laplacian) and recent results for the Ornstein-Uhlenbeck operator to the non-linear Gaussian $p$-Laplacian.

Result 4: Equivalence of Solution Concepts

Theorem 3.6: A bounded, non-negative, upper semi-continuous viscosity subsolution of the Gaussian $p$-Laplacian equation is also a local weak subsolution.

• Significance: This is a standalone technical contribution that facilitates the use of viscosity solution techniques (which are robust for handling non-smooth domains and geometric constructions) in conjunction with variational methods (which are required for eigenvalue comparisons).

4. Significance and Impact

• Unification: The paper successfully unifies the geometric analysis of the Gaussian measure with non-linear elliptic operators ($p$-Laplacian).
• Regularity Handling: The authors provide a rigorous framework for handling the singular/degenerate nature of the $p$-Laplacian in the context of viscosity solutions, allowing for weaker regularity assumptions on the domain boundaries (Lipschitz vs. $C^{1,\alpha}$).
• Geometric Analysis: The results deepen the understanding of how the geometry of the domain (convexity, Minkowski addition) influences the spectral properties of operators in non-Euclidean (weighted) spaces.
• Future Directions: The paper suggests that these methods could be extended to other measures satisfying specific concavity conditions, opening avenues for research in weighted geometric inequalities.

In summary, this work provides a comprehensive extension of classical Brunn-Minkowski and log-concavity results to the non-linear Gaussian setting, relying on a sophisticated interplay between viscosity theory, sup-convolutions, and variational principles.