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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]

In this paper we are concerned with a two-penalty boundary obstacle problem of interest in thermics, fluid dynamics and electricity. Specifically, we prove existence, uniqueness and optimal regularity of the solutions, and we establish structural properties of the free boundary. A central role is played by the monotonicity of ad hoc Almgren- and Monneau-type functionals.

In this paper we revisit the proof of the Alt-Caffarelli-Friedman monotonicity formula. Then, in the framework of the Heisenberg group, we discuss the existence of an analogous monotonicity formula introducing a necessary condition for its existence, recently proved in [18].

This note focuses on the geometric-theoretic analysis of the nodal set of solutions to specific degenerate or singular equations. As they belong to the Muckenhoupt class A_2, these operators appear in the seminal works of Fabes, Kenig, Jerison and Serapioni. In particular, they have recently attracted a lot of attention in the last decade due to their link to the local realization of the fractional Laplacian. The goal is to get a glimpse of the complete theory of the nodal set of solutions of such equations in the spirit of the seminal works of Hardt, Simon, Han and Lin.

This work is devoted to the geometric-theoretic analysis of the nodal set of solutions to degenerate or singular equations involving a class of operators including L a = div (| y | a ∇) , with a ∈ (− 1 , 1) and their perturbations. As they belong to the Muckenhoupt class A 2 , these operators appear in the seminal works of Fabes, Kenig, Jerison and Serapioni [1–3] and have recently attracted a lot of attention in the last decade due to their link to the localization of the fractional Laplacian via the extension in one more dimension [4]. Our goal in the present paper is to develop a complete theory of the stratification properties for the nodal set of solutions of such equations in the spirit of the seminal works of Hardt, Simon, Han and Lin [5–7]. [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]

We provide monotonicity formulas for solutions to the p -Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely on the conformal splitting technique introduced by the second author in collaboration with V. Agostiniani. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R N be an open set. In this work we consider solutions of the following gradient elliptic system − div (A (x) ∇ u i , β) = f i (x , u i , β) + a (x) β | u i , β | γ − 1 u i , β ∑ j = 1 j ≠ i l | u j , β | γ + 1 , for i = 1 , ... , l. We work in the competitive case, namely β < 0. Under suitable assumptions on A , a , f i and on the exponent γ , we prove that uniform L ∞ –bounds on families of positive solutions { u β } β < 0 = { (u 1 , β , ... , u l , β) } β < 0 imply uniform Lipschitz bounds (which are optimal). One of the main points in the proof are suitable generalizations of Almgren's and Alt–Caffarelli–Friedman's monotonicity formulas for solutions of such systems. Our work generalizes previous results, where the case A (x) = I d (i.e. the operator is the Laplacian) was treated. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for every nonnegative initial data in L 1 ( R 2 ) , the Patlak–Keller–Segel equation is globally well-posed if and only if the total mass M ≤ 8 π . Our proof is based on some monotonicity formulas of nonnegative mild solutions. [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]

We establish the C 1 + γ -Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss in [15] for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research [12] , where two of us established the optimal interior regularity of solutions. [ABSTRACT FROM AUTHOR]

Weiss' and Monneau's type quasi-monotonicity formulas are established for quadratic energies having matrix of coefficients which are Dini, double-Dini continuous, respectively. Free boundary regularity for the corresponding classical obstacle problems under Hölder continuity assumptions is then deduced. [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]

In this paper we are concerned with a two phase boundary obstacle-type problem for the bi-Laplace operator in the upper unit ball. The problem arises in connection with unilateral phenomena for flat elastic plates. It can also be seen as an extension problem to an obstacle problem for the fractional Laplacian (− Δ) 3 / 2 , as first observed in Yang (2013). We establish the well-posedness and the optimal regularity of the solution, and we study the structure of the free boundary. Our proofs are based on monotonicity formulas of Almgren- and Monneau-type. [ABSTRACT FROM AUTHOR]

We study the regularity of segregated profiles arising from competition–diffusion models, where the diffusion process is of nonlocal type and is driven by the fractional Laplacian of power s ∈ (0 , 1). Among others, our results apply to the regularity of the densities of an optimal partition problem involving the eigenvalues of the fractional Laplacian. More precisely, we show C 0 , α ∗ regularity of the density, where the exponent α ∗ is explicit and is given by α ∗ = s for s ∈ (0 , 1 ∕ 2 ] 2 s − 1 for s ∈ (1 ∕ 2 , 1). Under some additional assumptions, we then show that solutions are C 0 , s. These results are optimal in the class of Hölder continuous functions. Thus, we find a complete correspondence with known results in case of the standard Laplacian. [ABSTRACT FROM AUTHOR]