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In this note, we prove two monotonicity formulas for solutions of $\Delta _H f = c$ and $\Delta _H f - \partial _t f = c$ in Carnot groups. Such formulas involve the right-invariant carré du champ of a function and they are false for the left-invariant one. The main results, theorems 1.1 and 1.2, display a resemblance with two deep monotonicity formulas respectively due to Alt–Caffarelli–Friedman for the standard Laplacian, and to Caffarelli for the heat equation. In connection with this aspect we ask the question whether an 'almost monotonicity' formula be possible. In the last section, we discuss the failure of the nondecreasing monotonicity of an Almgren type functional. [ABSTRACT FROM AUTHOR]

We give an overview of our recent new proof of the Riemannian Penrose inequality in the case of a single black hole. The proof is based on a new monotonicity formula, holding along the level sets of the p-capacitary potential of the connected boundary of an asymptotically flat 3-manifold, with nonnegative scalar curvature. [ABSTRACT FROM AUTHOR]

We establish Weiss' and Monneau's type quasi-monotonicity formulas for quadratic energies having matrix of coefficients in a Sobolev space with summability exponent larger than the space dimension and provide an application to the corresponding free boundary analysis for the related classical obstacle problems. [ABSTRACT FROM AUTHOR]

We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, min{(-Δ)su, u - φ} = 0 in Rn, for general obstacles φ. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in those of Garofalo-Petrosyan to all s ∈ (0, 1). [ABSTRACT FROM AUTHOR]

In this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. This includes gradient estimates, monotonicity formulas, and Liouville theorems under curvature and energy assumptions. [ABSTRACT FROM AUTHOR]

Motivated and inspired by the recent work of Colding (Acta Math 209(2):229-263, ) and Colding and Minicozzi (Calc Var Partial Differ Equ 49(3-4):1045-1059, ) we derive several families of monotonicity formulas for manifolds with nonnegative Bakry-Emery Ricci curvature, extending the formulas in Colding () and Colding and Minicozzi (). [ABSTRACT FROM AUTHOR]

We consider the parabolic obstacle type problem where Ω is an unknown open subset of . This problem has its origin in parabolic potential theory. When f is merely Hölder continuous, the usual method based on the use of a monotonicity formula does not apply. Nevertheless, we can, under a combination of energetic and geometric assumptions, prove the optimal regularity of the solution. [ABSTRACT FROM PUBLISHER]