Search results for "boundary"

A short proof of a theorem of Ahlfors

1988

In [1] Ahlfors proved that the Weil-Petersson metric of the Teichmfiller space is K~hler. A new proof was given by Fischer and Tromba [5] in a purely Riemannian setting of Teichmfiller theory [3]. We shall provide yet another proof that slightly shortens the argument of Fischer and Tromba. We begin with a brief review of the Fischer-Tromba approach to Teichmiiller theory [3-5]. Let M be a compact connected oriented 2-dimensional manifold without boundary.

Rank structured approximation method for quasi--periodic elliptic problems

2016

We consider an iteration method for solving an elliptic type boundary value problem $\mathcal{A} u=f$, where a positive definite operator $\mathcal{A}$ is generated by a quasi--periodic structure with rapidly changing coefficients (typical period is characterized by a small parameter $\epsilon$) . The method is based on using a simpler operator $\mathcal{A}_0$ (inversion of $\mathcal{A}_0$ is much simpler than inversion of $\mathcal{A}$), which can be viewed as a preconditioner for $\mathcal{A}$. We prove contraction of the iteration method and establish explicit estimates of the contraction factor $q$. Certainly the value of $q$ depends on the difference between $\mathcal{A}$ and $\mathcal…

Resolvent Estimates for Non-Selfadjoint Operators via Semigroups

2009

We consider a non-selfadjoint h-pseudodifferential operator P in the semiclassical limit (h → 0). If p is the leading symbol, then under suitable assumptions about the behavior of p at infinity, we know that the resolvent (z–P)–1 is uniformly bounded for z in any compact set not intersecting the closure of the range of p. Under a subellipticity condition, we show that the resolvent extends locally inside the range up to a distance \(\mathcal{O}(1)((h\ln \frac{1}{h})^{k/(k + 1)} )\) from certain boundary points, where \(k \in \{ 2,4, \ldots \} \). This is a slight improvement of a result by Dencker, Zworski, and the author, and it was recently obtained by W. Bordeaux Montrieux in a model sit…

On the solutions to 1-Laplacian equation with L1 data

2009

AbstractIn the present paper we study the behaviour, as p goes to 1, of the renormalized solutions to the problems(0.1){−div(|∇up|p−2∇up)=finΩ,up=0on∂Ω, where p>1, Ω is a bounded open set of RN (N⩾2) with Lipschitz boundary and f belongs to L1(Ω). We prove that these renormalized solutions pointwise converge, up to “subsequences,” to a function u. With a suitable definition of solution we also prove that u is a solution to a “limit problem.” Moreover we analyze the situation occurring when more regular data f are considered.

On the boundary spectrum of dominatedC o-Semigroups

1989

Functional Calculus and Fredholm Criteria for Boundary Value Problems on Noncompact Manifolds

1992

A Boutet de Monvel type calculus is developed for boundary value problems on (possibly) noncompact manifolds. It is based on a class of weighted symbols and Sobolev spaces. If the underlying manifold is compact, one recovers the standard calculus. The following is proven:

Quasihyperbolic boundary conditions and Poincaré domains

2002

We prove that a domain in ${\Bbb R}^n$ whose quasihyperbolic metric satisfies a logarithmic growth condition with coefficient $\beta\le 1$ is a (q,p)-\Poincare domain for all p and q satisfying $p\in[1,\infty)\cap(n-n\beta,n)$ and $q\in[p,\beta p^*)$ , where $p^*=np/(n-p)$ denotes the Sobolev conjugate exponent. An elementary example shows that the given ranges for p and q are sharp. The proof makes use of estimates for a variational capacity. When p=2 we give an application to the solvability of the Neumann problem on domains with irregular boundaries. We also discuss the relationship between this growth condition on the quasihyperbolic metric and the s-John condition.

Completeness number of families of subsets of convergence spaces

2016

International audience; Compactoid and compact families generalize both convergent filters and compact sets. This concept turned out to be useful in various quests, like Scott topologies, triquotient maps and extensions of the Choquet active boundary theorem.The completeness number of a family in a convergence space is the least cardinality of collections of covers for which the family becomes complete. 0-completeness amounts to compactness, finite completeness to relative local compactness and countable completeness to Čech completeness. Countably conditional countable completeness amounts to pseudocompleteness of Oxtoby. Conversely, each completeness class of families can be represented a…

On Different Type Solutions of Boundary Value Problems

2016

We consider boundary value problems of the type x'' = f(t, x, x'), (∗) x(a) = A, x(b) = B. A solution ξ(t) of the above BVP is said to be of type i if a solution y(t) of the respective equation of variations y'' = fx(t, ξ(t), ξ' (t))y + fx' (t, ξ(t), ξ' (t))y' , y(a) = 0, y' (a) = 1, has exactly i zeros in the interval (a, b) and y(b) 6= 0. Suppose there exist two solutions x1(t) and x2(t) of the BVP. We study properties of the set S of all solutions x(t) of the equation (∗) such that x(a) = A, x'1(a) ≤ x' (a) ≤ x'2(a) provided that solutions extend to the interval [a, b].

Energy localization in a nonlinear discrete system

1996

International audience; We show that, in the weak amplitude and slow time limits, the discrete equations describing the dynamics of a one-dimensional lattice can be reduced to a modified Ablowitz-Ladik equation. The stability of a continuous wave solution is then investigated without and with periodic boundary conditions; Energy localization via modulational instability is predicted. Our numerical simulations, performed on a cyclic system of six oscillators, agree with our theoretical predictions.