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We are concerned about the coarse and precise aspects of a priori estimates for Green's function of a regular domain for the Laplacian-Betrami operator on any $3\le n$-dimensional complete non-compact boundary-free Riemannian manifold…

In a complete metric space equipped with a doubling measure supporting a $p$-Poincar\'e inequality, we prove sharp growth and integrability results for $p$-harmonic Green functions and their minimal $p$-weak upper gradients. We show that…

In this paper, a new proof of the Positive Mass Theorem is established through a newly discovered monotonicity formula, holding along the level sets of the Green's function of an asymptotically flat $3$-manifold. In the same context and for…

We study the Neumann Green's function for second order parabolic systems in divergence form with time-dependent measurable coefficients in a cylindrical domain $\mathcal{Q}=\Omega\times (-\infty,\infty)$, where $\Omega\subset \mathbb{R}^n$…

Consider an asymptotically flat Riemannian manifold $(M,g)$ of dimension $n \geq 3$ with nonempty compact boundary. We recall the harmonic conformal class $[g]_h$ of the metric, which consists of all conformal rescalings given by a harmonic…

We develop a comprehensive study on sharp potential type Riemannian Sobolev inequalities of order 2 by means of a local geometric Sobolev inequality of same kind and suitable De Giorgi-Nash-Moser estimates. In particular we discuss…

In this paper, we study Riemannian zeroth-order optimization in settings where the underlying Riemannian metric $g$ is geodesically incomplete, and the goal is to approximate stationary points with respect to this incomplete metric. To…

For a given finite subset $S$ of a compact Riemannian manifold $(M,g)$ whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on $M…

We establish that the $p$-conformal energy, $p\geq 1$, defined by the $L^p$-norms of the distortion of Sobolev mappings, is a proper functional on the Teichm\"uller space of Riemann surfaces of a fixed genus. This result is an application…

We prove optimal annealed decay estimates on the derivative and mixed second derivative of the elliptic Green functions on $\mathbb{R}^d$ for random stationary measurable coefficients that satisfy a certain logarithmic Sobolev inequality…

Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $C^{4,\gamma}$-smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green…

This article develops a unified and intrinsic framework for the theory of Sobolev spaces on vector bundles over Riemannian manifolds. The analytical core of our approach is an explicit higher-order geometric integration by parts formula,…

On a complete $p$-nonparabolic $3$-dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper $p$-Green function along its level sets for $1<p<3$. This can…

We introduce a natural definition of $L^p$-convergence of maps, $p \ge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a…

Let $\Sigma$ be a compact manifold without boundary whose first homology is nontrivial. Hodge decomposition of the incompressible Euler's equation in terms of 1-forms yields a coupled PDE-ODE system. The $L^2$-orthogonal components are a…

We propose a general method for finding sharp constants in the imbeddings of the Hilbert Sobolev spaces of order m defined on a n-dimensional Riemann manifold into the space of bounded continuous functions, where m>n/2. The method is based…

We extend monotonicity-based inversion methods to an inverse coefficient problem for the isotropic nonlocal elliptic equation \[ (-\nabla \cdot \sigma \nabla)^s u = 0 \quad \text{in } \Omega \subset \mathbb{R}^n, \] where $0 < s < 1$, $n…

We provide a full resolution of the Yamabe problem on closed 3-manifolds for Riemannian metrics of Sobolev class $W^{2,q}$ with $q > 3$. This requires developing an elliptic theory for the conformal Laplacian for rough metrics and…

We resolve the problem of optimal regularity and Uhlenbeck compactness for affine connections in General Relativity and Mathematical Physics. First, we prove that any affine connection $\Gamma$, with components $\Gamma \in L^{2p}$ and…

Given a complete doubling metric measure space $X$ that supports a $2$-Poincar\'e inequality, we approximate harmonic functions on a bounded domain $\Omega$ with a prescribed Newton-Sobolev boundary data. Our approach is based on the…