共查询到20条相似文献,搜索用时 531 毫秒
1.
David Arcoya Lucio Boccardo Luigi Orsina 《Calculus of Variations and Partial Differential Equations》2013,47(1-2):159-180
For a bounded, open subset Ω of ${\mathbb{R}^{N}}$ with N > 2, and a measurable function a(x) satisfying 0 < α ≤ a(x) ≤ β, a.e. ${x \in \Omega}$ , we study the existence of positive solutions of the Euler–Lagrange equation associated to the non-differentiable functional $$\begin{array}{ll}J(v) = \frac{1}{2} \int \limits_{\Omega} [a(x)+|v|^{\gamma}]| \nabla v|^{2}- \frac{1}{p} \int \limits_{\Omega}(v_{+})^p,\end{array}$$ if γ > 0 and p > 1. Special emphasis is placed on the case ${2^{*} < p < \frac{2^{*}}{2} ( \gamma +2 )}$ . 相似文献
2.
Yu. K. Dem'yanovich 《Mathematical Notes》2005,78(5-6):615-630
With each infinite grid X: ? < x ?1 < x 0 < x 1 < ? we associate the system of trigonometric splines $\{ \mathfrak{T}_j^B \}$ of class C 1(α, β), the linear space $$T^B (X)\mathop = \limits^{def} \{ \tilde u|\tilde u = \sum\limits_j {c_j \mathfrak{T}_j^B } \quad \forall c_j \in \mathbb{R}^1 \} ,$$ and the functionals g (i) ∈ (C 1(α, β))* with the biorthogonality property: $\left\langle {g(i),\mathfrak{T}_j^B } \right\rangle = \delta _{i,j}$ (here $\alpha \mathop = \limits^{def} \lim _{j \to - \infty } x_j ,\quad \beta \mathop = \limits^{def} \lim _{j \to + \infty } x_j$ ). For nested grids $\bar X \subset X$ , we show that the corresponding spaces $T^B (\bar X)$ are embedded in $T^B (X)$ and obtain decomposition and reconstruction formulas for the spline-wavelet expansion $T^B (X) = T^B (\bar X)\dot + W$ derived with the help of the system of functionals indicated above. 相似文献
3.
Miroslav Pavlović 《Annali di Matematica Pura ed Applicata》2013,192(5):745-762
Let ${\mathcal L(r) = \sum_{n=0}^\infty a_nr^{\lambda_n}}$ be a lacunary series converging for 0 < r < 1, with coefficients in a quasinormed space. It is proved that $$\int_0^1 F(1-r,\|\mathcal L(r)\|)(1-r)^{-1}\,{\rm d}r < \infty $$ if and only if $$ \sum_{n=0}^\infty F(1/\lambda_n,\|a_n\|) < \infty, $$ where F is a “normal function” of two variables. In the case when p ≥ 1 and F(x, y) = x y p , this reduces to a theorem of Gurariy and Matsaev. As an application we prove that if ${f(r\zeta) = \sum_{n=0}^\infty r^{\lambda_n}f_{\lambda_n}(\zeta)}$ is a function harmonic in the unit ball of ${\mathbb R^N,}$ then $$\int_0^1M_p^q(r,f)(1-r)^{q\alpha-1} \,{\rm d}r <\infty\quad (p,\,q,\,\alpha >0 ) $$ if and only if $$\sum_{n=0}^\infty \|f_{\lambda_n} \|^q_{L^p(\partial B_N)}(1/\lambda_n)^{q\alpha} <\infty. $$ 相似文献
4.
R. A. Hibschweiler 《Complex Analysis and Operator Theory》2012,6(4):897-911
For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C ?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C ?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded. 相似文献
5.
Mehdi Hassani 《The Ramanujan Journal》2012,28(3):435-442
We discuss on the sign of $\mathcal{R}_{\alpha }(x):=\pi(x)^{2}-\frac{\alpha x}{\log x}\pi(\frac{x}{\alpha })$ for x sufficiently large, and for various values of ??>0. The case ??=e refers to a result due to Ramanujan asserting that $\mathcal{R}_{e}(x)<0$ . Related by this inequality, we obtain a conditional result that gives the number N>530.2 such that $\mathcal{R}_{e}(x)<0$ is valid for x>e N . Moreover, we show that under assumption of validity of the Riemann hypothesis, the inequality $\mathcal{R}_{e}(x)<0$ holds for x>138,766,146,692,471,228. Then, in various cases for ??, we find numerical values of x ?? in which $\mathcal{R}_{\alpha }(x)$ is strictly positive or negative for x??x ?? . 相似文献
6.
A. B. Shapoval 《Mathematical Notes》1996,60(4):415-424
We consider the solutions of the inequalityLu≤?(¦gradu¦), whereL is a uniformly elliptic homogeneous operator and ? is a function increasing faster than any linear function but not faster thanξ lnξ, in the unbounded domain $$\left\{ {x \in \mathbb{R}^n |\sum\limits_{i = 2}^n {x_i^2< (\psi (x_1 ))^2 ,} {\text{ }} - \infty< x_1< \infty } \right\},$$ , whereψ is a bounded function with bounded derivative. We estimate the growth of the solutions in terms of $\int_0^{x_1 } {(1/\psi (r))dr}$ . For the special case in which?(ξ)=aξ lnξ+C, the solutionsu(x 1,x 2,...,x n ) grow as $\left( {\int_0^{x_1 } {(1/\psi (r))dr} } \right)^N$ , whereN is any given number anda=a(N). 相似文献
7.
V. I. Arnautov 《Journal of Mathematical Sciences》2012,185(2):176-183
Let R(+, ·) be a nilpotent ring and $ \left( {\mathfrak{M}, < } \right) $ be the lattice of all ring topologies on R(+, ·) or the lattice of all such ring topologies on R(+, ·) in each of which the ring R possesses a basis of neighborhoods of zero consisting of subgroups. Let ?? and ??? be ring topologies from $ \mathfrak{M} $ such that $ \tau = {\tau_0}{ \prec_\mathfrak{M}}{\tau_1}{ \prec_\mathfrak{M}} \cdots { \prec_\mathfrak{M}}{\tau_n} = \tau ^{\prime} $ . Then k????n for every chain $ \tau = {\tau ^{\prime}_0} < {\tau ^{\prime}_1} < \cdots < {\tau ^{\prime}_k} = \tau ^{\prime} $ of topologies from $ \mathfrak{M} $ , and also n?=?k if and only if $ {\tau ^{\prime}_i}{ \prec_\mathfrak{M}}{\tau ^{\prime}_{i + 1}} $ for all 0????i?<?k. 相似文献
8.
Michael Winkler 《Mathematische Annalen》2013,355(2):519-549
This work deals with positive classical solutions of the degenerate parabolic equation $$u_t=u^p u_{xx} \quad \quad (\star)$$ when p > 2, which via the substitution v = u 1?p transforms into the super-fast diffusion equation ${v_t=(v^{m-1}v_x)_x}$ with ${m=-\frac{1}{p-1} \in (-1,0)}$ . It is shown that ( ${\star}$ ) possesses some entire positive classical solutions, defined for all ${t \in \mathbb {R}}$ and ${x \in \mathbb {R}}$ , which connect the trivial equilibrium to itself in the sense that u(x, t) → 0 both as t → ?∞ and as t → + ∞, locally uniformly with respect to ${x \in \mathbb {R}}$ . Moreover, these solutions have quite a simple structure in that they are monotone increasing in space. The approach is based on the construction of two types of wave-like solutions, one of them being used for ?∞ < t ≤ 0 and the other one for 0 < t < + ∞. Both types exhibit wave speeds that vary with time and tend to zero as t → ?∞ and t → + ∞, respectively. The solutions thereby obtained decay as x → ?∞, uniformly with respect to ${t \in \mathbb {R}}$ , but they are unbounded as x → + ∞. It is finally shown that within the class of functions having such a behavior as x → ?∞, there does not exist any bounded homoclinic orbit. 相似文献
9.
A classical result states that every lower bounded superharmonic function on ${\mathbb{R}^{2}}$ is constant. In this paper the following (stronger) one-circle version is proven. If ${f : \mathbb{R}^{2} \to (-\infty,\infty]}$ is lower semicontinuous, lim inf|x|→∞ f (x)/ ln |x| ≥ 0, and, for every ${x \in \mathbb{R}^{2}}$ , ${1/(2\pi) \int_0^{2\pi} f(x + r(x)e^{it}) \, dt \le f(x)}$ , where ${r : \mathbb{R}^{2} \to (0,\infty)}$ is continuous, ${{\rm sup}_{x \in \mathbb{R}^{2}} (r(x) - |x|) < \infty},$ , and ${{\rm inf}_{x \in \mathbb{R}^{2}} (r(x)-|x|)=-\infty}$ , then f is constant. Moreover, it is shown that, assuming r ≤ c| · | + M on ${\mathbb{R}^d}$ , d ≤ 2, and taking averages on ${\{y \in \mathbb{R}^{d} : |y-x| \le r(x)\}}$ , such a result of Liouville type holds for supermedian functions if and only if c ≤ c 0, where c 0 = 1, if d = 2, whereas 2.50 < c 0 < 2.51, if d = 1. 相似文献
10.
The instability property of the standing wave uω(t, x) = eiωtφ(x) for the Klein–Gordon– Hartree equation 相似文献
11.
Jonathan Di Cosmo Jean Van Schaftingen 《Calculus of Variations and Partial Differential Equations》2013,47(1-2):243-271
We study positive bound states for the equation ${- \varepsilon^2 \Delta u + Vu = u^p, \quad {\rm in} \quad \mathbb{R}^N}$ , where ${\varepsilon > 0}$ is a real parameter, ${\frac{N}{N-2} < p < \frac{N+2}{N-2}}$ and V is a nonnegative potential. Using purely variational techniques, we find solutions which concentrate at local maxima of the potential V without any restriction on the potential. 相似文献
12.
Alan Carey Fritz Gesztesy Denis Potapov Fedor Sukochev Yuri Tomilov 《Integral Equations and Operator Theory》2014,79(3):389-447
We study the analog of semi-separable integral kernels in \({\mathcal {H}}\) of the type $$ K(x, x') = \left\{\begin{array}{ll} F_1(x) G_1(x'), \quad& a < x' < x < b,\\ F_2 (x)G_2(x'), \quad& a < x < x' < b,\end{array}\right.$$ where \({-\infty \leqslant a < b \leqslant \infty}\) , and for a.e. \({x \in (a, b)}\) , \({F_j (x) \in \mathcal{B}_2(\mathcal{H}_j, \mathcal{H})}\) and \({G_j(x) \in \mathcal {B}_2(\mathcal {H},\mathcal {H}_j)}\) such that F j (·) and G j (·) are uniformly measurable, and $$\begin{array}{ll} || F_j ( \cdot) ||_{\mathcal {B}_2(\mathcal {H}_j,\mathcal {H})} \in L^2((a, b)), ||G_j (\cdot)||_{\mathcal {B}_2(\mathcal {H},\mathcal {H}_j)} \in L^2((a, b)), \quad j=1,2, \end{array}$$ with \({\mathcal {H}}\) and \({\mathcal {H}_j}\) , j = 1, 2, complex, separable Hilbert spaces. Assuming that K(·, ·) generates a trace class operator K in \({L^2((a, b);\mathcal {H})}\) , we derive the analog of the Jost–Pais reduction theory that succeeds in proving that the Fredholm determinant \({{\rm det}_{L^2((a,b);\mathcal{H})}}\) (I ? α K), \({\alpha \in \mathbb{C}}\) , naturally reduces to appropriate Fredholm determinants in the Hilbert spaces \({\mathcal{H}}\) (and \({\mathcal{H}_1 \oplus \mathcal{H}_2}\) ). Explicit applications of this reduction theory to Schrödinger operators with suitable bounded operator-valued potentials are made. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator. 相似文献
13.
Let A be a left and right coherent ring and C A (resp., $C_{A^{\mathrm{op}}}$ ) a minimal cogenerator for right (resp., left) A-modules. We show that $\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}}$ whenever flat dim C A ?<?∞ and $\mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty$ , and that $\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty$ if and only if the finitely presented right A-modules have bounded Gorenstein dimension. 相似文献
14.
Horst Alzer 《Advances in Computational Mathematics》2010,33(3):349-379
We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ \delta_{\alpha} < \frac{ \mbox{erf} \left( x+ \mbox{erf}(y)^{\alpha}\right) +\mbox{erf}\left( y+ \mbox{erf}(x)^{\alpha}\right) } {\mbox{erf}\left( \mbox{erf}(x)+\mbox{erf}(y)\right) } < \Delta_{\alpha} $$ with the best possible bounds $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$ 相似文献
15.
Let {X n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X 1|2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x + = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function. 相似文献
16.
Elena Prestini 《Mathematische Zeitschrift》2012,271(1-2):271-291
We prove that the operator ${Tf(x,y)=\int^\pi_{-\pi}\int_{|x^{\prime}|<|y^{\prime}|} \frac{e^{iN(x,y) x^{\prime}}}{x^{\prime}}\frac{e^{iN(x,y) y^{\prime}}}{y^{\prime}}f(x-x^{\prime}, y-y^{\prime}) dx^{\prime} dy^{\prime}}$ , with ${x,y \in[0,2\pi]}$ and where the cut off ${|x^{\prime}|<|y^{\prime}|}$ is performed in a smooth and dyadic way, is bounded from L 2 to weak- ${L^{2-\epsilon}}$ , any ${\epsilon > 0 }$ , under the basic assumption that the real-valued measurable function N(x, y) is “mainly” a function of y and the additional assumption that N(x, y) is non-decreasing in x, for every y fixed. This is an extension to 2D of C. Fefferman’s proof of a.e. convergence of Fourier series of L 2 functions. 相似文献
17.
K. Sreenadh Sweta Tiwari 《NoDEA : Nonlinear Differential Equations and Applications》2013,20(6):1831-1850
Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ. 相似文献
18.
Francesca Alessio Piero Montecchiari 《NoDEA : Nonlinear Differential Equations and Applications》2013,20(3):1317-1346
This paper concerns the existence and asymptotic characterization of saddle solutions in ${\mathbb {R}^{3}}$ for semilinear elliptic equations of the form $$-\Delta u + W'(u) = 0,\quad (x, y, z) \in {\mathbb {R}^{3}} \qquad\qquad\qquad (0.1)$$ where ${W \in \mathcal{C}^{3}(\mathbb {R})}$ is a double well symmetric potential, i.e. it satisfies W(?s) = W(s) for ${s \in \mathbb {R},W(s) > 0}$ for ${s \in (-1,1)}$ , ${W(\pm 1) = 0}$ and ${W''(\pm 1) > 0}$ . Denoted with ${\theta_{2}}$ the saddle planar solution of (0.1), we show the existence of a unique solution ${\theta_{3} \in {\mathcal{C}^{2}}(\mathbb {R}^{3})}$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies ${0 < \theta_{3}(x,y,z) < 1}$ for x, y, z > 0 and ${\theta_{3}(x, y, z) \to_{z \to + \infty} \theta_{2}(x, y)}$ uniformly with respect to ${(x, y) \in \mathbb {R}^{2}}$ . 相似文献
19.
Yajun Zhou 《The Ramanujan Journal》2014,34(3):373-428
A closed-form formula is derived for the generalized Clebsch–Gordan integral \(\int_{-1}^{1} {[}P_{\nu}(x){]}^{2}P_{\nu}(-x)\,\mathrm {d}x\) , with P ν being the Legendre function of arbitrary complex degree \(\nu\in\mathbb{C}\) . The finite Hilbert transform of P ν (x)P ν (?x), ?1<x<1 is evaluated. An analytic proof is provided for a recently conjectured identity \(\int_{0}^{1}[\mathbf{K}( \sqrt{1-k^{2}} )]^{3}\,\mathrm {d}k=6\int_{0}^{1}[\mathbf{K}(k)]^{2}\mathbf{K}( \sqrt{1-k^{2}} )k\,\mathrm {d}k=[\Gamma (\frac{1}{4})]^{8}/(128\pi^{2}) \) involving complete elliptic integrals of the first kind K(k) and Euler’s gamma function Γ(z). 相似文献
20.
We consider the double Walsh orthonormal system $$\{w_m(x)w_n(y):\, m,n \in \mathbb{N}\}$$ on the unit square $\mathbb{I}^{2}$ , where {w m (x)} is the ordinary Walsh system on the unit interval $\mathbb{I}:=[0,1)$ in the Paley enumeration. Our aim is to give sufficient conditions for the absolute convergence of the double Walsh?CFourier series of a function $f \in L^{p}(\mathbb{I}^{2})$ for some 1<p?Q2. More generally, we give best possible sufficient conditions for the finiteness of the double series $$\sum_{m=1}^{\infty}\ \sum_{n=1}^{\infty} a_{mn} {|\hat{f}(m,n)|}^r,$$ where {a mn } is a given double sequence of nonnegative real numbers satisfying a mild assumption and 0<r<2. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity of?f. 相似文献