Superfici non parametriche a curvatura media assegnata con ostacoli

Riassunto In questo lavoro studio il problema dell’esistenza e regolarità di superfici non parametriche a curvatura mediaH(x) con ostacoli discontinui e sottili nelle ipotesi che e ∂Ω∈C ³ verifichino rispettivamente le condizioni (0.1), (0.2) e (0.3).

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Summary In this paper existence and regularity for surfaces of prescribed mean curvature with discontinuous and thin obstacles is proved when and verifies conditions (0.1) and (0.2), and ∂Ω∈C ³ and verifies condition (0.3).

     

Higher integrability for minimizers of variational integrals with nonstandard growth

We consider a (possibly) vector-valued function u: Ω→R ^N, Ω⊂R ⁿ, minimizing the integral , whereD _iu=∂u/∂x _i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D ₁u,…,D_n−1u∈L^q, under suitable assumptions ona _i(x).

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Sunto Consideriamo una funzione u: Ω→R ^N, Ω⊂R ⁿ che minimizzi l'integrale , doveD _iu=∂u/∂x_i, o un funzionale con un comportamento simile; sotto opportune ipotesi sua _i(x), dimostriamo la seguente maggiore integrabilità perDu:D ₁u,…,D_n−1uεL^q.

     

粗糙平方函数及极大算子的弱型估计

§ 1　 PreliminariesWe considerψ( x)∈ L1 ( Rn) satisfying the mean valuezero,i.e.∫Rnψdx=0 ,and definethe square function g( f) on Rnbyg( f) ( x) =( k|ψk* f|2 ) 1 2 ( x)for f∈ S( Rn) ,the Schwartz space,whereψk( x) =ψ2 k( x) .　　 Whenψ has some smooth property,one can obtain the weak type estimate by viewingthe square function g( f) as the vector-valued singularintegrals,which the readercan referto [1 ,2 ] .As for the results aboutthe Lp-estimates,see [3,4 ] .In this paper,we sha…     

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

Regolarità hölderiana delle derivate dei minimi di funzionali con parte principale quadratica

Riassunto In questo lavoro si prova la regolarità h?lderiana delle derivate, fino all'ordinek, dei minimi locali dei funzionali sotto opportune ipotesi suA _ij ^αβ e sug.

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Summary In this paper we prove h?lder-continuity of the derivates, up to orderk, of local minima of functionals under suitable hypotheses forA _ij ^αβ andg.

     

On minimizing partitions with infinitely many components

It is known [8] that, wheng ∈L ⁿ (Ω) (Ω open and bounded inR ⁿ , with ≪regular≫ boundary∂Ω), any minimizer (K, w) of the functional among relatively closed subsetsC ofΩ and piecewise-constant functionsu onΩ/C, gives rise to a finite decomposition ofΩ/K. Here we exhibit a piecewise-constant functiong on the unit diskD ofR ², with radial symmetry, such thatg ∈L ^q (D) for all 1 ⩽q < 2 and the unique minimizer of F has infinitely many components. We also fill a gap occurred in the proof of Proposition 5.2 of [8].

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Sunto è noto [8] che quandog ∈L ⁿ (Ω (Ω aperto limitato diR ⁿ , con frontiera sufficientemente regolare) i minimi (K, w) del funzionale , doveC è relativamente chiuso in Ω eu è costante a tratti suΩ/C, danno luogo a decomposizioni finite diΩ/K. In questo lavoro mostriamo un controesempio relativo ad un datog ∈L ^q (D) per ogni 1 ⩽q < 2 (D è il disco unitario diR ²), a simmetria radiale e costante a tratti. Viene inoltre corretto un errore occorso nella dimostrazione della Prop. 5.2 di [8].

     

The individual ergodic theorem forp-sequences

Let (Ω,f,P) be a probability space and letT be a measure-preserving weak mixing transformation. We define a large class of sequences of integers calledp-sequences, such that iff∈L ₁ there exists a set Ω′⊂Ω of probability one and for ω∈Ω′ we have for everyp-sequence {k_n}.     

Elliptic Equations with Degenerate Coercivity： Gradient Regularity

In this paper, we prove higher integrability results for the gradient of the solutions of some elliptic equations with degenerate coercivity whose prototype is where for example, a(x,u)=(1+|u|)^−θ with θ ∈ (0,1). We study the same problem for minima of functionals closely related to the previous equation.     

Cauchy-Rassias Stability of Cauchy-Jensen Additive Mappings in Banach Spaces

Let X, Y be vector spaces. It is shown that if a mapping f ： X → Y satisfies f（（x＋y）/2＋z）＋f（（x-y）/2＋z=f（x）＋2f（z）,（0.1） f（（x＋y）/2＋z）-f（（x-y）/2＋z）f（y）,（0.2） or 2f（（x＋y）/2＋x）=f（x）＋f（y）＋2f（z）（0.3）for all x, y, z ∈ X, then the mapping f ： X →Y is Cauchy additive. Furthermore, we prove the Cauchy-Rassias stability of the functional equations （0.1）, （0.2） and （0.3） in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras.     

Regularity of Weak Solutions of Nonlinear Equations with Discontinuous Coefficients

In this paper, we prove that the weak solutions u∈Wloc^1, p （Ω） （1 〈p〈∞） of the following equation with vanishing mean oscillation coefficients A（x）： -div[（A（x）△↓u·△↓u）p-2/2 A（x）△↓u＋│F（x）│^p-2 F（x）]=B（x, u, △↓u）, belong to Wloc^1, q （Ω）（A↓q∈（p, ∞）, provided F ∈ Lloc^q（Ω） and B（x, u, h） satisfies proper growth conditions where Ω ∪→R^N（N≥2） is a bounded open set, A（x）=（A^ij（x）） N×N is a symmetric matrix function.     

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

Sul comportamento quasi periodico del resto di una certa classe di funzioni fortemente moltiplicative

Riassunto Siaf(x) una funzione periodica di periodo 1 ed a variazione limitata. Nel presente lavoro si dimostrano risultati diB ² quasi periodicità per convoluzioni aritmetiche del tipo dove la successione soddisfa opportuna ipotesi. I risultati ottenuti si possono applicare ai resti di certe funzioni fortemente moltiplicative.

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Summary Letf(x) be a periodic function with period 1 and of bounded variation. In this paper we prove results ofB ² almost periodicity for arithmetical convolutions of the type , where the sequence satisfies certain conditions. The results obtained can be applied to the remainder term of strongly multiplicative functions.

     

Problem of the first passage time for <Emphasis Type="Italic">p</Emphasis>-adic diffusion

This work is a continuation of paper [1], where was considered analog of the problem of the first return for ultrametric diffusion. The main result of this paper consists in construction and investigation of stochastic quantity $ \tau _{B_r (a)} $ \tau _{B_r (a)} (ω), which has meaning of the first passage time into domain B _r (a) by trajectories of the Markov stochastic process ζ(t, ω).Markov stochastic process is given by distribution density f(x, t), x ∈ ℚ _p , t ∈ R ₊, which is solution of the Cauchy problem

On the Boundary Behaviour,Including Second Order Effects,of Solutions to Singular Elliptic Problems

For γ≥1 we consider the solution u=u（x） of the Dirichlet boundary value problem Δu ＋ u^-γ=0 in Ω, u=0 on δΩ. For γ= 1 we find the estimate u（x）=p（δ（x））[1＋A（x）（log 1/δ（x）^-6], where p（r） ≈ r r√2 log（1/r） near r = 0,δ（x） denotes the distance from x to δΩ, 0 〈ε 〈 1/2, and A（x） is a bounded function. For 1 〈 γ 〈 3 we find u（x）=（γ＋1/√2（γ-1）δ（x））^2/γ＋[1＋A（x）（δ（x））2γ-1/γ＋1] For γ3= we prove that u（x）=（2δ（x））^1/2[1＋A（x）δ（x）log 1/δ（x）]     

Blow-up criterion for 2-D Boussinesq equations in bounded domain

We extend the results for 2-D Boussinesq equations from ℝ² to a bounded domain Ω. First, as for the existence of weak solutions, we transform Boussinesq equations to a nonlinear evolution equation U _t + A(t, U) = 0. In stead of using the methods of fundamental solutions in the case of entire ℝ², we study the qualities of F(u, υ) = (u · ▽)υ to get some useful estimates for A(t, U), which helps us to conclude the local-in-time existence and uniqueness of solutions. Second, as for blow-up criterions, we use energy methods, Sobolev inequalities and Gronwall inequality to control and by and . Furthermore, can control by using vorticity transportation equations. At last, can control . Thus, we can find a blow-up criterion in the form of .      

A sharp form of an embedding into multiple exponential spaces

Let Θ be a bounded open set in ℝ ⁿ , n ⩾ 2. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of K such that

Characterizations of Hardy-Orlicz and Bergman-Orlicz spaces

Let and τ denote the invariant gradient and invariant measure on the unit ball B of ℂⁿ, respectively. Assume that f is a holomorphic function on B and ϕ ∈ C²(ℝ) is a nonnegative, nondecreasing, convex function. Then f belongs to the Hardy-Orlicz space H _ϕ(B>) if and only if

Precise estimate of total deficiency of meromorphic derivatives

Let f(z) be a transcendental meromorphic function in the finite plane andk be a positive integer. Then we have . Moreover, if the order of f(z) is finite, then we also have , where δ(a, f^(k)) denotes the deficiency of the valuea with respect to f^(k) and θ(∞,f) is the ramification index of ∞ with respect tof.     

Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.