共查询到20条相似文献,搜索用时 452 毫秒
1.
Raúl Ferreira Mayte Pérez-Llanos 《NoDEA : Nonlinear Differential Equations and Applications》2016,23(2):14
The purpose of this work is the analysis of the solutions to the following problems related to the fractional p-Laplacian in a Lipschitzian bounded domain \({\Omega \subset \mathbb{R}^N}\),where \({\alpha\in(0,1)}\) and the exponent p goes to infinity. In particular we will analyze the cases: We show the convergence of the solutions to certain limit as \({p\to\infty}\) and identify the limit equation. In both cases, the limit problem is closely related to the Infinity Fractional Laplacian:where
相似文献
$$\left\{\begin{array}{lll}-\int_{\mathbb{R}^N}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{\alpha p}}\;dy=f(x,u)\;\;&x\in \Omega,\\ u=g(x) &x\in\mathbb{R}^N\setminus \Omega,\end{array}\right.$$
- (i)\({f=f(x).}\)
- (ii)\({f=f(u)=|u|^{\theta(p)-1} u \, {\rm with} \, 0 < \theta(p) < p -1 \, {\rm and} \, \lim_{p\to\infty}\frac{\theta(p)}{p-1}=\Theta < 1 \, {\rm with} \, g \geq 0.}\)
$$\mathcal{L}_\infty v(x)=\mathcal{L}_\infty^+ v(x)+\mathcal{L}_\infty^- v(x),$$
$$\mathcal{L}_\infty^+ v(x)=\sup_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}, \quad \mathcal{L}_\infty^- v(x)=\inf_{y\in\mathbb{R}^N}\frac{v(y)-v(x)}{|y-x|^\alpha}.$$
2.
We find conditions under which the system of root functions of the operator is a Riesz basis in L 2[0, 1].
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$$L_y = l[y] = ay'(x) + y'(1 - x) + p_1 (x)y(x) + p_2 (x)y(1 - x),x \in [0,1],U_1 (y) = \int\limits_0^1 {y(t)d\sigma (t) = 0,} $$
3.
Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated towhere \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)
相似文献
$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$
$$\mathcal{L}f + qf = 0 $$
4.
Choonkil Park Dong Yun Shin Jung Rye Lee 《Journal of Fixed Point Theory and Applications》2016,18(3):569-586
In this paper, we solve the additive \({\rho}\)-functional equations where \({\rho}\) is a fixed non-Archimedean number or a fixed real or complex number with \({\rho \neq 1}\). Using the fixed point method, we prove the Hyers–Ulam stability of the above additive \({\rho}\)-functional equations in non-Archimedean Banach spaces and in Banach spaces.
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$$\begin{aligned} f(x+y)-f(x)-f(y)= & {} \rho(2f(\frac{x+y}{2})-f(x)-f(y)), \\ 2f(\frac{x+y}{2})-f(x)-f(y)= & {} \rho(f(x+y)-f(x)-f(y)), \end{aligned}$$
5.
A. Bahraini G. Askari M. Eshaghi Gordji R. Gholami 《Journal of Fixed Point Theory and Applications》2018,20(2):89
Recently Eshaghi et al. introduced orthogonal sets and proved the real generalization of the Banach fixed point theorem on these sets. In this paper, we prove the real generalization of Diaz–Margolis fixed point theorem on orthogonal sets. By using this fixed point theorem, we study the stability of orthogonally \(*\)-m-homomorphisms on Lie \(C^*\)-algebras associated with the following functional equation: for each \(m=1,2,3,4.\). Moreover, we establish the hyperstability of these functional equations by suitable control functions.
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$$\begin{aligned} \begin{aligned}&f(2x+y)+f(2x-y)+(m-1)(m-2)(m-3)f(y)\\&\quad =2^{m-2}[f(x+y)+f(x-y)+6f(x)]. \end{aligned} \end{aligned}$$
6.
We present the conditions under which every nonoscillator solution x(t) of the forced fractional differential equation where \(y(t)= ( {a(t) ( {{x}'(t)} )^{\delta }})^{\prime },c_0 =\frac{y(c)}{\Gamma (1)}= y(c)\), is a real constant which satisfies It is shown that the technique can be applied to some related fractional differential equations. Examples are inserted to illustrate the relevance of the obtained results.
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$$\begin{aligned} ^{\mathrm{C}}D_{\mathrm{c}}^{\alpha } y ( t ) = e ( t ) +f ( {t, x ( t )} ), c > 1,\alpha \in ( {0,1} ), \quad \mathrm{{and}} \,\, \delta \ge 1, \end{aligned}$$
$$\begin{aligned} |x(t)|=O\left( {t^{1/\delta }e^{t}\int _{\mathrm{c}}^{t} {a^{-1/\delta }} (s)\mathrm{d}s} \right) , \quad t \rightarrow \infty \end{aligned}$$
7.
We establish the weak Harnack estimates for locally bounded sub- and superquasiminimizers u of with f subject to the general structural conditions where p : Ω →] 1, ∞[ is a variable exponent. The upper weak Harnack estimate is proved under the assumption that b, g ∈ L t (Ω) for some t > n/p ?, and the lower weak Harnack estimate is proved under the stronger assumption that b, g ∈ L ∞(Ω). As applications we obtain the Harnack inequality for quasiminimizers and the fact that locally bounded quasisuperminimizers have Lebesgue points everywhere whenever b, g ∈ L ∞(Ω). Throughout the paper, we make the standard assumption that the variable exponent p is logarithmically Hölder-continuous.
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$${\int}_{\Omega} f(x,u,\nabla u)\,dx $$
$$|z|^{p(x)} - b(x)|y|^{p(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} + b(x)|y|^{p(x)} + g(x), $$
8.
Huixue Lao 《Acta Appl Math》2010,110(3):1127-1136
Let L(sym j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that and
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$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$
$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$
9.
Let X be a nonempty set containing at least two elements and let \({\circ :X^2\to X}\) be a symmetric binary operation. Furthermore, let A, B, C be real parameters and let \({f,g:X\to\mathbb{R}_+}\) be unknown functions. We investigate the functional equation
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$f(x\circ y)[Ag(y)-Bg(x)]=(A+C)f(x)g(y)-(B+C)f(y)g(x)\quad {\rm for\,\,all}\,x,y \in X.$
10.
Let \(n\ge 2\) and \(g_{\lambda }^{*}\) be the well-known high-dimensional Littlewood–Paley function which was defined and studied by E. M. Stein, where \(P_tf(y,t)=p_t*f(y)\), \(p_t(y)=t^{-n}p(y/t)\), and \(p(x) = (1+|x|^2)^{-(n+1)/2}\), \(\nabla =(\frac{\partial }{\partial y_1},\ldots ,\frac{\partial }{\partial y_n},\frac{\partial }{\partial t})\). In this paper, we give a characterization of two-weight norm inequality for \(g_{\lambda }^{*}\)-function. We show that \(\big \Vert g_{\lambda }^{*}(f \sigma ) \big \Vert _{L^2(w)} \lesssim \big \Vert f \big \Vert _{L^2(\sigma )}\) if and only if the two-weight Muckenhoupt \(A_2\) condition holds, and a testing condition holds: where \(\widehat{Q}\) is the Carleson box over Q and \((w, \sigma )\) is a pair of weights. We actually prove this characterization for \(g_{\lambda }^{*}\)-function associated with more general fractional Poisson kernel \(p^\alpha (x) = (1+|x|^2)^{-{(n+\alpha )}/{2}}\). Moreover, the corresponding results for intrinsic \(g_{\lambda }^*\)-function are also presented.
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$$\begin{aligned} g_{\lambda }^{*}(f)(x) =\bigg (\iint _{\mathbb {R}^{n+1}_{+}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda } |\nabla P_tf(y,t)|^2 \frac{\mathrm{d}y \mathrm{d}t}{t^{n-1}}\bigg )^{1/2}, \ \quad \lambda > 1, \end{aligned}$$
$$\begin{aligned} \sup _{Q : \text {cubes}~\mathrm{in} \ {\mathbb {R}^n}} \frac{1}{\sigma (Q)} \int _{{\mathbb {R}^n}} \iint _{\widehat{Q}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda }|\nabla P_t(\mathbf {1}_Q \sigma )(y,t)|^2 \frac{w \mathrm{d}x \mathrm{d}t}{t^{n-1}} \mathrm{d}y < \infty , \end{aligned}$$
11.
We develop the method of lower and upper solutions for the fourth-order differential equation which models the stationary states of the deflection of an elastic beam, whose both ends simply supported under the condition \(0<k_1<k_2<x_1^2\approx 4.11585\), where \(x_1\) is the first positive solution of the equation \(x\cos (x)+\sin (x)=0\). The main tools are Schauder fixed point theorem and the Elias inequality.
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$$\begin{aligned}&y^{(4)}(x)+(k_1+k_2) y''(x)+k_1k_2 y(x)=f(x,y(x)), \ \ \ \ x\in (0,1),\\&y(0) = y(1) = y''(0) = y''(1) = 0\\ \end{aligned}$$
12.
A. H. Babayan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2007,42(4):177-183
A boundary value problem for the Bitsadze equation in the interior of the unit disc is considered. It is proved that the problem is Noetherian and its index is calculated, and solvability conditions for the non-homogeneous problem are proposed. Some solutions of the homogeneous problem are explicitely found.
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$\frac{{\partial ^2 }}{{\partial \bar z^2 }}u(x,y) \equiv \frac{1}{4}\left( {\frac{\partial }{{\partial x}} + i\frac{\partial }{{\partial y}}} \right)^2 u(x,y) = 0$
13.
Prondanai Kaskasem Chakkrid Klin-eam Yeol Je Cho 《Journal of Fixed Point Theory and Applications》2018,20(2):76
In this paper, we prove the Hyers–Ulam–Rassias stability of the generalized Cauchy–Jensen set-valued functional equation defined by for all \(x,y,z \in X\) and \(\alpha \ge 2\) on a Banach space by using the fixed point alternative theorem.
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$$\begin{aligned} \alpha f\left( \frac{x+y}{\alpha } + z\right) = f(x) \oplus f(y)\oplus \alpha f(z) \end{aligned}$$
14.
We prove Hölder estimates for viscosity solutions of a class of possibly degenerate and singular equations modelled by the fractional p-Laplace equationwhere \({s \in (0,1)}\) and \({p > 2}\) or \({1/(1-s) < p < 2}\). Our results also apply for inhomogeneous equations with more general kernels, when p and s are allowed to vary with x, without any regularity assumption on p and s. This complements and extends some of the recently obtained Hölder estimates for weak solutions.
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$$PV {\int_{\mathbb{R}^{n}}\frac{|u(x)-u(x + y)|^{p-2}(u(x) - u(x + y))}{|y|^{n+sp}}\,dy = 0},$$
15.
We establish conditions under which three-dimensional relaxational systems of the form where 0 ≤ ε ? 1, |µ| ? 1, and f, g ∈ C ∞, exhibit the so-called blue sky catastrophe [the appearance of a stable relaxational cycle whose period and length tend to infinity as µ tends to some critical value µ*(ε), µ*(0) = 0].
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$$\dot x = f(x,y,\mu ),\varepsilon \dot y = g(x,y),x = (x_1 ,x_2 ) \in \mathbb{R}^2 ,y \in \mathbb{R},$$
16.
Frédéric Bernicot 《Mathematische Annalen》2011,351(1):1-49
In this work, we study some non-smooth bilinear analogues of linear Littlewood–Paley square functions on the real line. We prove boundedness-properties in Lebesgue spaces for them. Let us consider the functions \({\phi_{n}}\) satisfying \({\widehat{\phi_n}(\xi)={\bf 1}_{[n,n+1]}(\xi)}\) and define the bilinear operator \({S_n(f,g)(x):=\int f(x+y)g(x-y) \phi_n(y) dy}\) . These bilinear operators are closely related to the bilinear Hilbert transforms. Then for exponents \({p,q,r'\in[2,\infty)}\) satisfying \({\frac{1}{p}+\frac{1}{q}=\frac{1}{r}}\) , we prove that
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$\left\| \left( \sum_{n\in \mathbb{Z}}\left|S_n(f,g) \right|^2 \right)^{1/2}\right\|_{L^{r}(\mathbb{R})}\lesssim \|f\|_{L^p(\mathbb{R})}\|g\|_{L^q(\mathbb{R})}.$
17.
In this article, we establish some new criteria for the oscillation of fourth-order nonlinear delay differential equations of the formprovided that the second-order equationis nonoscillatory or oscillatory.
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$$(r_2(t)(r_1(t)(y''(t))^\alpha)')' + p(t)(y''(t))^\alpha + q(t)f(y(g(t))) = 0$$
$$(r_2(t)z'(t))') + \frac{p(t)}{r_1(t)}z(t) = 0$$
18.
Jay Taylor 《Israel Journal of Mathematics》2017,217(1):435-475
In this paper we establish the following estimate: where ω ≥ 0, 0 < ε < 1 and Φ(t) = t(1 + log+(t)). This inequality relies upon the following sharp L p estimate: where 1 < p < ∞, ω ≥ 0 and 0 < δ < 1. As a consequence we recover the following estimate essentially contained in [18]: We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.
相似文献
$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant \frac{{{c_T}}}{{{\varepsilon ^2}}}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right){M_{L{{\left( {\log L} \right)}^{1 + \varepsilon }}}}} \omega \left( x \right)dx$$
$${\left\| {\left[ {b,T} \right]f} \right\|_{{L^p}\left( \omega \right)}} \leqslant {c_T}{\left( {p'} \right)^2}{p^2}{\left( {\frac{{p - 1}}{\delta }} \right)^{\frac{1}{{p'}}}}{\left\| b \right\|_{BMO}}{\left\| f \right\|_{{L^p}\left( {{M_{L{{\left( {{{\log }_L}} \right)}^{2p - 1 + {\delta ^\omega }}}}}} \right)}}$$
$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant {c_T}{\left[ \omega \right]_{{A_\infty }}}{\left( {1 + {{\log }^ + }{{\left[ \omega \right]}_{{A_\infty }}}} \right)^2}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)M} \omega \left( x \right)dx.$$
19.
Classification of positive solutions to an integral system with the poly-harmonic extension operator
In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R~+_n:{u(x)=c_n,a∫_?R_+~n(x_n~(1-a_v)(y)/|x-y|~(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n~(1-a_uθ)(x)/|x-y|~(n-a))dx,y∈ ?R_+~n,where n 2, 2-n a 1, κ, θ 0. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen(2014). The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case κ =n-2+a/n-a,θ =n+2-a/ n-2+a. Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system. 相似文献
20.
Given a continuous strictly monotone function \(\varphi \) defined on an open real interval I and a probability measure \(\mu \) on the Borel subsets of [0, 1], the Makó–Páles mean is defined by Under some conditions on the functions \(\varphi \) and \(\psi \) defined on I, the quotient mean is given by In this paper, we study some invariance of the quotient mean with respect to Makó–Páles means.
相似文献
$$\begin{aligned} {\mathcal {M}}_{\varphi ,\mu }(x,y):=\varphi ^{-1}\left( \int ^1_0\varphi (tx+(1-t)y)\, d\mu (t)\right) ,\quad x,y\in I. \end{aligned}$$
$$\begin{aligned} Q_{\varphi ,\psi }(x,y):=\left( \frac{\varphi }{\psi }\right) ^{-1}\left( \frac{\varphi (x)}{\psi (y)}\right) , \quad x,y\in I. \end{aligned}$$