N-BODY HAMILTONIANS WITH HARD-CORE INTERACTIONS

1994 ◽  
Vol 06 (04) ◽  
pp. 515-596 ◽  
Author(s):  
ANNE BOUTET de MONVEL-BERTHIER ◽  
VLADIMIR GEORGESCU ◽  
AVY SOFFER
Keyword(s):  
Maximal Regularity ◽  
Hard Core ◽  
Regularity Result ◽  
Weak Sense ◽  
Compact Sets ◽  
Dirichlet Laplacian ◽  
Irregular Boundary ◽  
Open Set ◽  
Dilation Group ◽  
Threshold Energies

We study N-body hamiltonians with short and long range potentials which are infinite on compact sets of non-zero measure. We show that the generator of the dilation group is locally conjugated to them away from the threshold energies. The notion of conjugacy has to be interpreted in a very weak sense, but this is enough to deduce an optimal form of the limiting absorption principle, and so absence of singular continuous spectrum and local decay. One of the main technical steps of our approach requires a maximal regularity result for the Dirichlet Laplacian in an open set with irregular boundary. We prove it for a large class of non-smooth domains.

AnL2-maximal regularity result for the evolutionary Stokes–Fourier system

2011 ◽  
Vol 90 (1) ◽  
pp. 31-45 ◽  
Author(s):  
Miroslav Bulíček ◽  
Petr Kaplický ◽  
Josef Málek
Keyword(s):  
Maximal Regularity ◽  
Regularity Result

Descriptive Borel sets

1965 ◽  
Vol 286 (1407) ◽  
pp. 455-478 ◽  
Keyword(s):  
Metric Space ◽  
General Class ◽  
Hausdorff Space ◽  
Countable Union ◽  
Compact Sets ◽  
Borel Sets ◽  
Closed Sets ◽  
Open Set ◽  
Descriptive Theory ◽  
Satisfactory Theory

The descriptive theory of Borel sets is developed for a fairly general class of spaces. For a satisfactory theory it seems to be necessary to work with a Hausdorff space subject to the condition that each open set can be expressed as a countable union of closed sets. Under this condition it is shown that the descriptive Borel sets form a Borel ring of analytic absolutely Borel sets containing the compact sets. It is shown that a set in a metric space is descriptive Borel if and only if it is Lindelöf and absolutely Borel.

scholarly journals Partial regularity for mass-minimizing currents in Hilbert spaces

2018 ◽  
Vol 2018 (734) ◽  
pp. 99-144 ◽  
Author(s):  
Luigi Ambrosio ◽  
Camillo De Lellis ◽  
Thomas Schmidt
Keyword(s):  
Partial Regularity ◽  
Metric Spaces ◽  
Harmonic Approximation ◽  
Regularity Result ◽  
Acta Math ◽  
Target Space ◽  
Existence Theory ◽  
Infinite Dimensional ◽  
Open Set ◽  
Recent Developments

AbstractRecently, the theory of currents and the existence theory for Plateau’s problem have been extended to the case of finite-dimensional currents in infinite-dimensional manifolds or even metric spaces; see [Acta Math. 185 (2000), 1–80] (and also [Proc. Lond. Math. Soc. (3) 106 (2013), 1121–1142], [Adv. Calc. Var. 7 (2014), 227–240] for the most recent developments). In this paper, in the case when the ambient space is Hilbert, we provide the first partial regularity result, in a dense open set of the support, forn-dimensional integral currents which locally minimize the mass. Our proof follows with minor variants [Indiana Univ. Math. J. 31 (1982), 415–434], implementing Lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimensionnand not on codimension or dimension of the target space.

scholarly journals Maximal Regularity for Non-autonomous Evolutionary Equations

2021 ◽  
Vol 93 (3) ◽  
Author(s):  
Sascha Trostorff ◽  
Marcus Waurick
Keyword(s):  
Differential Equations ◽  
Maximal Regularity ◽  
Normal Operator ◽  
Time Derivative ◽  
Parabolic Type ◽  
Regularity Result ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Partial Differential ◽  
Evolutionary Equations

AbstractWe discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible, normal operator admitting a functional calculus with the Fourier–Laplace transformation providing the spectral representation. Here, the main result is then a regularity result for well-posed evolutionary equations solely based on an assumed parabolic-type structure of the equation and estimates of the commutator of the coefficients with the square root of the time derivative. We thus simultaneously generalise available results in the literature for non-smooth domains. Examples for equations in divergence form, integro-differential equations, perturbations with non-autonomous and rough coefficients as well as non-autonomous equations of eddy current type are considered.

scholarly journals A Note on the Three Dimensional Dirac Operator with Zigzag Type Boundary Conditions

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Markus Holzmann
Keyword(s):  
Dirac Operator ◽  
Boundary Conditions ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Dirichlet Laplacian ◽  
Open Set ◽  
Type Boundary ◽  
Infinite Multiplicity ◽  
Zigzag Type

AbstractIn this note the three dimensional Dirac operator $$A_m$$ A m with boundary conditions, which are the analogue of the two dimensional zigzag boundary conditions, is investigated. It is shown that $$A_m$$ A m is self-adjoint in $$L^2(\Omega ;{\mathbb {C}}^4)$$ L 2 ( Ω ; C 4 ) for any open set $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 and its spectrum is described explicitly in terms of the spectrum of the Dirichlet Laplacian in $$\Omega $$ Ω . In particular, whenever the spectrum of the Dirichlet Laplacian is purely discrete, then also the spectrum of $$A_m$$ A m consists of discrete eigenvalues that accumulate at $$\pm \infty $$ ± ∞ and one additional eigenvalue of infinite multiplicity.

scholarly journals Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

2020 ◽  
Vol 2020 (766) ◽  
pp. 195-228 ◽  
Author(s):  
Rupert L. Frank ◽  
Simon Larson
Keyword(s):  
Asymptotic Formula ◽  
Convex Domain ◽  
Lipschitz Domain ◽  
Spectral Asymptotics ◽  
Lipschitz Boundary ◽  
Dirichlet Laplacian ◽  
Open Set ◽  
Universal Bound

AbstractWe prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics.

scholarly journals A regularity result for minimal configurations of a free interface problem

2021 ◽  
Author(s):  
Lorenzo Lamberti
Keyword(s):  
Regularity Result ◽  
Variational Problems ◽  
Energy Functional ◽  
Interface Problem ◽  
Hölder Continuous ◽  
Free Interface ◽  
Open Set ◽  
Minimal Configuration ◽  
Open Region ◽  
Bulk Energy

AbstractWe prove a regularity result for minimal configurations of variational problems involving both bulk and surface energies in some bounded open region $$\varOmega \subseteq {\mathbb {R}}^n$$ Ω ⊆ R n . We will deal with the energy functional $${\mathscr {F}}(v,E):=\int _\varOmega [F(\nabla v)+1_E G(\nabla v)+f_E(x,v)]\,dx+P(E,\varOmega )$$ F ( v , E ) : = ∫ Ω [ F ( ∇ v ) + 1 E G ( ∇ v ) + f E ( x , v ) ] d x + P ( E , Ω ) . The bulk energy depends on a function v and its gradient $$\nabla v$$ ∇ v . It consists in two strongly quasi-convex functions F and G, which have polinomial p-growth and are linked with their p-recession functions by a proximity condition, and a function $$f_E$$ f E , whose absolute valuesatisfies a q-growth condition from above. The surface penalization term is proportional to the perimeter of a subset E in $$\varOmega $$ Ω . The term $$f_E$$ f E is allowed to be negative, but an additional condition on the growth from below is needed to prove the existence of a minimal configuration of the problem associated with $${\mathscr {F}}$$ F . The same condition turns out to be crucial in the proof of the regularity result as well. If (u, A) is a minimal configuration, we prove that u is locally Hölder continuous and A is equivalent to an open set $${\tilde{A}}$$ A ~ . We finally get $$P(A,\varOmega )={\mathscr {H}}^{n-1}(\partial {\tilde{A}}\cap \varOmega $$ P ( A , Ω ) = H n - 1 ( ∂ A ~ ∩ Ω ).

scholarly journals Maximal Regularity for Flexible Structural Systems in Lebesgue Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Claudio Fernández ◽  
Carlos Lizama ◽  
Verónica Poblete
Keyword(s):  
Maximal Regularity ◽  
Flexible Structures ◽  
Fourier Multipliers ◽  
Smooth Domain ◽  
Structural Systems ◽  
Material Damping ◽  
Lebesgue Spaces ◽  
Dirichlet Laplacian ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

We study abstract equations of the formλu′′′(t)+u′′(t)=c2Au(t)+c2μAu′(t)+f(t),0<λ<μwhich is motivated by the study of vibrations of flexible structures possessing internal material damping. We introduce the notion of(α;β;γ)-regularized families, which is a particular case of(a;k)-regularized families, and characterize maximal regularity inLp-spaces based on the technique of Fourier multipliers. Finally, an application with the Dirichlet-Laplacian in a bounded smooth domain is given.

scholarly journals ANALYTIC REGULARITY FOR LINEAR ELLIPTIC SYSTEMS IN POLYGONS AND POLYHEDRA

2012 ◽  
Vol 22 (08) ◽  
pp. 1250015 ◽  
Author(s):  
MARTIN COSTABEL ◽  
MONIQUE DAUGE ◽  
SERGE NICAISE
Keyword(s):  
A Priori Estimates ◽  
Elliptic Boundary ◽  
A Priori ◽  
Three Dimensional ◽  
Elliptic Systems ◽  
Regularity Result ◽  
Open Set ◽  
Singular Functions ◽  
New Formulation ◽  
Analytic Regularity

We prove weighted anisotropic analytic estimates for solutions of second-order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.

Sign in / Sign up

Export Citation Format

Share Document