Results 301 to 310 of about 634,419 (341)

WEAK SOLUTIONS FOR WEAK SINGULARITIES

International Journal of Modern Physics A, 2002
We revisit the problem of the development of singularities in the gravitational collapse of an inhomogeneous dust sphere. As shown by Yodzis et al1, naked singularities may occur at finite radius where shells of dust cross one another. These singularities are gravitationally weak 2, and it has been claimed that at these singularities, the metric may ...
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Weak Solutions of SDEs

2015
So far, we have focussed on solutions of SDEs where we are simply given a filtration, and with it the Brownian motion W and the random measure μ. We then construct the solution to our equation ( 17.2). In essence, we have used no properties of the filtration except the fact that W and μ are adapted.
Robert J. Elliott, Robert J. Elliott, Samuel N. Cohen   +2 more
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Boundedness of weak solutions

1993
Let u be a weak solution of equations of the type of (1.1) of Chap. II in Ω T We will establish local and global bounds for u in. Ω T . Global bounds depend on the data prescribed on the parabolic boundary of Ω T . Local bounds are given in terms of local integral norms of u. Consider the cubes K ρ ⊂ K 2ρ .
Emmanuele DiBenedetto, Emmanuele DiBenedetto   +1 more
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Regularity of Weak Solutions

1998
It is shown that under appropriate ellipticity assumptions, weak solutions of partial differential equations (PDEs) are smooth. This applies in particular to the Laplace equation for harmonic functions, thereby justifying Dirichlet’s principle introduced in the previous paragraph.
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Extension of weak solutions

1997
Let 0 < T < +∞, A and B be c.n.o. in H. In the previous chapter, we have answered the following question: what must be A and B for each bounded weak solution of equation (1) on [0,T) to have a limit in H as t → T?
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Weak Solutions for Semi-Martingales

Canadian Journal of Mathematics, 1981
The fundamental theorem of this paper is stated in Section 8. In this theorem, the stochastic differential equation dX = a(X)dZ is studied when Z is a *-dominated (cf. [15]) Banach space valued process and a is a predictable functional which is continuous for the uniform norm.For such an equation, the existence of a “weak solution” is stated; actually,
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Weak Solutions for Hyperbolic Equations

2018
In principle, in this chapter, we will study the wave equation, which constitutes the prototype of the hyperbolic equations. Let \(\Omega \) be an open set from \(\mathrm{I}\!\mathrm{R}^n\) and T a real number \(T>0\). Then, the Cauchy problem, associated with the wave equation, consists of $$\begin{aligned}&\frac{\partial ^2u}{\partial t^2}(t,x)- \
Andreas Öchsner, Marin Marin
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Weak Solutions and Related Topics

2014
In the previous chapter we discussed strong solutions, which for their definition require smooth boundaries and smooth dependence on t. This section is devoted to weak solutions, more precisely variational inequality weak solutions, which are closely related to potential theory (they can be viewed as instances of partial balayage) and also to ...
Björn Gustafsson, Razvan Teodorescu, Alexander Vasil’ev   +2 more
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Stability and stabilization of weak solutions [PDF]

, 1997
Definition 10.1. Let A and B he c.n.o. in H. Equation (1) is said to be stable on R+ if for every weak solution y(t) of (1) on R+: \( \mathop {\sup }\limits_{t \in R_ + } \left\| {y(t)} \right\| < + \infty \)\( \left( {i.e.,\exists C_y < + \infty :\left\| {y(t)} \right\| \leqslant C_y ,\forall t \in R_ + } \right). \)
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Weak Solutions of Classical Problems

2018
The Sobolev spaces, which will be defined in the following, are spaces on which weak solutions can be defined (in a sense to be defined later) for classical boundary value problems.
Andreas Öchsner, Marin Marin
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