共查询到20条相似文献,搜索用时 78 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.
In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L¥loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem
inf[`(u)] + W1,¥0(W) òW (ID(?u) + g(u)) dx, (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1) 相似文献
3.
We show that any entropy solution u of a convection diffusion equation ?t u + div F(u)-Df(u) = b{\partial_t u + {\rm div} F(u)-\Delta\phi(u) =b} in Ω × (0, T) belongs to C([0,T),L1loc(W)){C([0,T),L^1_{\rm loc}({\Omega}))} . The proof does not use the uniqueness of the solution. 相似文献
4.
Mikhail Perepelitsa 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,114(1):267-276
We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain
W = \mathbbR3+{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density
ρ
0 is bounded and the magnitude of the initial velocity u
0 is suitably restricted in the norm ||?{r0(·)}u0(·)||L2(W) + ||?u0(·)||L2(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}. 相似文献
5.
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
6.
M. Fuchs 《Journal of Mathematical Sciences》2010,167(3):418-434
We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields
u:\mathbbR2 é W? \mathbbR2 u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation
in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form
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