共查询到20条相似文献,搜索用时 46 毫秒
1.
Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions
Hans-Christoph Grunau Frédéric Robert 《Archive for Rational Mechanics and Analysis》2010,195(3):865-898
In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped
plate boundary value problem, neither a maximum principle nor a comparison principle or—equivalently—a positivity preserving
property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem
from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for
bounded smooth domains
W ì \mathbbRn{\Omega \subset\mathbb{R}^n} , the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided
n\geqq 3{n\geqq 3} . Moreover, the biharmonic Green’s function in balls
B ì \mathbbRn{B\subset\mathbb{R}^n} under Dirichlet (that is, clamped) boundary conditions is known explicitly and is positive. It has been known for some time
that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for
n\geqq 3{n\geqq 3} . 相似文献
2.
Hammadi Abidi Guilong Gui Ping Zhang 《Archive for Rational Mechanics and Analysis》2012,204(1):189-230
We prove the local wellposedness of three-dimensional incompressible inhomogeneous Navier–Stokes equations with initial data
in the critical Besov spaces, without assumptions of small density variation. Furthermore, if the initial velocity field is
small enough in the critical Besov space
[(B)\dot]1/22,1(\mathbbR3){\dot B^{1/2}_{2,1}(\mathbb{R}^3)} , this system has a unique global solution. 相似文献
3.
Mauro Fabrizio Claudio Giorgi Vittorino Pata 《Archive for Rational Mechanics and Analysis》2010,198(1):189-232
We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing convex function ${G : \mathbb{R}^+ \to \mathbb{R}^+}We discuss a novel approach to the mathematical analysis of equations with memory, based on a new notion of state. This is the initial configuration of the system at time t = 0 which can be unambiguously determined by the knowledge of the dynamics for positive times. As a model, for a nonincreasing
convex function
G : \mathbbR+ ? \mathbbR+{G : \mathbb{R}^+ \to \mathbb{R}^+} such that
$G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0 $G(0) = \lim_{s\to 0}G(s) > \lim_{s\to\infty}G(s) >0 相似文献
4.
We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several
properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary
and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application
to the Navier–Stokes equations, our approach unifies several results known in the literature, partly with different proofs.
Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in
\mathbbR3{\mathbb{R}}^{3} and irregular domains in
\mathbbRn{\mathbb{R}}^{n}. 相似文献
5.
S. I. Maksymenko 《Nonlinear Oscillations》2009,12(4):522-542
Let
D2 ì \mathbbR2 {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin
O ì \mathbbR2 O \subset {\mathbb{R}^2} and let F be a smooth vector field such that O is the unique singular point of F, and all other orbits of F are simple closed curves wrapping once around O: Thus, topologically, O is a “center” singularity. Let D+ (F) {\mathcal{D}^{+} }(F) be the group of all diffeomorphisms of D
2 that preserve the orientation and orbits of F. Recently, the author described the homotopy type of D+ (F) {\mathcal{D}^{+} }(F) under the assumption that the 1-jet j
1
F(O) of F at O is nondegenerate. In this paper, the degenerate case j
1
F(O) is considered. Under additional “nondegeneracy assumptions” on F, the path components of D+ (F) {\mathcal{D}^{+} }(F) with respect to distinct weak topologies are described. These conditions imply that, for each h ? D+ (F) h \in {\mathcal{D}^{+} }(F) , its path component in D+ (F) {\mathcal{D}^{+} }(F) is uniquely determined by the 1-jet of h at O. 相似文献
6.
We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}
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