共查询到20条相似文献,搜索用时 453 毫秒
1.
We consider degenerate parabolic equations of the form $$\left. \begin{array}{ll}\,\,\, \partial_t u = \Delta_\lambda u + f(u) \\u|_{\partial\Omega} = 0, u|_{t=0} = u_0\end{array}\right.$$ in a bounded domain ${\Omega\subset\mathbb{R}^N}$ , where Δλ is a subelliptic operator of the type $$\quad \Delta_\lambda:= \sum_{i=1}^{N} \partial_{x_i}(\lambda_{i}^{2} \partial_{x_i}),\qquad \lambda = (\lambda_1,\ldots, \lambda_N).$$ We prove global existence of solutions and characterize their longtime behavior. In particular, we show the existence and finite fractal dimension of the global attractor of the generated semigroup and the convergence of solutions to an equilibrium solution when time tends to infinity. 相似文献
2.
We extend classical volume formulas for ellipsoids and zonoids to p-sums of segments $${vol}\left( {\sum\limits_{i=1}^m { \oplus_p } [ -x_i ,x_i ]} \right)^{1/n} \sim_{c_p} n^{ - \frac{1}{{p'}}} \left( {\sum\limits_{card(I) = n} {|\det (x_i)_i |^p}} \right)^{\frac{1}{{pn}}}$$ where x1,...,xm are m vectors in $\mathbb{R}^n ,\frac{1}{p} + \frac{1}{{p\prime }} = 1$ . According to the definition of Firey, the Minkowski p-sum of segments is given by $$\sum\limits_{i = 1}^m { \oplus _p [ - x_{i,} x_i ]} = \left\{ {\sum\limits_{i = 1}^m {\alpha _i } x_i \left| {\left( {\sum\limits_{i = 1}^m {|\alpha _i |^{p^\prime } } } \right)} \right.^{\frac{1}{{p^\prime }}} \leqslant 1} \right\}.$$ We describe related geometric properties of the Lewis maps associated to classical operator norms. 相似文献
3.
We consider integral functionals in which the density has growth p i with respect to ${\frac{\partial u}{\partial x_i}}$ , like in $$\int\limits_{\Omega}\left( \left| \frac{\partial u}{\partial x_1}(x) \right|^{p_1} + \left|\frac{\partial u}{\partial x_2}(x)\right|^{p_2} + \cdots + \left|\frac{\partial u}{\partial x_n}(x) \right|^{p_n} \right) dx.$$ We show that higher integrability of the boundary datum forces minimizer to be more integrable. 相似文献
4.
Yifeng Yu 《Calculus of Variations and Partial Differential Equations》2010,37(1-2):63-74
The Dirichlet problem $$ \left\{ \begin{array}{l}\Delta _\infty u - |Du|^2 = 0 \quad {\rm on} \, \Omega \subset {{\mathbb R}^n} \\ u|\partial \Omega = g \\\end{array} \right. $$ might have many solutions, where ${\Delta_{\infty}u=\sum_{1\leq i,j\leq n}u_{x_i}u_{x_j}u_{x_ix_j}}$ . In this paper, we prove that the maximal solution is the unique absolute minimizer for ${H(p,z)={\frac{1}{2}}|p|^2-z}$ from calculus of variations in L ∞ and the minimal solution is the continuum value function from the “tug-of-war” game. We will also characterize graphes of solutions which are neither an absolute minimizer nor a value function. A remaining interesting question is how to interpret those intermediate solutions. Most of our approaches are based on an idea of Barles–Busca (Commun Partial Differ Equ 26(11–12):2323–2337, 2001). 相似文献
5.
I. E. Egorov 《Mathematical Notes》1978,23(3):211-217
Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W 2 1 (QT) and W 2 2 (QT). 相似文献
6.
D. Leviatan 《Israel Journal of Mathematics》1970,8(2):138-146
The purpose of this paper is to prove that for a large set of absolute Hausdorff and quasi-Hausdorff methods the condition $$\sum\limits_{k = 1}^\infty {\left| {\lambda _n a_n - \lambda _{n - 1} a_{n - 1} } \right|< } \infty $$ is a Tauberian condition, i.e., its fulfillment together with the absolute summability of \(\sum\limits_{n = 0}^\infty {a_n } \) tos implies that \(\sum\limits_{n = 0}^\infty {\left| {a_n } \right|}< \infty \) and \(\sum\limits_{n = 0}^\infty {a_n } = s.\) a n =s. 相似文献
7.
For quasilinear doubly degenerate parabolic equations it has been possible to establish the existence of nonnegative generalized solutions to the first boundary-value problem that are Hölder continuous up to the boundary of the cylinder QT=Ωx(0, T]. A typical example of an admissible equation is the equation of Newtonian polytropic filtration $$\frac{{\partial u}}{{\partial t}} - \frac{\partial }{{\partial x_i }}\left\{ {a_0 \left| u \right|^l \left| {u_x } \right|^{m - 2} u_{x_i } } \right\} = 0, a_0 > 0,l > 0,m > 2.$$ . 相似文献
8.
Chang-Shou Lin 《Milan Journal of Mathematics》2011,79(1):81-94
In this article, we discuss the recent work of Lin and Zhang on the Liouville system of mean field equations: $$\Delta{u}_i+\sum_{j}a_{ij}\rho_{j} ({\frac{{h_j}e^{u_{j}}}{\int_{M}{h_{j}e^{u_{j}}}}-{\frac{1}{|M|}}})=0\,\, \quad{\rm on}\, M,$$ where M is a compact Riemann surface and |M| is the area, or $$\Delta{u}_i+\sum_{j}a_{ij}\rho_{j} \frac{{h_j}e^{u_{j}}}{\int_{\Omega}{h_{j}e^{u_{j}}}}=0\,\, \quad{\rm in}\, \Omega,$$ $${u_j}=0,\,\, \quad{\rm on}\, \partial\Omega, $$ where ?? is a bounded domain in ${\mathbb{R}^2}$ . Among other things, we completely determine the set of non-critical parameters and derive a degree counting formula for these systems. 相似文献
9.
10.
П. В. Задерей 《Analysis Mathematica》1989,15(3):245-262
Assume that the coefficients of the series $$\mathop \sum \limits_{k \in N^m } a_k \mathop \Pi \limits_{i = 1}^m \sin k_i x_i $$ satisfy the following conditions: a) ak → 0 for k1 + k2 + ...+km →∞, b) \(\delta _{B,G}^M (a) = \mathop {\mathop \sum \limits_{k_i = 1}^\infty }\limits_{i \in B} \mathop {\mathop \sum \limits_{k_j = 2}^\infty }\limits_{j \in G} \mathop {\mathop \sum \limits_{k_v = 0}^\infty }\limits_{v \in M\backslash (B \cup G)} \mathop \Pi \limits_{i \in B} \frac{1}{{k_i }}|\mathop \sum \limits_{I_j = 1}^{[k_j /2]} (\nabla _{l_G }^G (\Delta _1^{M\backslash B} a_k ))\mathop \Pi \limits_{j \in G} l_j^{ - 1} |< \infty ,\) for ∨B?M, ∨G?M,B∩G, where M={1,2, ...,m}, $$\begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\Delta _1^j a_k = a_k - a_{k_{M\backslash \{ j\} } ,k_{j + 1} } ,\Delta _1^B a_k = \Delta _1^{B\backslash \{ j\} } (\Delta _1^j a_k ), \hfill \\ \Delta _{l_j }^j a_k = a_{k_{M\backslash \{ j\} } ,k_j - l_j } - a_{k_{M\backslash \{ j\} } ,k_j + l_j } ,\nabla _{l_G }^G a_k = \nabla _{l_{G\backslash \{ j\} } }^{G\backslash \{ j\} } (\nabla _{l_j }^j a_k ). \hfill \\ \end{gathered} $$ Then for all n∈Nm the following asymptotic equation is valid: $$\mathop \smallint \limits_{{\rm T}_{\pi /(2n + 1)}^m } |\mathop \sum \limits_{k \in N^m } a_k \mathop \Pi \limits_{i \in M} \sin k_i x_i |dx = \mathop \sum \limits_{k = 1}^n \left| {a_k } \right|\mathop \Pi \limits_{i \in M} k^{ - 1} + O(\mathop {\mathop \sum \limits_{B,{\mathbf{ }}G \subset M} }\limits_{B \ne M} \delta _{B,G}^M (a)).$$ Here \(T_{\pi /(2n + 1)}^m = \left\{ {x = (x1,x2,...,xm):\pi /(2n + 1) \leqq xi \leqq \pi ;i = \overline {1,m} } \right\}\) . In the one-dimensional case such an equation was proved by S. A. Teljakovskii. 相似文献
11.
12.
V. Totik 《Analysis Mathematica》1979,5(4):287-299
Пустьf 2π-периодическ ая суммируемая функц ия, as k (x) еë сумма Фурье порядк аk. В связи с известным ре зультатом Зигмунда о сильной суммируемости мы уст анавливаем, что если λn→∞, то сущес твует такая функцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _{2n} } } \right\}^{1/\lambda _{2n} } = \infty .$$ Отсюда, в частности, вы текает, что если λn?∞, т о существует такая фун кцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } } \right\}^{1/\lambda _n } = \infty .$$ Пусть, далее, ω-модуль н епрерывности и $$H^\omega = \{ f:\parallel f(x + h) - f(x)\parallel _c \leqq K_f \omega (h)\} .$$ . Мы доказываем, что есл и λ n ?∞, то необходимым и достаточным условие м для того, чтобы для всехf∈H ω выполнялос ь соотношение $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _n } } \right\}^{1/\lambda _n } = 0(x \in [0;2\pi ])$$ является условие $$\omega \left( {\frac{1}{n}} \right) = o\left( {\frac{1}{{\log n}} + \frac{1}{{\lambda _n }}} \right).$$ Это же условие необхо димо и достаточно для того, чтобы выполнялось соотнош ение $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } = 0(f \in H^\omega ,x \in [0;2\pi ]).$$ 相似文献
13.
Suppose that $${g\left( n \right)}$$ is an additive real-valued function, W(N) = 4+ $$\mathop {\min }\limits_\lambda $$ ( λ2 + $$\sum\limits_{p < N} {\frac{1}{2}} $$ min (1, ( g(p) - λlog p)2), E(N) = 4+1 $$\sum\limits_{\mathop {p < N,}\limits_{g(p) \ne 0} } {\frac{1}{p}.} $$ In this paper, we prove the existence of constants C1, C2 such that the following inequalities hold: $\mathop {\sup }\limits_a \geqslant \left| {\left\{ {n, m, k: m, k \in \mathbb{Z},n \in \mathbb{N},n + m^2 + k^2 } \right.} \right. = \left. {\left. {N,{\text{ }}g(n) \in [a,a + 1)} \right\}} \right| \leqslant \frac{{C_1 N}}{{\sqrt {W\left( N \right)} }},$ $\mathop {\sup }\limits_a \geqslant \left| {\left\{ {n, m, k: m, k \in \mathbb{Z},n \in \mathbb{N},n + m^2 + k^2 } \right.} \right. = \left. {\left. {N,{\text{ }}g(n) = a} \right\}} \right| \leqslant \frac{{C_2 N}}{{\sqrt {E\left( N \right)} }},$ . The obtained estimates are order-sharp. 相似文献
14.
Ryozo Yokoyama 《Analysis Mathematica》1983,9(1):79-84
Пусть {Xj} - строго стац ионарная последоват ельностьс ?перемешиванием, EXj-Q,E¦-X j¦r<∞ для некоторогоr>2. Положим \(S_n = \mathop \sum \limits_{j = 1}^n X_j \) . Ибрагимов (1962) доказал, что если приn →∞, то 1 $$\mathop {\lim }\limits_{n \to \infty } P\{ S_n /\sigma _n< x\} = (2\pi )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \mathop \smallint \limits_{ - \infty }^x e^{{{ - u^2 } \mathord{\left/ {\vphantom {{ - u^2 } 2}} \right. \kern-\nulldelimiterspace} 2}} du.$$ В работе установлено, что при указанных выш е условиях в этой центральной пр едельной теореме имеет место т акже и сходимостьr-ых абсолютных моментов, т.е. если σ n 2 →∞ приn→ ∞, то $$\mathop {\lim }\limits_{n \to \infty } E|S_n /\sigma _n |^r = (2\pi )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \mathop \smallint \limits_{ - \infty }^{ + \infty } |u|^r e^{ - u^2 /2} du.$$ Этот результат обобщ ает один более ранний результат автора (1980 г.). 相似文献
15.
B. B. Manna P. N. Srikanth 《NoDEA : Nonlinear Differential Equations and Applications》2014,21(6):915-927
Let \({A=\{x\in \mathbb{R}^{2m}: 0 < a < |x| < b\}}\) be an annulus. We consider the following singularly perturbed elliptic problem on A $$\left\{\begin{array}{lll}-\varepsilon ^2{\Delta u} + |x|^{\eta}u =|x|^{\eta}u^p, \quad {\rm in} A,\\ u > 0, \quad \quad \quad \quad \quad \quad \quad {\rm in} A, \\ u=0, \quad \quad \quad \quad \quad \quad \quad {\rm on}\partial A,\end{array}\right. $$ where \({1 < p < \frac{m+3}{m-1}}\) . We shall prove the existence of a positive solution \({u_\epsilon }\) which concentrates on two different orthogonal spheres of dimension (m?1) as \({\varepsilon \to 0}\) . We achieve this by studying a reduced problem on an annular domain in \({\mathbb{R}^{m+1}}\) and analysing the profile of a two point concentrating solution in this domain. 相似文献
16.
Daomin Cao Shusen Yan 《Calculus of Variations and Partial Differential Equations》2010,38(3-4):471-501
In this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth and a Hardy potential: $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2}u+a u\quad {\rm in}\;\Omega,\quad u=0 \quad {\rm on}\; \partial\Omega,\qquad (*)$$ under the assumptions that N ≥ 7, ${\mu\in \left[0,\frac{(N-2)^2}4-4\right)}$ and a > 0, where ${2^{\ast}=\frac{2N}{N-2}}$ , and Ω is an open bounded domain in ${\mathbb{R}^N}$ which contains the origin. To achieve this goal, we consider the following perturbed problem of (*), which is of subcritical growth, $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2-\varepsilon_n}u+au \quad {\rm in}\,\Omega, \quad u=0 \quad {\rm on}\;\partial\Omega,\qquad(\ast\ast)_n$$ where ${\varepsilon_{n} > 0}$ is small and ${\varepsilon_n \to 0}$ as n → + ∞. By the critical point theory for the even functionals, for each fixed ${\varepsilon_{n} > 0}$ small, (**) n has a sequence of solutions ${u_{k,\varepsilon_{n}} \in H^{1}_{0}(\Omega)}$ . We obtain the existence of infinitely many solutions for (*) by showing that as n → ∞, ${u_{k,\varepsilon_{n}}}$ converges strongly in ${H^{1}_{0}(\Omega)}$ to u k , which must be a solution of (*). Such a convergence is obtained by applying a local Pohozaev identity to exclude the possibility of the concentration of ${\{u_{k,\varepsilon_n}\}}$ . 相似文献
17.
Felipe J. Zó 《Analysis Mathematica》1978,4(2):153-158
Пустьl 1 иl 2 — неотрицательные убывающие функции на (0, ∞). Допустим, что $$\int\limits_0^\infty {S^{n_i - 1} l_i (S)\left( {1 + \log + \frac{1}{{S^{n_i } l_i (S)}}} \right)dS}< \infty ,$$ , гдеn 1 иn 2 — натуральные числа. Тогда для каждой функции \(f \in L^1 (R^{n_1 + n_2 } )\) при почти всех (x0, у0) мы имеем $$\mathop {\lim }\limits_{\lambda \to \infty } \lambda ^{n_1 + n_2 } \int\limits_{R^{n_1 } } {\int\limits_{R^{n_2 } } {l_1 } } (\lambda |x|)l_2 (\lambda |y|)f(x_0 - x,y_0 - y)dx dy = f(x_0 ,y_0 )\int\limits_{R^{n_1 } } {\int\limits_{R^{n_2 } } {l_i (|x|)l_2 } } (|y|)dx dy.$$ 相似文献
18.
M. A. Chahkiev 《Analysis Mathematica》2008,34(3):177-185
Let $ \mathcal{P}_n $ denote the set of algebraic polynomials of degree n with the real coefficients. Stein and Wpainger [1] proved that $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \leqslant C_n , $$ where C n depends only on n. Later A. Carbery, S. Wainger and J. Wright (according to a communication obtained from I. R. Parissis), and Parissis [3] obtained the following sharp order estimate $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \sim \ln n. $$ . Now let $ \mathcal{T}_n $ denote the set of trigonometric polynomials $$ t(x) = \frac{{a_0 }} {2} + \sum\limits_{k = 1}^n {(a_k coskx + b_k sinkx)} $$ with real coefficients a k , b k . The main result of the paper is that $$ \mathop {\sup }\limits_{t( \cdot ) \in \mathcal{T}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{it(x)} }} {x}dx} } \right| \leqslant C_n , $$ with an effective bound on C n . Besides, an analog of a lemma, due to I. M. Vinogradov, is established, concerning the estimate of the measure of the set, where a polynomial is small, via the coefficients of the polynomial. 相似文献
19.
J. Prestin 《Analysis Mathematica》1987,13(3):251-259
ПустьM m,α - множество 2π-п ериодических функци йf с конечной нормой $$||f||_{p,m,\alpha } = \sum\limits_{k = 1}^m {||f^{(k)} ||_{_p } + \mathop {\sup }\limits_{h \ne 0} |h|^{ - \alpha } ||} f^{(m)} (o + h) - f^{(m)} (o)||_{p,} $$ где1 ≦ p ≦ ∞, 0≦α≦1. Рассмотр им средние Bалле Пуссе на $$(\sigma _{n,1} f)(x) = \frac{1}{\pi }\int\limits_0^{2x} {f(u)K_{n,1} (x - u)du} $$ и $$(L_{n,1} f)(x) = \frac{2}{{2n + 1}}\sum\limits_{k = 1}^{2n} {f(x_k )K_{n,1} } (x - x_k ),$$ де0≦l≦n и x k=2kπ/(2n+1). В работе по лучены оценки для вел ичин \(||f - \sigma _{n,1} f||_{p,r,\beta } \) и $$||f - L_{n,1} f||_{p,r,\beta } (r + \beta \leqq m + \alpha ).$$ 相似文献
20.
In the present paper, we consider the following stochastic control problem: to minimize the average expected total cost $$J(x,u) = \mathop {\lim \inf }\limits_{T \to \infty } (1/T)E_x^u \int_0^T {\left[ {\phi (\xi _t ) + |u_t (\xi )|} \right]} dt,$$ 〈subject to $$d\xi _t = u_1 (\xi )dt + dw_t , \xi _0 = x, |u| \leqslant 1,$$ (w t) a Wiener process, with all measurable functions on the past of the state process {ξ s ;s≤t} and bounded by unity, admissible as controls. It is proved that, under very mild conditions on the running cost function φ(·), the optimal law is of the form $$\begin{gathered} u_t^* (\xi ) = - sign\xi _t , |\xi _t | > b, \hfill \\ u_t^* (\xi ) = 0, |\xi _t | > b. \hfill \\ \end{gathered} $$ The cutoff pointb and the performance rate of the optimal lawu* are simultaneously determined in terms of the function φ(·) through a simple system of integrotranscendental equations. 相似文献