Monotonicity Formula Research Articles

Quantitative uniqueness of solutions to a class of Schrödinger equations with inverse square potentials

This paper is devoted to proving the quantitative unique continuation property for solutions to a class of Schrödinger equations with inverse square potentials. The argument is to introduce a frequency function and show an almost monotonicity formula and three-ball inequalities by combining the Hardy's inequality.

The Yang–Mills–Higgs functional on complex line bundles: Asymptotics for critical points

Abstract We consider a gauge-invariant Ginzburg–Landau functional (also known as Abelian Yang–Mills–Higgs model), on Hermitian line bundles over closed Riemannian manifolds of dimension n ≥ 3 {n\geq 3} . Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the London limit. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, thanks to a suitable monotonicity formula, we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, converge to the weight measure of a stationary, rectifiable varifold of codimension 2.

Uniqueness of blowup at singular points for superconductivity problem

In this paper, we investigate the uniqueness of blowup at singular points of the free boundary in the superconductivity problem. We provide a sufficient condition and demonstrate that this condition can be verified in certain special cases. The proof of the main results in this paper is primarily based on Weiss-type and Monneau-type monotonicity formulas, and is inspired by the recent paper [Chen et al. arXiv: 2204.11426v2 (2022)].

Some counterexamples to Alt–Caffarelli–Friedman monotonicity formulas in Carnot groups

Some counterexamples to Alt–Caffarelli–Friedman monotonicity formulas in Carnot groups

On the fractional powers of a Schrödinger operator with a Hardy-type potential

AbstractStrong unique continuation properties and a classification of the asymptotic profiles are established for the fractional powers of a Schrödinger operator with a Hardy-type potential, by means of an Almgren monotonicity formula combined with a blow-up analysis.

Asymptotic upper bound life span estimates for L1-solutions of the 2-D Patlak–Keller–Segel equation

We show asymptotic upper bound life span estimates for L1-solutions of the 2-D Patlak–Keller–Segel equation with large initial data. The proof is based on a modification of the monotonicity formula introduced by Wei in [17] recently.

Liouville theorems and optimal regularity in elliptic equations

AbstractThe objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic partial differential equations with measurable coefficients and the Liouville property at infinity. Initially, we address the two‐dimensional case by proving an Alt–Caffarelli–Friedman‐type monotonicity formula, enabling the proof of optimal regularity and the Liouville property for multiphase problems. In higher dimensions, we delve into the role of monotonicity formulas in characterizing optimal regularity. By employing a hole‐filling technique, we present a distinct “almost‐monotonicity” formula that implies Hölder regularity of solutions. Finally, we explore the interplay between the least growth at infinity and the exponent of regularity by combining blow‐up and ‐convergence arguments.

Monotonicity of thep-Green Functions

AbstractOn 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 be viewed as a generalization of the recent result by Munteanu-Wang [ 43] in the case of $p=2$. No smoothness assumption is made on the $p$-Green function when $1<p\leq 2$. Several rigidity results are also proven.

The first nonzero eigenvalue of the (p,q)-Laplace system along the inverse mean curvature flow with forced term

In this paper, we study the first nonzero eigenvalue of the (p,q)-Laplace system along the inverse mean curvature flow with forced term in Euclidean space. Some monotonicity formulas for the first nonzero eigenvalue are obtained.

Almost minimizers for the thin obstacle problem with variable coefficients

We study almost minimizers for the thin obstacle problem with variable Hölder continuous coefficients and zero thin obstacle, and establish their C^{1,\beta} regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.

A Green’s Function Proof of the Positive Mass Theorem

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-manifolds. In the same context and for 1<p<3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<3$$\\end{document}, a Geroch-type calculation is performed along the level sets of p-harmonic functions, leading to a new proof of the Riemannian Penrose Inequality under favourable assumptions. A new characterisation of scalar curvature lower bounds in terms of the monotonicity formulas is also given.

Monotonicity formulas and $\text{(S}_+)$-property: Old and new

Monotonicity formulas and $\text{(S}_+)$-property: Old and new

A new proof of the Riemannian Penrose inequality

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.

A Short Proof of Allard’s and Brakke’s Regularity Theorems

Abstract We give new short proofs of Allard’s regularity theorem for varifolds with bounded first variation and Brakke’s regularity theorem for integral Brakke flows with bounded forcing. They are based on a decay of flatness, following from weighted versions of the respective monotonicity formulas, together with a characterization of non-homogeneous blow-ups using the viscosity approach introduced by Savin.

A counterexample to the monotone increasing behavior of an Alt–Caffarelli–Friedman formula in the Heisenberg group

In this paper, we provide a counterexample about the existence of an increasing monotonicity behavior of a function introduced by Ferrari and Forcillo (2020), companion of the celebrated Alt–Caffarelli–Friedman monotonicity formula, in the noncommutative framework.

Alt–Caffarelli–Friedman monotonicity formula and mean value properties in Carnot groups with applications

Alt–Caffarelli–Friedman monotonicity formula and mean value properties in Carnot groups with applications

Almost monotonicity formula for H‐minimal Legendrian surfaces in the Heisenberg group

AbstractWe prove an almost monotonicity formula for H‐minimal Legendrian Surfaces (also called contact stationary Legendrian immersions or Hamiltonian stationary immersions) in the Heisenberg Group . From this formula we deduce a Bernstein‐Liouville type theorem for H‐minimal Legendrian Surfaces. We also present some possible range of applications of this formula.

A vectorial problem with thin free boundary

We consider the vectorial analogue of the thin free boundary problem introduced by Caffarelli et al. (J Eur math Soc 12:1151-1179, 2010) as a realization of a nonlocal version of the classical Bernoulli problem. We study optimal regularity, nondegeneracy, and density properties of local minimizers. Via a blow-up analysis based on a Weiss type monotonicity formula, we show that the free boundary is the union of a “regular” and a “singular” part. Finally we use a viscosity approach to prove C^{1,alpha } regularity of the regular part of the free boundary.

Optimal uniform bounds for competing variational elliptic systems with variable coefficients

Let Ω⊂RN be an open set. In this work we consider solutions of the following gradient elliptic system −div(A(x)∇ui,β)=fi(x,ui,β)+a(x)β|ui,β|γ−1ui,β∑j=1j≠il|uj,β|γ+1,for i=1,…,l. We work in the competitive case, namely β<0. Under suitable assumptions on A, a, fi and on the exponent γ, we prove that uniform L∞–bounds on families of positive solutions {uβ}β<0={(u1,β,…,ul,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)=Id (i.e. the operator is the Laplacian) was treated.

Relative entropy of hypersurfaces in hyperbolic space

Abstract We study a notion of relative entropy for certain hypersurfaces in hyperbolic space. We relate this quantity to the renormalized area introduced by Graham–Witten. We also obtain a monotonicity formula for relative entropy applied to mean curvature flows in hyperbolic space.