Displaying similar documents to “Regularity properties of the equilibrium distribution”

Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

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We establish an explicit connection between the perimeter measure of an open set E with C 1 boundary and the spherical Hausdorff measure S Q 1 restricted to E , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and Q denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of E is less than or equal to S Q 1 ( E ) up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

On a bifurcation problem arising in cholesteric liquid crystal theory

Carlo Greco (2017)

Commentationes Mathematicae Universitatis Carolinae

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In a cholesteric liquid crystal the director field n ( x , y , z ) tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known...

Continuous pluriharmonic boundary values

Per Åhag, Rafał Czyż (2007)

Annales Polonici Mathematici

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Let D j be a bounded hyperconvex domain in n j and set D = D × × D s , j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco, Patrícia Gonçalves, Adriana Neumann (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the exclusion process in the one-dimensional discrete torus with N points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance N - β , with β [ 0 , ) . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter β . If β [ 0 , 1 ) , the hydrodynamic limit is given by the usual heat equation. If β = 1 , it is given by a parabolic equation involving an operator...

Equivalence of measures of smoothness in L p ( S d - 1 ) , 1 < p < ∞

F. Dai, Z. Ditzian, Hongwei Huang (2010)

Studia Mathematica

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Suppose Δ̃ is the Laplace-Beltrami operator on the sphere S d - 1 , Δ ρ k f ( x ) = Δ ρ Δ ρ k - 1 f ( x ) and Δ ρ f ( x ) = f ( ρ x ) - f ( x ) where ρ ∈ SO(d). Then ω m ( f , t ) L p ( S d - 1 ) s u p Δ ρ m f L p ( S d - 1 ) : ρ S O ( d ) , m a x x S d - 1 ρ x · x c o s t and K ̃ ( f , t m ) p i n f f - g L p ( S d - 1 ) + t m ( - Δ ̃ ) m / 2 g L p ( S d - 1 ) : g ( ( - Δ ̃ ) m / 2 ) are equivalent for 1 < p < ∞. We note that for even m the relation was recently investigated by the second author. The equivalence yields an extension of the results on sharp Jackson inequalities on the sphere. A new strong converse inequality for L p ( S d - 1 ) given in this paper plays a significant role in the proof.

Finite element variational crimes in the case of semiregular elements

Alexander Ženíšek (1996)

Applications of Mathematics

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The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain Ω whose boundary Ω is formed by two circles Γ 1 , Γ 2 with the same center S 0 and radii R 1 , R 2 = R 1 + ϱ , where ϱ R 1 . On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for u = 0 are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus...

Essential norms of the Neumann operator of the arithmetical mean

Josef Král, Dagmar Medková (2001)

Mathematica Bohemica

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Let K m ( m 2 ) be a compact set; assume that each ball centered on the boundary B of K meets K in a set of positive Lebesgue measure. Let C 0 ( 1 ) be the class of all continuously differentiable real-valued functions with compact support in m and denote by σ m the area of the unit sphere in m . With each ϕ C 0 ( 1 ) we associate the function W K ϕ ( z ) = 1 σ m m K g r a d ϕ ( x ) · z - x | z - x | m x of the variable z K (which is continuous in K and harmonic in K B ). W K ϕ depends only on the restriction ϕ | B of ϕ to the boundary B of K . This gives rise to a linear operator W K ...

A complete characterization of R-sets in the theory of differentiation of integrals

G. A. Karagulyan (2007)

Studia Mathematica

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Let s be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis s differentiates the integral of f if s ∉ S, and D ̅ s f ( x ) = l i m s u p d i a m ( R ) 0 , x R s | R | - 1 R f = almost everywhere if s ∈ S. If the condition D ̅ s f ( x ) = holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a G δ (resp. a G δ σ ).

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point x in a metric measure space ( X , d , μ ) is called a generalized Lebesgue point of a measurable function f if the medians of f over the balls B ( x , r ) converge to f ( x ) when r converges to 0 . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

Functions with prescribed singularities

Giovanni Alberti, S. Baldo, G. Orlandi (2003)

Journal of the European Mathematical Society

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The distributional k -dimensional Jacobian of a map u in the Sobolev space W 1 , k 1 which takes values in the sphere S k 1 can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M of codimension k can be realized as Jacobian of a Sobolev map valued in S k 1 . In case M is polyhedral, the...

A density version of the Carlson–Simpson theorem

Pandelis Dodos, Vassilis Kanellopoulos, Konstantinos Tyros (2014)

Journal of the European Mathematical Society

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We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer k 2 and every set A of words over k satisfying lim sup n | A [ k ] n | / k n > 0 there exist a word c over k and a sequence ( w n ) of left variable words over k such that the set c { c w 0 ( a 0 ) . . . w n ( a n ) : n and a 0 , . . . , a n [ k ] } is contained in A . While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Jacques Giacomoni, Ian Schindler, Peter Takáč (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We investigate the following quasilinear and singular problem, t o 2 . 7 c m - Δ p u = λ u δ + u q in Ω ; u | Ω = 0 , u &gt; 0 in Ω , t o 2 . 7 c m (P) where Ω is an open bounded domain with smooth boundary, 1 &lt; p &lt; , p - 1 &lt; q p * - 1 , λ &gt; 0 , and 0 &lt; δ &lt; 1 . As usual, p * = N p N - p if 1 &lt; p &lt; N , p * ( p , ) is arbitrarily large if p = N , and p * = if p &gt; N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 0 1 , p ( Ω ) . While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle...

An interpolatory estimate for the UMD-valued directional Haar projection

Richard Lechner

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We prove an interpolatory estimate linking the directional Haar projection P ( ε ) to the Riesz transform in the context of Bochner-Lebesgue spaces L p ( ; X ) , 1 < p < ∞, provided X is a UMD-space. If ε i = 1 , the result is the inequality | | P ( ε ) u | | L p ( ; X ) C | | u | | L p ( ; X ) 1 / | | R i u | | L p ( ; X ) 1 - 1 / , (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of L p ( ; X ) . In order to obtain the interpolatory result (1) we analyze stripe operators S λ , λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

Symmetry of minimizers with a level surface parallel to the boundary

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi (2015)

Journal of the European Mathematical Society

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We consider the functional Ω ( v ) = Ω [ f ( | D v | ) - v ] d x , where Ω is a bounded domain and f is a convex function. Under general assumptions on f , Crasta [Cr1] has shown that if Ω admits a minimizer in W 0 1 , 1 ( Ω ) depending only on the distance from the boundary of Ω , then Ω must be a ball. With some restrictions on f , we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss...

Perron-Frobenius operators and the Klein-Gordon equation

Francisco Canto-Martín, Håkan Hedenmalm, Alfonso Montes-Rodríguez (2014)

Journal of the European Mathematical Society

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For a smooth curve Γ and a set Λ in the plane 2 , let A C ( Γ ; Λ ) be the space of finite Borel measures in the plane supported on Γ , absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ . Following [12], we say that ( Γ , Λ ) is a Heisenberg uniqueness pair if A C ( Γ ; Λ ) = { 0 } . In the context of a hyperbola Γ , the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly...

Interpolating sequences, Carleson measures and Wirtinger inequality

Eric Amar (2008)

Annales Polonici Mathematici

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Let S be a sequence of points in the unit ball of ℂⁿ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure μ S : = a S ( 1 - | a | ² ) δ a is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of such that any δ -separated sequence S has its associated measure μ S bounded by C/δⁿ, then X is the zero set of a function in the Nevanlinna class of . As an easy consequence, we prove that if S is a dual...

Subclasses of typically real functions determined by some modular inequalities

Leopold Koczan, Katarzyna Trąbka-Więcław (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let T be the family of all typically real functions, i.e. functions that are analytic in the unit disk Δ : = { z : | z | < 1 } , normalized by f ( 0 ) = f ' ( 0 ) - 1 = 0 and such that Im z Im f ( z ) 0 for z Δ . Moreover, let us denote: T ( 2 ) : = { f T : f ( z ) = - f ( - z ) for z Δ } and T M , g : = { f T : f M g in Δ } , where M > 1 , g T S and S consists of all analytic functions, normalized and univalent in Δ .We investigate  classes in which the subordination is replaced with the majorization and the function g is typically real but does not necessarily univalent, i.e. classes { f T : f M g in Δ } , where M > 1 , g T , which we denote...