共查询到20条相似文献,搜索用时 15 毫秒
1.
2.
Summary. An elliptic obstacle problem
is approximated by piecewise linear finite elements
with numerical integration on the penalty and forcing terms. This leads
to diagonal nonlinearities and thereby to a practical scheme.
Optimal error estimates in the maximum norm are derived.
The proof is based on constructing suitable super and subsolutions that
exploit the special structure of the penalization, and using quite
precise pointwise error estimates for an associated linear
elliptic problem with quadrature via the discrete maximum principle.
Received March 19, 1993 相似文献
3.
Given a connected open set
and a function w ∈LN/p(Ω) if 1 < p < N and w ∈Lr (Ω) for some r ∈(1, ∞) if p ≧ N, with
we prove that the positive principal eigenvalue of the problem
is unique and simple. This improves previous works all of which assumed w in a smaller space than LN/p (Ω) to ensure that Harnack’s inequality holds. Our proof does not rely on Harnack’s inequality, which may fail in our case.
Received: 18 March 2005; revised: 7 April 2005 相似文献
4.
In one space dimension we address the homogenization of the spectral problem for a singularly perturbed diffusion equation
in a periodic medium. Denoting by ε the period, the diffusion coefficient is scaled as ε2. The domain is made of two purely periodic media separated by an interface. Depending on the connection between the two cell
spectral equations, three different situations arise when ε goes to zero. First, there is a global homogenized problem as
in the case without an interface. Second, the limit is made of two homogenized problems with a Dirichlet boundary condition
on the interface. Third, there is an exponential localization near the interface of the first eigenfunction.
Received: January 10, 2001; in final form: July 9, 2001?Published online: June 11, 2002 相似文献
5.
A cascadic multigrid algorithm for semilinear elliptic problems 总被引:12,自引:0,他引:12
Gisela Timmermann 《Numerische Mathematik》2000,86(4):717-731
Summary. We propose a cascadic multigrid algorithm for a semilinear elliptic problem. The nonlinear equations arising from linear
finite element discretizations are solved by Newton's method. Given an approximate solution on the coarsest grid on each finer
grid we perform exactly one Newton step taking the approximate solution from the previous grid as initial guess. The Newton
systems are solved iteratively by an appropriate smoothing method. We prove that the algorithm yields an approximate solution
within the discretization error on the finest grid provided that the start approximation is sufficiently accurate and that
the initial grid size is sufficiently small. Moreover, we show that the method has multigrid complexity.
Received February 12, 1998 / Revised version received July 22, 1999 / Published online June 8, 2000 相似文献
6.
Frank Müller 《Calculus of Variations and Partial Differential Equations》2002,15(2):257-288
This paper deals with systems , , where the right hand side is a -valued, real analytic function. We prove that a solution of such a system can be continued across a straight line segment , if one prescribe certain nonlinear, mixed boundary conditions on , which are assumed to be real analytic too. This continuation will be constructed by solving certain hyperbolic initial boundary
value problems, generalizing an idea of H. Lewy. We apply this result to surfaces of prescribed mean curvature and to minimal
surfaces in Riemannian manifolds spanned into a regular Jordan curve : Supposing analyticity of all data, we show that both types of surfaces can be continued across .
Received: 29 December 2000 / Accepted: 11 July 2001 / Published online: 29 April 2002 相似文献
7.
Giovanna Cerami Mónica Clapp 《Calculus of Variations and Partial Differential Equations》2007,30(3):353-367
We prove the existence of a sign changing solution to the semilinear elliptic problem , in an exterior domain Ω having finite symmetries. 相似文献
8.
We prove the existence of solutions of nonlinear elliptic equations with first-order terms having “natural growth” with respect
to the gradient. The assumptions on the source terms lead to the existence of possibly unbounded solutions (though with exponential
integrability). The domain Ω is allowed to have infinite Lebesgue measure.
Received: April 13, 2001; in final form: September 29, 2001?Published online: July 9, 2002 相似文献
9.
Ralf Kornhuber 《Numerische Mathematik》1996,72(4):481-499
Summary.
We derive globally convergent multigrid methods
for discrete elliptic
variational inequalities of the second kind
as obtained from
the approximation of related continuous
problems by piecewise linear finite elements.
The coarse grid corrections are computed
from certain obstacle problems.
The actual constraints are fixed by the
preceding nonlinear fine grid smoothing.
This new approach allows the implementation
as a classical V-cycle and preserves
the usual multigrid efficiency.
We give estimates
for the asymptotic convergence rates.
The numerical results indicate a significant improvement
as compared with previous multigrid approaches.
Received
March 26, 1994 / Revised version received September 22, 1994 相似文献
10.
Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems 总被引:1,自引:0,他引:1
M. F. Betta F. Chiacchio A. Ferone 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(1):37-52
We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is
with Dirichlet boundary condition, where γ is the normalized Gaussian function in
. To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian
measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality
with respect to Gaussian measure.
Partially supported by GMAMPA - INDAM, Progetto “Proprietà analitico geometriche di soluzioni di equazioni ellittiche e paraboliche”. 相似文献
11.
Zhaoli Liu Jingxian Sun 《Calculus of Variations and Partial Differential Equations》2002,14(3):319-327
This paper concerns the existence of four (or six) solutions of semilinear elliptic boundary value problems provided that
two disorderly solutions are known. The results are obtained under very generic conditions.
Received: 26 August 2000 / Accepted: 23 February 2001 / Published online: 23 July 2001 相似文献
12.
Adimurthi Jacques Giacomoni 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(1):1-20
This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem
We consider a function h which is smooth and changes sign. 相似文献
13.
14.
Christian Bär 《Inventiones Mathematicae》1999,138(1):183-202
Consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over
an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution
is contained in a countable union of smooth (n−2)-dimensional submanifolds. Hence it is countably (n−2)-rectifiable and its Hausdorff dimension is at most n−2. Moreover, it has locally finite (n−2)-dimensional Hausdorff measure. We show by example that every real number between 0 and n−2 actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic
equations of second order.
Oblatum 22-V-1998 & 26-III-1999 / Published online: 10 June 1999 相似文献
15.
We consider local minimizers a domain in , of the variational integral with integrand f of upper (lower) growth rate q (s). We show using a lemma due to Frehse and Seregin that u has H?lder continuous first derivatives provided that .
Received: 2 October 2001 / Accepted: 25 October 2001 / Published online: 28 February 2002 相似文献
16.
Giovanni Cimatti 《Annali di Matematica Pura ed Applicata》2002,181(2):181-212
The stability of the stationary solution of the thermistor as a circuit element is studied using a Liapunov functional and
the Hale–LaSalle invariance principle. The asymptotic stability of a class of periodic solutions is also considered.
Received: November 22, 1999; in final form: May 23, 2001?Published online: May 29, 2002 相似文献
17.
Summary. In this paper we study the numerical behaviour of elliptic
problems in which a small parameter is involved and an example
concerning the computation of elastic arches is analyzed using this
mathematical framework. At first, the statements of the problem and its
Galerkin approximations are defined and an asymptotic
analysis is performed. Then we give general conditions ensuring that
a numerical scheme will converge uniformly with respect to the small
parameter. Finally we study an example in
computation of arches working in linear elasticity conditions. We build one
finite element scheme giving a locking behaviour, and another one
which does not.
Revised version received October 25, 1993 相似文献
18.
Boundary regularity for nonlinear elliptic systems 总被引:3,自引:0,他引:3
J.F. Grotowski 《Calculus of Variations and Partial Differential Equations》2002,15(3):353-388
We consider questions of boundary regularity for solutions of certain systems of second-order nonlinear elliptic equations.
We obtain a general criterion for a weak solution to be regular in the neighbourhood of a given boundary point. The proof
yields directly the optimal regularity for the solution in this neighbourhood. This result is new for the situation under
consideration (general nonlinear second order systems in divergence form, with inhomogeneity obeying the natural growth conditions).
Received: 6 July 2001 / Accepted: 27 September 2001 / Published online: 28 February 2002 相似文献
19.
Olivier Guibé 《Annali di Matematica Pura ed Applicata》2002,180(4):441-449
We give a partial uniqueness result concerning comparable renormalized solutions of the nonlinear elliptic problem -div(a(x,Du))=μ in Ω, u=0 on ∂Ω, where μ is a Radon measure with bounded variation on Ω.
Received: December 27, 2000 Published online: December 19, 2001 相似文献
20.
We prove the existence of an unbounded sequence of solutions for an elliptic equation of the form \({-\Delta u=\lambda u + a(x)g(u)+f(x), u\in H^1_0(\Omega)}\), where \({\lambda \in \mathbb{R}, g(\cdot)}\) is subcritical and superlinear at infinity, and a(x) changes sign in Ω; moreover, g( ? s) = ? g(s) \({\forall s}\). The proof uses Rabinowitz’s perturbation method applied to a suitably truncated problem; subsequent energy and Morse index estimates allow us to recover the original problem. We consider the case of \({\Omega\subset \mathbb{R}^N}\) bounded as well as \({\Omega=\mathbb{R}^N, \, N\geqslant 3}\). 相似文献