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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., To appear in Mathematics in Engineering, special issue in honor of Sandro Salsa

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.

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 \cite{FeFo}., arXiv admin note: text overlap with arXiv:2001.04393

We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, $\min \left\{(-\Delta)^s u, u-\varphi\right\}=0$ in $\mathbb{R}^n$, for general obstacles $\varphi$. 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 $s \in(0,1)$.

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., Any suggestions and comments are welcome!

We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a Holder continuous linear term. With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli (J Fourier Anal Appl 4(4–5), 383–402, 1998), Monneau (J Geom Anal 13(2), 359–389, 2003), and Weiss (Invent Math 138(1), 23–50, 1999).

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] , [2] , [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] , [6] , [7] .

We establish the $C^{1+\gamma}$-H\"older 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 for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research, where two of us established the optimal interior regularity of solutions., Comment: 40 pages

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.

Using the electrostatic potential $u$ due to a uniformly charged body $\Omega\subset\mathbb R^n$, $n\geq 3$, we introduce a family of monotone quantities associated with the level set flow of $u$. The derived monotonicity formulas are exploited to deduce a new quantitative version of the classical Willmore inequality., Comment: This is a new version of the paper, with new proofs and enhanced results. For a detailed description of these changes we refer the reader to the Added note after the Introduction

We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely ∫ Ω | ∇ u ( x ) | 2 d x + Per σ ( { u > 0 } , Ω ) , with σ ∈ ( 0 , 1 ) . We obtain regularity results for the minimizers and for their free boundaries ∂ { u > 0 } using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler–Lagrange equations and extension problems.