An estimate of the solutions of a stationary problem for the Navier-Stokes equations

It is shown that the solutions of a nonlinear stationary problem for the Navier-Stokes equations in a bounded domain Ω ? ?³ with boundary conditions $\vec \upsilon \left| {_{\partial \Omega } } \right. = \vec a(x)$ satisfy the inequality $\left. {_{x \in \Omega }^{\sup } } \right|\left. {\vec v(x)} \right| \leqslant c\left( {\left. {_{x \in \partial \Omega }^{\sup } } \right|\left. {\vec a(x)} \right|} \right)$ for arbitrary Reynolds numbers. Bibliography: 9 titles.     

On the discrepancy function in arbitrary dimension, close to L 1

Let A _N to be N points in the unit cube in dimension d, and consider the discrepancy function $$ D_N (\vec x): = \sharp \left( {\mathcal{A}_N \cap \left[ {\vec 0,\vec x} \right)} \right) - N\left| {\left[ {\vec 0,\vec x} \right)} \right| $$ Here, $$ \vec x = \left( {\vec x,...,x_d } \right),\left[ {0,\vec x} \right) = \prod\limits_{t = 1}^d {\left[ {0,x_t } \right),} $$ and $ \left| {\left[ {0,\vec x} \right)} \right| $ denotes the Lebesgue measure of the rectangle. We show that necessarily $$ \left\| {D_N } \right\|_{L^1 (log L)^{(d - 2)/2} } \gtrsim \left( {log N} \right)^{\left( {d - 1} \right)/2} . $$ In dimension d = 2, the ‘log L’ term has power zero, which corresponds to a Theorem due to [11]. The power on log L in dimension d ≥ 3 appears to be new, and supports a well-known conjecture on the L ¹ norm of D _N . Comments on the discrepancy function in Hardy space also support the conjecture.     

Lower bounds for the modulus of the logarithmic derivative of a polynomial

Estimates are given for the measure of a section of an arbitrary straight line of the set $$E_\delta = \left\{ {z:\left| {P' {{\left( z \right)} \mathord{\left/ {\vphantom {{\left( z \right)} {\left( {nP \left( z \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {nP \left( z \right)} \right)}} \leqslant \delta } \right|} \right\} \left( {\delta > 0} \right)$$ where P (z) is a polynomial of degree n. THEOREM. Suppose P (x) = (x ? x₁) ... (x ? x_n) is a polynomial with real zeros. Then, for any δ > 0, on any intervala ?x ?b, containing all of the x_k (k=1, 2, ..., n), outside an exceptional set E_δ?[a,b] such that $$mes E_\delta \leqslant \left( {\sqrt {1 + \delta ^2 \left( {b - a} \right)^2 } - 1} \right)/\delta $$ , we have the inequality $$\left| {P' {{\left( x \right)} \mathord{\left/ {\vphantom {{\left( x \right)} {\left( {nP \left( x \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {nP \left( x \right)} \right)}}} \right| > \delta $$ . A similar estimate is given for polynomials whose roots lie either in Imz ? 0 or in Imz ? 0.     

A priori estimates for a class of degenerate elliptic equations

In this paper we investigate the regularity of solutions for the following degenerate partial differential equation $$\left \{\begin{array}{ll} -\Delta_p u + u = f \qquad {\rm in} \,\Omega,\\ \frac{\partial u}{\partial \nu} = 0 \qquad \qquad \,\,\,\,\,\,\,\,\,\, {\rm on} \,\partial \Omega, \end{array}\right.$$ when ${f \in L^q(\Omega), p > 2}$ and q ≥ 2. If u is a weak solution in ${W^{1, p}(\Omega)}$ , we obtain estimates for u in the Nikolskii space ${\mathcal{N}^{1+2/r,r}(\Omega)}$ , where r = q(p ? 2) + 2, in terms of the L ^q norm of f. In particular, due to imbedding theorems of Nikolskii spaces into Sobolev spaces, we conclude that ${\|u\|^r_{W^{1 + 2/r - \epsilon, r}(\Omega)} \leq C(\|f\|_{L^q(\Omega)}^q + \| f\|^{r}_{L^q(\Omega)} + \|f\|^{2r/p}_{L^q(\Omega)})}$ for every ${\epsilon > 0}$ sufficiently small. Moreover, we prove that the resolvent operator is continuous and compact in ${W^{1,r}(\Omega)}$ .     

The metric projection on the classH(Ω) satisfies Lipschitz condition

Устанавливается лиш шщивость и односторо нняя дифференцируемость метрической проекции $$P_H^t :x \to P_H^t \left( x \right) = \left\{ {y \in H: \parallel x - y\parallel \leqq t + \varrho \left( {x, H} \right)} \right\}, t \geqq 0$$ на класс из пространстваB(Q) огр аниченных на множест веQ функцийx c нормой ∥x∥=sup {∣x(q)∣:q∈Q}, гдеΩ—метрика наQ. В частн ости, дляy _i ∈B(Q),t _i ≧0, метрикΩ _i и \(P_i = P_{H(\Omega _i )}^{t_i } \left( {y_i } \right), i = 1, 2\) доказано неравенство $$h\left( {P_1 , P_2 } \right) \leqq \frac{3}{2}\mathop {\sup }\limits_{q_1 , q_2 \in Q} \left| {\Omega _1 \left( {q_1 , q_2 } \right) - \Omega _2 \left( {q_1 , q_2 } \right)} \right| + 2\parallel y_1 - y_2 \parallel + 1\left| {t_1 - t_2 } \right|,$$ гдеh — расстояние Хау сдорфа. Константы 3/2, 2, 1 в э том неравенстве неулучш аемы.     

On an over-determined problem of free boundary of a degenerate parabolic equation

This work is concerned with the inverse problem of determining initial value of the Cauchy problem for a nonlinear diffusion process with an additional condition on free boundary. Considering the flow of water through a homogeneous isotropic rigid porous medium, we have such desire: for every given positive constants K and T ₀, to decide the initial value u ₀ such that the solution u(x, t) satisfies $\mathop {\sup }\limits_{x \in H_u (T_0 )} |x| \geqslant K$ , where H _u(T ₀) = {x, ?^N: u(x, T ₀) > 0}. In this paper, we first establish a priori estimate u _t ? C(t)u and a more precise Poincaré type inequality $\left\| \phi \right\|_{L^2 (B_\varrho )}^2 \leqslant \varrho \left\| {\nabla \phi } \right\|_{L^2 (B_\varrho )}^2 $ , and then, we give a positive constant C ₀ and assert the main results are true if only $\left\| {u_0 } \right\|_{L^2 (\mathbb{R}^N )} \geqslant C_0 $ .     

A weighted estimate of best approximations in L2(Ω)

The best approximation [in the space L₂(Ω)] of a function f, satisfying a Lipschitz condition with exponent α, 0?α?1, with the aid of certain spaces of local functions, dependent on a parameter h, is discussed. We obtain the estimate $$\left\| {f - \tilde f} \right\|_\beta \leqslant \tilde C\left( f \right)h\min \left\{ {\alpha , \beta } \right\}$$ , where $$\left\| u \right\|_\beta = \mathop {\max }\limits_{x \in \bar \Omega } \left| {r^\beta u\left( x \right)} \right|,\beta \geqslant 0, u \in C\left( {\bar \Omega } \right)$$ and r = r(x) is the distance of the point x from the boundary of the domain Ω.     

Strong approximation of Riesz means at critical index on Hp(T) (0<p≤1)

This paper is a continuation of [3]. Suppose f∈H^p(T), 0σ _r ^σ f,σ=1/p?1. When p=1, it is just the partial Fourier sums S_kf. In this paper we establish the sharp estimations on the degree of approximation: $$\left\{ { - \frac{1}{{logR}}\int\limits_1^R {\left\| {\sigma _r^\delta f - f} \right\|_{H^p (T)}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqq C{\mathbf{ }}{}_p\omega \left( {f,{\mathbf{ }}( - \frac{1}{{logR}})^{1/p} } \right)_{H^p (T)} ,0< p< 1,$$ and \(\frac{1}{{\log L}}\sum\limits_{k - 1}^L {\frac{{\left\| {S_k f - f} \right\|_H 1_{(T)} }}{k} \leqq Cp\omega (f; - \frac{1}{{\log L}})_H 1_{(T)} } \) Where $$\omega (f,{\mathbf{ }}h)_{H^p (T)} \begin{array}{*{20}c} { = Sup} \\ {0 \leqq \left| u \right| \leqq h} \\ \end{array} \left\| {f( \cdot + u) - f( \cdot )} \right\|_{H^p (T).} $$ .     

On a multilinear singular integral operator

In this paper, we prove that the maximal operatorsatisfiesis homogeneous of degree 0, has vanishing moment up to order M and satisfies Lq-Dini condition for some     

New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

We obtain Hardy type inequalities $$\int_0^\infty {M\left( {\omega \left( r \right)\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr} \leqslant C_1 \int_0^\infty {M\left( {\left| {u\left( r \right)} \right|} \right)\rho \left( r \right)dr + C_2 \int_0^\infty {M\left( {\left| {u'\left( r \right)} \right|} \right)\rho \left( r \right)dr,} }$$ and their Orlicz-norm counterparts $$\left\| {\omega u} \right\|_{L^M (\mathbb{R}_ + ,\rho )} \leqslant \tilde C_1 \left\| u \right\|_{L^M (\mathbb{R}_ + ,\rho )} + \tilde C_2 \left\| {u'} \right\|_{L^M (\mathbb{R}_ + ,\rho )} ,$$ with an N-function M, power, power-logarithmic and power-exponential weights ??, ??, holding on suitable dilation invariant supersets of C ₀ ^?? (?₊). Maximal sets of admissible functions u are described. This paper is based on authors?? earlier abstract results and applies them to particular classes of weights.     

Estimation of the remainder of a cubature formula on a Chebyshev grid

Let C(Q) denote the space of continuous functions f(x, y) in the square Q = [?1, 1] × [?1, 1] with the norm $\begin{gathered} \left\| f \right\| = \max \left| {f(x,y)} \right|, \hfill \\ (x,y) \in Q. \hfill \\ \end{gathered} $ On a Chebyshev grid, a cubature formula of the form $\int\limits_{ - 1}^1 {\int\limits_{ - 1}^1 {\frac{1} {{\sqrt {(1 - x^2 )(1 - y^2 )} }}f(x,y)dxdy = \frac{{\pi ^2 }} {{mn}}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {f\left( {\cos \frac{{2i - 1}} {{2n}}\pi ,\cos \frac{{2j - 1}} {{2m}}\pi } \right)} + R_{m,n} (f)} } } $ is considered in some class H(r ₁, r ₂) of functions f ?? C(Q) defined by a generalized shift operator. The remainder R _{m, n} (f) is proved to satisfy the estimate $\mathop {\sup }\limits_{f \in H(r_1 ,r_2 )} \left| {R_{m,n} (f)} \right| = O(n^{ - r_1 + 1} + m^{ - r_2 + 1} ), $ where r ₁, r ₂ > 1; ??^?1 ?? n/m ?? ?? with ?? > 0; and the constant in O(1) depends on ??.     

On the best constants in the Khintchine inequality for Steinhaus variables

Let S _j : (Ω, P) → S ¹ ? ? be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S ¹. We determine the best constants a _p in the Khintchine-type inequality $${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$ for 0 < p < 1, verifying a conjecture of U. Haagerup that $${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$ . Both expressions are equal for p = p ₀ }~ 0.4756. For p ≥ 1 the best constants a _p have been known for some time. The result implies for a norm 1 sequence x ∈ ? ⁿ , ‖x‖₂ = 1, that $${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$ , answering a question of A. Baernstein and R. Culverhouse.     

Sharp weak-type inequalities for Hilbert transform and Riesz projection

The paper is devoted to the study of the weak norms of the classical operators in the vector-valued setting.

1. Let S, H denote the singular integral involution operator and the Hilbert transform on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {\mathcal{S}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p,$$ $$\left\| {\mathcal{H}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p.$$ Both inequalities are sharp.
2. Let P ₊ and P _? stand for the Riesz projection and the co-analytic projection on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {P + f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p ,$$ $$\left\| {P - f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p .$$ Both inequalities are sharp.
3. We establish the sharp versions of the estimates above in the nonperiodic case.

The results are new even if the operators act on complex-valued functions. The proof rests on the construction of an appropriate plurisubharmonic function and probabilistic techniques.     

{W^{1,1}_0} -solutions for elliptic problems having gradient quadratic lower order terms

In this paper we deal with solutions of problems of the type $$\left\{\begin{array}{ll}-{\rm div} \Big(\frac{a(x)Du}{(1+|u|)^2} \Big)+u = \frac{b(x)|Du|^2}{(1+|u|)^3} +f \quad &{\rm in} \, \Omega,\\ u=0 &{\rm on} \partial \, \Omega, \end{array} \right.$$ where ${0 < \alpha \leq a(x) \leq \beta, |b(x)| \leq \gamma, \gamma > 0, f \in L^2 (\Omega)}$ and Ω is a bounded subset of ${\mathbb{R}^N}$ with N ≥ 3. We prove the existence of at least one solution for such a problem in the space ${W_{0}^{1, 1}(\Omega) \cap L^{2}(\Omega)}$ if the size of the lower order term satisfies a smallness condition when compared with the principal part of the operator. This kind of problems naturally appears when one looks for positive minima of a functional whose model is: $$J (v) = \frac{\alpha}{2} \int_{\Omega}\frac{|D v|^2}{(1 + |v|)^{2}} + \frac{12}{\int_{\Omega}|v|^2} - \int_{\Omega}f\,v , \quad f \in L^2(\Omega),$$ where in this case a(x) ≡ b(x) = α > 0.     

Pointwise estimates for Lagrange interpolation polynomials

Let f ∈ C[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ _{n + 2}(f, x). In this paper we study the estimate of Δ _{n + 2}(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T _n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f ∈ C ^r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$      

Full regularity of a free boundary problem with two phases

Let ?? be a bounded domain in ${\mathbb{R}^{n}, n\geq2}$ . We use ${\mathcal{M}_{\Omega}}$ to denote the collection of all pairs of (A, u) such that ${A\subset\Omega}$ is a set of finite perimeter and ${u\in H^{1}\left( \Omega\right)}$ satisfies $$u\left( x\right) =0\quad\text{a.e.}x\in A.$$ We consider the energy functional $$E_{\Omega}\left( A,u\right) =\int\limits_{\Omega}\left\vert\triangledown u\right\vert ^{2}+P_{\Omega}\left( A\right)$$ defined on ${\mathcal{M}_{\Omega}}$ , where P _??(A) denotes the perimeter of A inside ??. Let ${\left( A,u\right)\in\mathcal{M}_{\Omega}}$ be a minimizer with volume constraint. Our main result is that when n????7, u is locally Lipschitz and the free boundary ?A is analytic in ??.     

On the best polynomial approximation of entire transcendental functions in banach spaces. I

We study the behavior of the best approximationsE _n (?) _p of entire transcendental functions ?(z) of the order ρ=∞ by polynomials of at mostn th degree in the metric of the Banach space E′_p(Ω) of functions /tf(z) analytic in a bounded simply connected domain Ω with rectifiable Jordan boundary and such that $$\left\| f \right\|_{E'_p } = \left\{ {\iint_\Omega {\left| {f\left( z \right)} \right|^p }dxdy} \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}}< \infty $$ . In particular, we describe the relationship between the best approximationsE _n (?)_p and theq-order andq-type of the function ?(z).     

ОцЕНкА (ε, δ)-ЁНтРОпИИ к лАссА цЕлых ФУНкцИИ В ИНтЕгРАльНОИ МЕтРИ кЕ

LetB _σ be the class of entire functions of exponential type σ, real valued and bounded in modulus by 1 in the real line. A setG of functions defined on the segment [-T-r, T+r], wherer is a fixed positive number, is called an (ε, δ)-net of the classB _σ on the segment [-т, т] if for any f?B _σ there existsg?G such that for anyx?[-T,T] $$\left| {f(x) - g(x)} \right| \leqq \frac{\varepsilon }{{2r}}\int\limits_{x - r}^{x + r} {\left| {f(t)} \right|dt + \delta .} $$ The main result consists in the following: For any positive σ, r, ε≦1, δ≦1 and sufficiently largeT we have $$H_{\varepsilon ,\delta } (B_\sigma ,T) \leqq \frac{{2\sigma T}}{\pi }\log \frac{{c(\sigma r)}}{{\max (\varepsilon ,\delta )}},$$ where c(σr) depends only on the product σr. The main tool of the proof of this inequality is the following estimate of the derivative of a polynomialP(x) with real coefficients: $$\left\| {P'(x)} \right\|_{L_p ( - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}) \leqq } c\left( {q + 1 + \sum\limits_{i = 1}^{n - q} {\frac{1}{{\left| {a_i } \right|^2 }}} } \right)\left\| {P(x} \right\|_{L_p ( - 1,1)} ,$$ whereq is the number of roots of the polynomialP(x) lying in the disk ¦z¦<1; a₁, ..., a_n?g are the other roots, с is an absolute constant, and 1≦p≦∞.