Article
Mathematics, Applied
Xavier Fernandez-Real, Xavier Ros-Oton
Summary: The study examines the regularity of the free boundary for the Signorini problem in Rn+1 and the structure of degenerate points. It is found that, despite the potential for large sets of non-regular points, the set of degenerate points is typically small. This work presents new insights into the characteristics of free boundaries with degenerate points.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2021)
Article
Mathematics, Applied
Matteo Focardi, Emanuele Spadaro
Summary: Building upon recent results, this paper provides a thorough description of the free boundary for solutions to the fractional obstacle problem in Rn+1 with obstacle function phi. If phi is analytic, results are similar to the zero obstacle case with local finiteness, rectifiability, and classification of frequencies and blowups. If phi is an element of Ck+1 (R-n), k >= 2, results hold for distinguished subsets of points with specific contact with the obstacle function.
ADVANCES IN CALCULUS OF VARIATIONS
(2022)
Article
Mathematics
Morteza Fotouhi, Henrik Shahgholian, Georg S. Weiss
Summary: This study examines solutions and the free boundary partial derivative of the sublinear system from a regularity point of view. Using the epiperimetric inequality approach, it demonstrates C-1, C-ss regularity for the free boundary at asymptotically flat points under certain conditions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Jonas Azzam
Summary: This study focuses on the properties of harmonic measure in semi-uniform domains, removing the John condition and providing alternative estimates for harmonic measure in nontangentially accessible domains. It also explores the relationship between semi-uniform domains with uniformly rectifiable boundary and the A(infinity)-property of harmonic measure, classifying geometrically all domains for which this property holds.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics, Applied
Robert Young
Summary: In this paper, we provide a necessary and sufficient condition for a graphical strip in the Heisenberg group to be area-minimizing in a specific region. We prove the necessity of our condition by introducing a family of deformations of graphical strips based on a varying vertical curve. Furthermore, we demonstrate the sufficiency of our condition by showing that strips satisfying the condition have monotone epigraphs. We also use this condition to prove that any area-minimizing ruled entire intrinsic graph in the Heisenberg group is a vertical plane and to find a boundary curve that can be filled with uncountably many area-minimizing surfaces.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Daniela De Silva, Seongmin Jeon, Henrik Shahgholian
Summary: This paper studies vector-valued almost minimizers of the energy functional. Using the epiperimetric inequality approach, the regularity for both almost minimizers and the set of regular free boundary points is proved.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Swati Yadav, Pratima Rai
Summary: In this article, a higher order parameter uniform numerical scheme is proposed for a class of two-dimensional parabolic singularly perturbed convection-diffusion problem. The scheme has second order convergence in both space and time directions.
Article
Mathematics
Luigi de Masi
Summary: This study establishes a partial rectifiability result for the free boundary of a k-varifold V, by refining a theorem of Grater and Jost and showing properties of the first variation and density of V. It is proven that under certain assumptions, the part of the first variation of V with positive and finite (k - 1)-density is (k - 1)-rectifiable.
INDIANA UNIVERSITY MATHEMATICS JOURNAL
(2021)
Article
Mathematics
Yuwei Hu, Jun Zheng
Summary: In this paper, we study the geometric properties of the set partial differential {x ∈ R(1)Ω:u(x)>0} for non-negative minimizers of the functional J(u) = p |backward difference u|p+q(u+)γ+hu dx over Ω K, where Ω ⊂Rn(n≥2) is an open bounded domain, p ∈ (1,+∞) and γ ∈ (0, 1] are constants, u+ is the positive part of u, and partial differential {x ∈ Ω:u(x)>0} is the so-called free boundary. The non-degeneracy of non-negative minimizers near the free boundary is established using the comparison principle of p-Laplacian equations, followed by the proof of the local porosity of the free boundary.
ELECTRONIC RESEARCH ARCHIVE
(2023)
Article
Mathematics, Applied
Seongmin Jeon, Arshak Petrosyan
Summary: This study focuses on Anzellotti-type almost minimizers for the thin obstacle problem, establishing their regularity on either side of the thin obstacle and analyzing the characteristics of the free boundary. The analysis is based on successful adaptations of energy methods and leads to several structural theorems.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Engineering, Civil
Xiao-Jian Xu
Summary: This study addresses the paradox of abnormal frequencies in the free vibration of nonlocal cantilever beams within the framework of nonlocal strain gradient theory. By updating the inconsistencies of reported boundary conditions and proposing a method for calibrating size-effect parameters, the numerical results demonstrate the model's capability in capturing the size-dependent mechanical properties of materials, whether exhibiting stiffness-hardening or stiffness-softening effects.
THIN-WALLED STRUCTURES
(2021)
Article
Mathematics, Applied
Antoine Henrot, Michiaki Onodera
Summary: The study of Bernoulli's free boundary problem involves elliptic and hyperbolic solutions, with the former being stable and tractable, and the latter being unstable and less understood. By introducing a new implicit function theorem and solving non-local geometric flows, the existence of hyperbolic and elliptic solutions is addressed by clarifying the spectral structure of the linearized operator through harmonic analysis.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2021)
Article
Mathematics
Steve Hofmann, Olli Tapiola
Summary: The study presents extensions of Varopoulos type for BMO functions on uniformly rectifiable sets. The results demonstrate the existence of a smooth function V meeting Carleson measure condition and converging in a non-tangential sense under certain conditions for BMO functions with compact support.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics, Applied
Jun Zheng, Leandro S. Tavares
Summary: This paper establishes regularities for the free boundary problem in Orlicz spaces, including continuity of minimizers in subcritical cases, growth rates near the free boundary for non-negative minimizers, and local Lipschitz continuity of non-negative minimizers under natural growth conditions. These results are new even for the free boundary problems of p-Laplacian.
ANNALI DI MATEMATICA PURA ED APPLICATA
(2022)
Article
Mathematics, Applied
Laura Prat, Carmelo Puliatti, Xavier Tolsa
Summary: This paper discusses the measurement problem related to elliptic PDEs, explores the relationship between measurement and the fundamental solution, and ultimately proves the rectifiability of some measurements under certain boundedness conditions.
Article
Mathematics
Nick Edelen, Max Engelstein
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2019)
Article
Mathematics, Applied
Matthew Badger, Max Engelstein, Tatiana Toro
Article
Mathematics
Quinn Maurmann, Max Engelstein, Anthony Marcuccio, Taryn Pritchard
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES
(2010)
Article
Computer Science, Theory & Methods
Max Engelstein
DISCRETE & COMPUTATIONAL GEOMETRY
(2010)
Article
Mathematics
Max Engelstein, Luca Spolaor, Bozhidar Velichkov
GEOMETRY & TOPOLOGY
(2019)
Article
Mathematics
Matthew Badger, Max Engelstein, Tatiana Toro
REVISTA MATEMATICA IBEROAMERICANA
(2020)
Article
Mathematics
Guy David, Max Engelstein, Svitlana Mayboroda
Summary: In this paper, the rectifiability of an Ahlfors regular set of arbitrary codimension is characterized by analyzing the behavior of a regularized distance function in the complement of the set. A certain version of the Riesz transform characterization of rectifiability for lower-dimensional sets is established. Additionally, a special situation is uncovered where the regularized distance itself is a solution to a degenerate elliptic operator in the complement of the set E. This leads to precise computation of the harmonic measure of sets associated with this degenerate operator, showing a contrast from the usual setting of codimension 1.
DUKE MATHEMATICAL JOURNAL
(2021)
Article
Mathematics
Guy David, Max Engelstein, Mariana Smit Vega Garcia, Tatiana Toro
Summary: The authors studied the regularity of almost minimizers for energy functionals with variable coefficients, proving Lipschitz regularity up to, and across, the free boundary, fully generalizing previous results.
MATHEMATISCHE ZEITSCHRIFT
(2021)
Article
Mathematics, Applied
Max Engelstein, Aapo Kauranen, Marti Prats, Georgios Sakellaris, Yannick Sire
Summary: The study focuses on the one-phase free boundary problem related to minimizing weighted Dirichlet energy and the area of the positivity set of a function. The research demonstrates full regularity of the free boundary in certain dimensions, almost everywhere regularity in arbitrary dimensions, and provides estimates on the singular set of the free boundary. The results are applicable across a range of relevant weights and present a novel approach distinct from standard methods.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2021)
Article
Mathematics
Max Engelstein, Luca Spolaor, Bozhidar Velichkov
DUKE MATHEMATICAL JOURNAL
(2020)
Article
Mathematics, Applied
Max Engelstein
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2017)
Article
Mathematics
Max Engelstein
ADVANCES IN MATHEMATICS
(2017)
Article
Mathematics
Max Engelstein
ANNALES SCIENTIFIQUES DE L ECOLE NORMALE SUPERIEURE
(2016)
Article
Mathematics
Matija Bucic, Richard Montgomery
Summary: This article improves upon previous research by showing that any n-vertex graph can be decomposed into O(n log* n) cycles and edges.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Laurentiu G. Maxim, Jose Israel Rodriguez, Botong Wang, Lei Wu
Summary: The paper investigates the relationship between linear optimization degree and geometric structure. By analyzing the geometric structure of the conormal variety of an affine variety, the Chern-Mather classes of the given variety can be completely determined. Additionally, the paper shows that these bidegrees coincide with the linear optimization degrees of generic affine sections.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
William Chan, Stephen Jackson, Nam Trang
Summary: Under the determinacy hypothesis, this paper completely characterizes the existence of nontrivial maximal almost disjoint families for specific cardinals kappa, considering the ideals of bounded subsets and subsets of cardinality less than kappa.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Zhenguo Liang, Zhiyan Zhao, Qi Zhou
Summary: This paper investigates the reducibility of the one-dimensional quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form. It provides a description and upper bound for the growth of the Sobolev norms of the solution, and demonstrates the optimality of the upper bound.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Zhao Yu Ma, Yair Shenfeld
Summary: This study provides a new approach to understanding the extremal cases of Stanley's inequalities by establishing a connection between the combinatorics of partially ordered sets and the geometry of convex polytopes.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Laurent Laurent, Rosa M. Miro-Roig
Summary: This paper discusses the problem of constructing matrices of linear forms of constant rank by focusing on vector bundles on projective spaces. It introduces important examples of classical Steiner bundles and Drezet bundles, and uses the classification of globally generated vector bundles to describe completely the indecomposable matrices of constant rank up to six.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Nicoletta Cantarini, Fabrizio Caselli, Victor Kac
Summary: In this paper, we construct a duality functor in the category of continuous representations to study the Lie superalgebra E(4, 4). By constructing a specific type of Lie conformal superalgebra, we obtain that E(4, 4) is its annihilation algebra. Furthermore, we also obtain an explicit realization of E(4, 4) on a supermanifold in the process of studying.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Rotem Assouline, Bo'az Klartag
Summary: This article studies the horocyclic Minkowski sum of two subsets in the hyperbolic plane and its properties. It proves an inequality relating the area of the subsets when they are Borel-measurable, and provides a connection to other inequalities.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Alessio Porretta
Summary: This article discusses Fokker-Planck equations driven by Levy processes in the entire Euclidean space, under the influence of confining drifts, similar to the classical Ornstein-Ulhenbeck model. A new PDE method is introduced to obtain exponential or sub-exponential decay rates of zero average solutions as time goes to infinity, under certain diffusivity conditions on the Levy process, including the fractional Laplace operator as a model example. The approach relies on long-time oscillation estimates of the adjoint problem and applies to both local and nonlocal diffusions, as well as strongly or weakly confining drifts.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Weichao Qian, Yong Li, Xue Yang
Summary: In this paper, we investigate the persistence of resonant invariant tori in Hamiltonian systems with high-order degenerate perturbation, and prove a quasiperiodic Poincare theorem under high degeneracy, answering a long-standing conjecture on the persistence of resonant invariant tori in general situations.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Julius Ross, David Witt Nystroem
Summary: This article extends Prekopa's Theorem and the Brunn-Minkowski Theorem from convexity to F-subharmonicity, and applies it to the interpolation problem of convex functions and convex sets, introducing a new notion of harmonic interpolation.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Airi Takeuchi, Lei Zhao
Summary: In this article, we explore the connection between several integrable mechanical billiards in the plane through conformal transformations. We discuss the equivalence of free billiards and central force problems, as well as the correspondence between integrable Hooke-Kepler billiards. We also investigate the integrability of Kepler billiards and Stark billiards, and the relationship between billiard systems and Euler's two-center problems.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Damiano Rossi
Summary: In this study, we prove new results in generalised Harish-Chandra theory by providing a description of the Brauer-Lusztig blocks using the p-adic cohomology of Deligne-Lusztig varieties. We then propose new conjectures for finite reductive groups by considering geometric analogues of the p-local structures. Our conjectures coincide with the counting conjectures for large primes, thanks to a connection established between p-structures and their geometric counterparts. Finally, we simplify our conjectures by reducing them to the verification of Clifford theoretic properties.
ADVANCES IN MATHEMATICS
(2024)
