共查询到20条相似文献,搜索用时 15 毫秒
1.
Tiziana Giorgi Robert Smits 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,59(4):600-618
We consider the principal eigenvalue λ
1Ω(α) corresponding to Δu = λ (α) u in on ∂Ω, with α a fixed real, and a C
0,1 bounded domain. If α > 0 and small, we derive bounds for λ
1Ω(α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature.
We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains
which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable.
Robert Smits: This author was partially supported by a grant of the National Security Agency, grant #H98230-05-1-0060. 相似文献
2.
It is proved that if Ω ⊂ Rn {R^n} is a bounded Lipschitz domain, then the inequality
|| u ||1 \leqslant c(n)\textdiam( W)òW | eD(u) | {\left\| u \right\|_1} \leqslant c(n){\text{diam}}\left( \Omega \right)\int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} is valid for functions of bounded deformation vanishing on ∂Ω. Here eD(u) {\varepsilon^D}(u) denotes the deviatoric part of the symmetric gradient and òW | eD(u) | \int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} stands for the total variation of the tensor-valued measure eD(u) {\varepsilon^D}(u) . Further results concern possible extensions of this Poincaré-type inequality. Bibliography: 27 titles. 相似文献
3.
Reinhard Farwig Christian Komo 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):303-321
Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain
W ì \mathbbR3{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u
0, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u
0 and f or on the solution u itself such as Serrin’s condition || u ||Ls(0,T; Lq(W)) < ¥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with
2 < s < ¥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math
J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g.
u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||Ls¢(t-d,t; Lq(W)){\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy
\frac12|| u(t) ||22{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}. 相似文献
4.
Let Ω and Π be two finitely connected hyperbolic domains in the complex plane
\Bbb C{\Bbb C}
and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities
Cn(W,P) := supf ? A(W,P)supz ? W\frac|f(n)(z)| R(f(z),P)n! (R(z,W))nC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n}
are finite for all
n ? \Bbb N{n \in {\Bbb N}}
if and only if ∂Ω and ∂Π do not contain isolated points. 相似文献
5.
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
6.
Christopher Hammond 《Mathematische Zeitschrift》2010,266(2):285-288
Let Ω be a domain in ${\mathbb{C}^{2}}Let Ω be a domain in
\mathbbC2{\mathbb{C}^{2}}, and let
p: [(W)\tilde]? \mathbbC2{\pi: \tilde{\Omega}\rightarrow \mathbb{C}^{2}} be its envelope of holomorphy. Also let W¢=p([(W)\tilde]){\Omega'=\pi(\tilde{\Omega})} with
i: W\hookrightarrow W¢{i: \Omega \hookrightarrow \Omega'} the inclusion. We prove the following: if the induced map on fundamental groups i*:p1(W) ? p1(W¢){i_{*}:\pi_{1}(\Omega) \rightarrow \pi_{1}(\Omega')} is a surjection, and if π is a covering map, then Ω has a schlicht envelope of holomorphy. We then relate this to earlier
work of Fornaess and Zame. 相似文献
7.
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
8.
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
9.
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
10.
We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}
11.
Let
W í \Bbb C\Omega \subseteq {\Bbb C}
be a simply connected domain in
\Bbb C{\Bbb C}
, such that
{¥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}]
is connected. If g is holomorphic in Ω and every derivative of g extends continuously on
[`(W)]\bar{\Omega}
, then we write g ∈ A∞ (Ω). For g ∈ A∞ (Ω) and
z ? [`(W)]\zeta \in \bar{\Omega}
we denote
SN (g,z)(z) = ?Nl=0\fracg(l) (z)l ! (z-z)lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l
. We prove the existence of a function f ∈ A∞(Ω), such that the following hold:
12.
The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}
13.
Inspired by the work of Marcellini and Papi, we consider local minima u: ℝ
n
⊃ Ω → ℝ
M
of variational integrals òW h( | ?u | ) \int\limits_\Omega {h\left( {\left| {\nabla u} \right|} \right)} dx and prove interior gradient bounds under rather general assumptions on h provided that u is locally bounded. Our requirements on the density h do not involve the dimension n. Bibliography: 18 titles. 相似文献
14.
N. M. Ivochkina 《Journal of Mathematical Sciences》2010,170(4):496-509
We construct an a priori estimate of the seminorm
á uxx
ña, [`(W)] {\left\langle {{u_{xx}}} \right\rangle_{\alpha, \bar{\Omega }}} for solutions to the problem
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