Existence and stability of solutions of the Dirichlet problem for the $p$-Poisson equation in metric measure spaces

Technical Summary: Existence and Stability of Solutions of the Dirichlet Problem for the p-Poisson Equation in Metric Measure Spaces

Problem Statement
This paper investigates the Dirichlet problem for the $p$-Poisson equation within the framework of metric measure spaces $(X, d, \mu)$. The equation is given by $-\Delta_p u = f$ in a domain $\Omega \subset X$, with boundary conditions $u = g$ on $\partial \Omega$. In the classical Euclidean setting, the $p$-Laplacian is defined as $\Delta_p u := \text{div}(|\nabla u|^{p-2}\nabla u)$. The authors aim to extend this problem to nonsmooth metric measure spaces equipped with a doubling measure and supporting a $(p, p)$-Poincaré inequality.

The primary challenge in this setting is the absence of a standard differential structure (i.e., a vector-valued gradient $\nabla u$). While the magnitude of the gradient can be generalized via upper gradients, the vector nature required for the standard weak formulation is more delicate. The paper addresses this by utilizing Cheeger's differential structure, which allows for a linear gradient operator in geodesic spaces, and alternatively, by defining solutions via energy minimization using minimal $p$-weak upper gradients, which does not require a geodesic structure.

Methodology
The authors employ a variational approach, defining solutions as minimizers of a specific energy functional rather than relying solely on weak formulations involving vector gradients.

1. Framework and Definitions:

   • The space is assumed to be a complete metric measure space with a doubling measure and a $(p, p)$-Poincaré inequality ($1 < p < \infty$).
   • Solutions are sought in the Newtonian space $N^{1,p}(\Omega)$, with boundary conditions handled via the subspace $N^{1,p}_0(\Omega)$.
   • Two definitions of solutions are proposed:
     • Cheeger's Setting (Geodesic Spaces): Using Cheeger's gradient $\nabla u$, a solution satisfies the weak formulation $\int_\Omega |\nabla u|_x^{p-2} \langle \nabla u, \nabla \phi \rangle_x d\mu = \int_\Omega f \phi d\mu$.
     • General Setting (Minimizer): A solution is defined as a function $u \in N^{1,p}(\Omega)$ minimizing the functional $J(u) = \int_\Omega g_u^p d\mu - p \int_\Omega f u d\mu$, where $g_u$ is the minimal $p$-weak upper gradient. This definition is shown to be equivalent to the Cheeger formulation when the space is geodesic.

2. Existence Proof:

   • The existence of a minimizer is established using the direct method in the calculus of variations.
   • The functional $J$ is shown to be bounded from below using Hölder's inequality and a Poincaré-type inequality (Lemma 2.8).
   • A minimizing sequence is shown to be bounded in the $N^{1,p}$ norm. By the reflexivity of $L^p$ spaces and Mazur's lemma, a sequence of convex combinations converges to a limit $u_0$.
   • Lower semicontinuity of the norm and the convergence of the linear term ensure that $u_0$ achieves the infimum, thus proving existence without requiring the space to be geodesic.

3. Stability and Uniqueness:

   • Stability is analyzed specifically in the Cheeger's setting (requiring the space to be geodesic).
   • The authors prove stability with respect to perturbations in the source term $f$ and the boundary data $g$.
   • Key to these proofs is Lemma 5.1, which establishes a uniform bound on the $p$-energy of the solution.
   • The stability estimates differ based on the value of $p$:
     • For $p \ge 2$, the stability is derived using the monotonicity properties of the $p$-Laplacian operator.
     • For $1 < p < 2$, the proofs utilize specific inequalities involving the sum of gradients and Hölder's inequality with exponents $2/p$ and $2/(2-p)$.
   • Uniqueness is derived as a direct corollary of the stability results.

Key Contributions and Results

• Existence: The paper proves the existence of solutions to the Dirichlet problem for the $p$-Poisson equation in general metric measure spaces (doubling + Poincaré) by defining solutions as minimizers of the energy functional $J$. This approach bypasses the need for a differentiable structure for the existence proof.
• Equivalence: It is demonstrated that in geodesic spaces, the variational definition (minimizer) is equivalent to the weak formulation using Cheeger's gradient.
• Stability Estimates: The authors provide explicit stability estimates for the solution in the $N^{1,p}$ norm with respect to changes in the source term $f$ and boundary data $g$. These estimates depend on the constant $p$ and the geometry of the space (via constants $c_1$ and $N$).
• Uniqueness: The paper establishes the uniqueness of the solution in the Cheeger's setting as a consequence of the stability analysis.

Significance and Claims
The paper claims to extend the theory of the $p$-Poisson equation from Euclidean and RCD spaces to the broader setting of metric measure spaces with doubling measures and Poincaré inequalities. By utilizing the minimizer approach, the authors provide a robust existence theory that does not rely on the specific differentiable structure of Cheeger, which is only guaranteed in geodesic spaces.

The significance lies in the rigorous treatment of the Dirichlet problem in nonsmooth settings, bridging the gap between the abstract potential theory of $p$-harmonic functions and the specific case of the $p$-Poisson equation with a source term. The authors explicitly state that their work focuses on the specific case of the source term being a function $f \in L^{p'}(\Omega)$, distinguishing it from more general measure-valued source terms studied in other literature. The stability results provide quantitative control over the solution's dependence on data, a property essential for numerical analysis and further theoretical developments in nonsmooth analysis.