共查询到20条相似文献,搜索用时 921 毫秒
1.
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. 相似文献
2.
Let
W ì \BbbR2\Omega \subset \Bbb{R}^2 denote a bounded domain whose boundary
?W\partial \Omega is Lipschitz and contains a segment G0\Gamma_0 representing
the austenite-twinned martensite interface. We prove
infu ? W(W) òW j(?u(x,y))dxdy=0\displaystyle{\inf_{{u\in \cal W}(\Omega)} \int_\Omega \varphi(\nabla
u(x,y))dxdy=0} 相似文献
3.
T. V. Malovichko 《Ukrainian Mathematical Journal》2009,61(3):435-456
We prove an analog of the Girsanov theorem for the stochastic differential equations with interaction
dz( u,t ) = a( z( u,t ),mt )dt + ò\mathbbR f( z( u,t ) - p )W( dp,dt ), dz\left( {u,t} \right) = a\left( {z\left( {u,t} \right),{\mu_t}} \right)dt + \int\limits_\mathbb{R} {f\left( {z\left( {u,t} \right) - p} \right)W\left( {dp,dt} \right)}, 相似文献
4.
K. Kh. Boimatov 《Journal of Mathematical Sciences》2002,108(4):543-573
Let
W ì \mathbbRn \Omega \subset \mathbb{R}^n
be an open set and l(x)
| u |p,l = ( òW lp (x)| u(x) |p dx )1/p \text (1 \leqslant p < + ¥\text),\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}} 相似文献
5.
V. M. Kaplitsky 《Journal of Mathematical Sciences》2010,165(4):455-462
We study sufficient conditions for exponential decay at infinity for eigenfunctions of a class of integral equations in unbounded
domains in ℝ
n
. We consider integral operators K whose kernels have the form
|