Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ ₁, … , ξ _n and a vector of scalars x = (x ₁, … , x _n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min_{1 ￡ i ￡ n} |x_ix_i|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max_{1 ￡ i ￡ n}|x_ix_i|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||y_x||_M{\|y_x\|_M} of the vector y _x  = (1/x ₁, … , 1/x _n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ ₁|, G(t) = \mathbb P ({ |x₁| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ ₁ is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò₀^t e^-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò₀^se^-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.     

Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where V(x)  ~  |x|^s, h(t)  ~  t^s{V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}. Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l ² + (n − 2)l = ω ₁. We prove that if m < p ￡ 1+(m-1)(1+s)+\frac2(s+1)+sn+l{m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}, then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if ${p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}${p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}, then the problem has global solutions for some u ₀ ≥ 0.     

Multiple solutions for a superlinear and periodic elliptic system on \mathbbRN{\mathbb{R}^N}

This paper is concerned with the following periodic Hamiltonian elliptic system

Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u ₀, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u ₀ and f or on the solution u itself such as Serrin’s condition || u ||_{L^s(0,T; L^q(W))} < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||_{L^s￠(t-d,t; L^q(W))}{\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||₂²{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.     

On expansions of numbers in alternating <Emphasis Type="Italic">s</Emphasis>-adic series and Ostrogradskii series of the first and second kind

We present expansions of real numbers in alternating s-adic series (1 < s ∈ N), in particular, s-adic Ostrogradskii series of the first and second kind. We study the “geometry” of this representation of numbers and solve metric and probability problems, including the problem of structure and metric-topological and fractal properties of the distribution of the random variable

Quasilinear elliptic equations on \mathbbRN{\mathbb{R}^{N}} with singular potentials and bounded nonlinearity

We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation

New separation axioms in generalized topological spaces

We introduce and study new separation axioms in generalized topological spaces, namely, m-T_\frac14\mu\mbox{-}T_{\frac{1}{4}}, m-T_\frac38\mu \mbox{-}T_{\frac{3}{8}} and m-T_\frac12\mu\mbox{-}T_{\frac{1}{2}}. m-T_\frac14\mu\mbox{-}T_{\frac{1}{4}} spaces are strictly placed between μ-T ₀ spaces and m-T_\frac38\mu\mbox{-}T_{\frac{3}{8}}, m-T_\frac38\mu\mbox{-}T_{\frac{3}{8}} spaces are strictly placed between m-T_\frac14\mu\mbox{-}T_{\frac{1}{4}} spaces and m-T_\frac12\mu \mbox{-}T_{\frac{1}{2}} spaces, and m-T_\frac12\mu\mbox{-}T_{\frac{1}{2}} spaces are strictly placed between m-T_\frac38\mu\mbox{-}T_{\frac{3}{8}} spaces and μ-T ₁ spaces.     

Inequalities for irregular Gabor frames

In [C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 (1993), 263–277], the authors proved that if {e^imbxg(x-na): m,n ? \Bbb Z}\{e^{imbx}g(x-na): m,n\in{\Bbb Z}\} is a Gabor frame for L²(\Bbb R)L^2({\Bbb R}) with frame bounds A and B, then the following two inequalities hold: A ￡ \frac2pb?_{n ? \Bbb Z}|g(x-na)|² ￡ B,     a.e.A\le \frac{2\pi}{b}\sum_{n\in{\Bbb Z}}\vert g(x-na)\vert^2\le B, \quad a.e. and A ￡ \frac1a?_{m ? \Bbb Z}|[^(g)](w-mb)|² ￡ B,     a.e.A\le \frac{1}{a}\sum_{m\in{\Bbb Z}}\vert \hat{g}(\omega-mb)\vert^2\le B, \quad a.e. . In this paper, we show that similar inequalities hold for multi-generated irregular Gabor frames of the form è_{1 ￡ k ￡ r}{e^{iáx, l?}g_k(x-m): m ? D_k, l ? L_k }\bigcup_{1\le k\le r}\{e^{i\langle x, \lambda\rangle}g_{k}(x-\mu):\, \mu\in \Delta_k, \lambda\in\Lambda_k \} , where Δ _k and Λ _k are arbitrary sequences of points in \Bbb R^d{\Bbb R}^d and g_k ? L²(\Bbb R^d)g_k\in{L^2{(\Bbb R}^d)} , 1 ≤ k ≤ r.     

An inverse spectral theorem for Kre?n strings with a negative eigenvalue

A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L²(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D_\mathfrakmD_x{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as

Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential

We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations

ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      12. Blow-up of solutions of nonlinear heat equations in unbounded domains for slowly decaying initial data      P. Rouchon 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2001,22(5):1017-1032 Consider the following equations: (E)  ut-Du=up{(E)\ \ u_t-\Delta u=u^p}, (E￠)  ut-Du=up-m | ?u | q{(E')\ \ u_t-\Delta u=u^p-\mu\mid\nabla u\mid^q}, (E")  ut-Du=up+a.?(uq){(E')\ \ u_t-\Delta u=u^p+a.\nabla (u^q)}, in W ì IRd{\Omega\subset I\!\!R^d}. For any unbounded domain W\Omega, intermediate between a cone and a strip, we obtain a sufficient condition on the decay at infinity of initial data to have blow-up. This condition is related to the geometric nature of W{\Omega}. For instance, if W\Omega is the interior of a revolution surface of the form | x￠d | < f( | xd | ){\mid x'_d\mid (x) > Cf( | x | )-2/(p-1){\Phi (x)>Cf(\mid x\mid )^{-2/(p-1)}} at infinity. Moreover, for a large class of domains W{\Omega}, we prove that those results are optimal (i.e. there exist global solutions with the same order of decay at infinity for their initial data).      13. Arithmetic mean of differences of Dedekind sums      Emre Alkan  Maosheng Xiong  Alexandru Zaharescu 《Monatshefte für Mathematik》2007,342(1):175-187 Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums \frac1j(N) ? 0 ￡ m < Ngcd(m,N)=1 |S(m,N)|\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form Ah(Q)=\frac1?\fracaq ? FQh(\fracaq) ×?\fracaq ? FQh(\fracaq) |s(a￠,q￠)-s(a,q)|A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\cal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\cal F}_{\!Q}}h\left(\frac{a}{q}\right) \vert s(a^{\prime},q^{\prime})-s(a,q)\vert , where h:[0,1] ? \Bbb Ch:[0,1] \rightarrow {\Bbb C} is a continuous function with ò01 h(t)  d t 1 0\int_0^1 h(t) \, {\rm d} t \ne 0 , \fracaq{\frac{a}{q}} runs over FQ{\cal F}_{\!Q} , the set of Farey fractions of order Q in the unit interval [0,1] and \fracaq < \fraca￠q￠{\frac{a}{q}}<\frac{a^{\prime}}{q^{\prime}} are consecutive elements of FQ{\cal F}_{\!Q} . We show that the limit lim Q→∞ A h (Q) exists and is independent of h.      14. Growth rate for beta-expansions      De-Jun Feng  Nikita Sidorov 《Monatshefte für Mathematik》2011,50(11):41-60 Let β > 1 and let m > β be an integer. Each x ? Ib:=[0,\fracm-1b-1]{x\in I_\beta:=[0,\frac{m-1}{\beta-1}]} can be represented in the form x=?k=1￥ ekb-k,x=\sum_{k=1}^\infty \epsilon_k\beta^{-k},      15. Harmonicity of functions satisfying a weak form of the mean value property      M. Hasson 《Archiv der Mathematik》2001,76(4):283-291 Let W ì \Bbb Rn\Omega \subset {\Bbb R}^n be a smooth domain and let u ? C0(W).u \in C^0(\Omega ). A classical result of potential theory states that¶¶-òSr([`(x)]) u(x)ds(x)=u([`(x)])-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x)=u(\bar x)¶¶for every [`(x)] ? W\bar x\in \Omega and r > 0r>0 if and only if¶¶Du=0 in W.\Delta u=0 \hbox { in } \Omega.¶¶Here -òSr([`(x)]) u(x)ds(x)-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x) denotes the average of u on the sphere Sr([`(x)])S_r(\bar x) of center [`(x)]\bar x and radius r. Our main result, which is a "localized" version of the above result, states:¶¶Theorem. Let u ? W2,1(W)u\in W^{2,1}(\Omega ) and let x ? Wx\in \Omega be a Lebesgue point of Du\Delta u such that¶¶-òSr([`(x)]) u d s- a = o(r2)-\kern-5mm\int\limits _{S_{r}(\bar x)} u d \sigma - \alpha =o(r^2)¶¶for some a ? \Bbb R\alpha \in \Bbb R and all sufficiently small r > 0.r>0. Then¶¶Du(x)=0.\Delta u(x)=0.      16. Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials      A. I. Podvysotskaya 《Ukrainian Mathematical Journal》2009,61(5):847-853 We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than ( n - 1 ) \mathord / \vphantom ( n - 1 ) ?{1 - x2} ?{1 - x2} {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} for all x ∈ (−1, 1) such that | x | ? èk = 0[ n \mathord / \vphantom n 2 2 ] [ cos\frac2k + 12( n - 1 )p, cos\frac2k + 12np ] \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} .      17. A Hyperplane Inequality for Measures of Convex Bodies in ? n , n≤4      Alexander Koldobsky 《Discrete and Computational Geometry》2012,47(3):538-547 Let 2≤n≤4. We show that for an arbitrary measure μ with even continuous density in ℝ n and any origin-symmetric convex body K in ℝ n , m(K) ￡ \fracnn-1\frac|B2n|\fracn-1n|B2n-1|maxx ? Sn-1 m(K?x^)\operatornameVoln(K)1/n,\mu(K) \le\frac{n}{n-1}\frac{|B_2^n|^{\frac{n-1}{n}}}{|B_2^{n-1}|}\max_{\xi\in S^{n-1}} \mu\bigl(K\cap\xi^\bot\bigr)\operatorname{Vol}_n(K)^{1/n},      18. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      19. A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations      Xiaochun Chen  Sadek Gala  Zhengguang Guo 《Acta Appl Math》2011,113(2):207-213 In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L\frac21-r( 0,T;[(X)\dot]r( \mathbbR3) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.      20. The numerical evaluation of the error term in a quadrature formula of Clenshaw-Curtis type for the Gegenbauer weight function      H. V. Smith  D. B. Hunter 《BIT Numerical Mathematics》2011,51(4):1031-1038 A method is derived for the numerical evaluation of the error term arising in a quadrature formula of Clenshaw-Curtis type for functions of the form (1-x2)l- \frac12f(x)(1-x^{2})^{\lambda - \frac{1}{2}}f(x) over the interval [−1,1]. The method is illustrated by an example.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Blow-up of solutions of nonlinear heat equations in unbounded domains for slowly decaying initial data

Consider the following equations: (E)  u_t-Du=u^p{(E)\ \ u_t-\Delta u=u^p}, (E￠)  u_t-Du=u^p-m | ?u | ^q{(E')\ \ u_t-\Delta u=u^p-\mu\mid\nabla u\mid^q}, (E")  u_t-Du=u^p+a.?(u^q){(E')\ \ u_t-\Delta u=u^p+a.\nabla (u^q)}, in W ì IR^d{\Omega\subset I\!\!R^d}. For any unbounded domain W\Omega, intermediate between a cone and a strip, we obtain a sufficient condition on the decay at infinity of initial data to have blow-up. This condition is related to the geometric nature of W{\Omega}. For instance, if W\Omega is the interior of a revolution surface of the form | x￠_d | < f( | x_d | ){\mid x'_d\mid (x) > Cf( | x | )^-2/(p-1){\Phi (x)>Cf(\mid x\mid )^{-2/(p-1)}} at infinity. Moreover, for a large class of domains W{\Omega}, we prove that those results are optimal (i.e. there exist global solutions with the same order of decay at infinity for their initial data).     

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums \frac1j(N) ? _{0 ￡ m < Ngcd(m,N)=1} |S(m,N)|\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form A_h(Q)=\frac1?_{\fracaq ? FQ}h(\fracaq) ×?_{\fracaq ? FQ}h(\fracaq) |s(a^￠,q^￠)-s(a,q)|A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\cal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\cal F}_{\!Q}}h\left(\frac{a}{q}\right) \vert s(a^{\prime},q^{\prime})-s(a,q)\vert , where h:[0,1] ? \Bbb Ch:[0,1] \rightarrow {\Bbb C} is a continuous function with ò₀¹ h(t)  d t 1 0\int_0^1 h(t) \, {\rm d} t \ne 0 , \fracaq{\frac{a}{q}} runs over F_Q{\cal F}_{\!Q} , the set of Farey fractions of order Q in the unit interval [0,1] and \fracaq < \fraca^￠q^￠{\frac{a}{q}}<\frac{a^{\prime}}{q^{\prime}} are consecutive elements of F_Q{\cal F}_{\!Q} . We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.     

Growth rate for beta-expansions

Let β > 1 and let m > β be an integer. Each x ? I_b:=[0,\fracm-1b-1]{x\in I_\beta:=[0,\frac{m-1}{\beta-1}]} can be represented in the form

Harmonicity of functions satisfying a weak form of the mean value property

Let W ì \Bbb Rⁿ\Omega \subset {\Bbb R}^n be a smooth domain and let u ? C⁰(W).u \in C^0(\Omega ). A classical result of potential theory states that¶¶-ò_{Sr([`(x)])</sub> u(x)ds(x)=u([`(x)])-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x)=u(\bar x)¶¶for every [`(x)] ? W\bar x\in \Omega and r > 0r>0 if and only if¶¶Du=0 in W.\Delta u=0 \hbox { in } \Omega.¶¶Here -ò<sub>Sr([`(x)])} u(x)ds(x)-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x) denotes the average of u on the sphere S_r([`(x)])S_r(\bar x) of center [`(x)]\bar x and radius r. Our main result, which is a "localized" version of the above result, states:¶¶Theorem. Let u ? W^2,1(W)u\in W^{2,1}(\Omega ) and let x ? Wx\in \Omega be a Lebesgue point of Du\Delta u such that¶¶-ò_Sr([`(x)]) u d s- a = o(r²)-\kern-5mm\int\limits _{S_{r}(\bar x)} u d \sigma - \alpha =o(r^2)¶¶for some a ? \Bbb R\alpha \in \Bbb R and all sufficiently small r > 0.r>0. Then¶¶Du(x)=0.\Delta u(x)=0.     

Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials

We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than ( n - 1 ) \mathord

A Hyperplane Inequality for Measures of Convex Bodies in ? n , n≤4

Let 2≤n≤4. We show that for an arbitrary measure μ with even continuous density in ℝ ⁿ and any origin-symmetric convex body K in ℝ ⁿ ,

The Hermite Property of a Causal Wiener Algebra Used in Control Theory

Let \mathbbC₊ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.     

The numerical evaluation of the error term in a quadrature formula of Clenshaw-Curtis type for the Gegenbauer weight function

A method is derived for the numerical evaluation of the error term arising in a quadrature formula of Clenshaw-Curtis type for functions of the form (1-x²)^{l- \frac12}f(x)(1-x^{2})^{\lambda - \frac{1}{2}}f(x) over the interval [−1,1]. The method is illustrated by an example.