共查询到20条相似文献,搜索用时 31 毫秒
1.
Mu‐Tao Wang 《纯数学与应用数学通讯》2004,57(2):267-281
Let Ω be a bounded C2 domain in ?n and ? ?Ω → ?m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : Ω → ?m with f|?Ω = ? and with the graph of f a minimal submanifold in ?n+m. For m = 1, the Dirichlet problem was solved more than 30 years ago by Jenkins and Serrin [12] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension m. We prove that if ψ : ¯Ω → ?m satisfies 8nδ supΩ |D2ψ| + √2 sup?Ω |Dψ| < 1, then the Dirichlet problem for ψ|?Ω is solvable in smooth maps. Here δ is the diameter of Ω. Such a condition is necessary in view of an example of Lawson and Osserman [13]. In order to prove this result, we study the associated parabolic system and solve the Cauchy‐Dirichlet problem with ψ as initial data. © 2003 Wiley Periodicals, Inc. 相似文献
2.
C.-I. Martin 《Applicable analysis》2013,92(5):843-859
We show existence and uniqueness for a linearized water wave problem in a two dimensional domain G with corner, formed by two semi-axes Γ1 and Γ2 which intersect under an angle α?∈?(0,?π]. The existence and uniqueness of the solution is proved by considering an auxiliary mixed problem with Dirichlet and Neumann boundary conditions. The latter guarantees the existence of the Dirichlet to Neumann map. The water wave boundary value problem is then shown to be equivalent to an equation like vtt ?+?gΛv?=?Pt with initial conditions, where t stands for time, g is the gravitational constant, P means pressure and Λ is the Dirichlet to Neumann map. We then prove that Λ is a positive self-adjoint operator. 相似文献
3.
E. A. Volkov 《Proceedings of the Steklov Institute of Mathematics》2010,269(1):57-64
We study the Dirichlet problem for the Laplace equation in an infinite rectangular cylinder. Under the assumption that the
boundary values are continuous and bounded, we prove the existence and uniqueness of a solution to the Dirichlet problem in
the class of bounded functions that are continuous on the closed infinite cylinder. Under an additional assumption that the
boundary values are twice continuously differentiable on the faces of the infinite cylinder and are periodic in the direction
of its edges, we establish that a periodic solution of the Dirichlet problem has continuous and bounded pure second-order
derivatives on the closed infinite cylinder except its edges. We apply the grid method in order to find an approximate periodic
solution of this Dirichlet problem. Under the same conditions providing a low smoothness of the exact solution, the convergence
rate of the grid solution of the Dirichlet problem in the uniform metric is shown to be on the order of O(h
2 ln h
−1), where h is the step of a cubic grid. 相似文献
4.
We establish a connection between the absolute continuity of elliptic measure associated with a second order divergence form
operator with bounded measurable coefficients with the solvability of an end-point BMO Dirichlet problem. We show that these
two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the BMO
Dirichlet problem implies L
p
solvability for all p>p
0. 相似文献
5.
J. H. Chabrowski 《Rendiconti del Circolo Matematico di Palermo》1988,37(2):307-320
In this paper we prove the multiplicity result for the Dirichlet problems (A
s
) and (B
t
) with a boundary data inL
2
(ϖQ) and with the nonlinearity interacting with the spectrum of the elliptic operatorL. The fact that the boundary data is inL
2
leads in a natural way to the Dirichlet problem in a weighted Sobolev space. We follow methods and arguments from the recent
papers of Walter and McKenna [11] and [12]. 相似文献
6.
P. Werner 《Mathematical Methods in the Applied Sciences》1986,8(1):134-156
We consider the Dirichlet problem for the reduced wave equation ΔUx + x2Ux = 0 in a two-dimensional exterior domain with boundary C, where C consists of a finite number of smooth closed curves C1,…,Cm. The question of interest is the behavior of Ux as ? → 0. We show that U converges to the solution of the corresponding exterior Dirichlet problem of potential theory if the boundary data converge to a limit uniformly on C. This generalizes a well-known result of R. C. MacCamy for the case m = 1. 相似文献
7.
Nikolai N. Tarkhanov 《Mathematische Nachrichten》1994,169(1):309-323
For an arbitrary differential operator P of order p on an open set X ? R n, the Laplacian is defined by Δ = P*P. It is an elliptic differential operator of order 2p provided the symbol mapping of P is injective. Let O be a relatively compact domain in X with smooth boundary, and Bj(j = 0…,p — 1) be a Dirichlet system of order p ? 1 on ?O. By {Cj} we denote the Dirichlet system on ?O adjoint for {Bj} with respect to the Green formula for P. The Hardy space H2(O) is defined to consist of all the solutions f of Δf = 0 in O of finite order of growth near the boundary such that the weak boundary values of the expression {Bjf} and {Cj(Pf)} belong to the Lebesgue space L2(?O). Then the Dirichlet problem consists of finding a solution f ? H2(O) with prescribed data {Bjf} on ?O. We develop the classical Fischer-Riesz equations method to derive a solvability condition of the Dirichlet problem as well as an approximate formula for solutions. 相似文献
8.
J.A. MONTERO 《Natural Resource Modeling》2001,14(1):139-146
ABSTRACT. In this work we consider the increase in benefit for a control problem when the size of domain increases. Our control problem involves the study of the profitability of a biological growing species whose growth is confined to a bounded domain Ω? RN and is modeled by a logistic elliptic equation with different boundary conditions (Dirichlet or Neumann). The payoff-cost functional considered, J, is of quadratic type. We prove that, under Dirichlet boundary conditions, the optimal benefit (sup J) increases when the domain ? increases. This is not true under Neumann boundary conditions. 相似文献
9.
T. Ya. Ershova 《Moscow University Computational Mathematics and Cybernetics》2009,33(4):171-180
The Dirichlet problem for a singularly perturbed reaction-diffusion equation in a square is solved with the help of the classic
five-point difference scheme and a grid that is the tensor product of 1D Bakhvalov grids. Without imposing additional matching
conditions in the corners of the domain, it is shown that the grid solution to the problem has the accuracy O(N
−2) in the norm L
∞
h
, where N is the number of grid nodes along each direction. The accuracy is uniform with respect to a small parameter. A simulation
confirms the theoretical prediction. 相似文献
10.
We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F (Hess u) = 0 on a smoothly bounded domain Ω ? ?n. In our approach the equation is replaced by a subset F ? Sym2(?n) of the symmetric n × n matrices with ?F ? { F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric “F‐convexity” assumption on the boundary ?Ω. We also study the topological structure of F‐convex domains and prove a theorem of Andreotti‐Frankel type. Two key ingredients in the analysis are the use of “subaffine functions” and “Dirichlet duality.” Associated to F is a Dirichlet dual set F? that gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F? is F, and in the analysis the roles of F and F? are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: all branches of the homogeneous Monge‐Ampère equation over ?, ?, and ?; equations appearing naturally in calibrated geometry, Lagrangian geometry, and p‐convex Riemannian geometry; and all branches of the special Lagrangian potential equation. © 2008 Wiley Periodicals, Inc. 相似文献
11.
M. K. Jain R. K. Jain R. K. Mohanty 《Numerical Methods for Partial Differential Equations》1992,8(1):21-31
We attempt to obtain a two-level implicit finite difference scheme using nine spatial grid points of O(k2 + kh2 + h4) for solving the 2D nonlinear parabolic partial differential equation v1uxx + v2uyy = f(x, y, t, u, ux, uy, u1) where v1 and v2 are positive constants, with Dirichlet boundary conditions. The method, when applied to a linear diffusion-convection problem, is shown to be unconditionally stable. Computational efficiency and the results of numerical experiments are discussed. 相似文献
12.
The uniqueness of solutions to two inverse Sturm–Liouville problems using three spectra is proven, based on the uniqueness of the solution-pair to an overdetermined Goursat–Cauchy boundary value problem. We discuss the uniqueness of the potential for a Dirichlet boundary condition at an arbitrary interior node, and for a Robin boundary condition at an arbitrary interior node, whereas at the exterior nodes we have Dirichlet boundary conditions in both situations. Here we are particularly concerned with potential functions that are L2(0,a). 相似文献
13.
Andreas Fleige 《Integral Equations and Operator Theory》2008,60(2):237-246
For the Sturm-Liouville eigenvalue problem − f′′ = λrf on [−1, 1] with Dirichlet boundary conditions and with an indefinite weight function r changing its sign at 0 we discuss the question whether the eigenfunctions form a Riesz basis of the Hilbert space L
2
|r|[− 1, 1]. So far a number of sufficient conditions on r for the Riesz basis property are known. However, a sufficient and necessary condition is only known in the special case of
an odd weight function r. We shall here give a generalization of this sufficient and necessary condition for certain generally non-odd weight functions
satisfying an additional assumption.
相似文献
14.
Constantin Bacuta Victor Nistor Ludmil T. Zikatanov 《Numerical Functional Analysis & Optimization》2013,34(6):613-639
ABSTRACT Let 𝒯 k be a sequence of triangulations of a polyhedron Ω ? ? n and let S k be the associated finite element space of continuous, piecewise polynomials of degree m. Let u k ∈ S k be the finite element approximation of the solution u of a second-order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence S k ensures optimal rates of convergence for the sequence u k . The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet–Neumann boundary conditions on a polygon. 相似文献
15.
We wish to solve the heat equation ut=Δu-qu in Id×(0,T), where I is the unit interval and T is a maximum time value, subject to homogeneous Dirichlet boundary conditions and to initial conditions u(·,0)=f over Id. We show that this problem is intractable if f belongs to standard Sobolev spaces, even if we have complete information about q. However, if f and q belong to a reproducing kernel Hilbert space with finite-order weights, we can show that the problem is tractable, and can actually be strongly tractable. 相似文献
16.
Iain Johnstone 《Probability Theory and Related Fields》1986,71(2):231-269
Summary We establish a connection between admissible simultaneous estimation and recurrence of reversible Markov chains on
+
p
. Specifically, to each generalized Bayes estimator of the mean of a vector of p independent Poisson variables for a weighted quadratic loss, we associate a variational problem and a reversible birth and death chain on
+
p
. The variational problem is closely related to the Dirichlet principle for reversible chains studied recently by Griffeath, Liggett and Lyons. Under side conditions, admissibility of the estimator is equivalent to zero infimal energy in the variational problem and to recurrence of the Markov chain. This yields analytic and probabilistic criteria for inadmissibility which are applied to establish a broad class of results and previous conjectures.Research supported by an Australian National University Scholarship and A.D. White Fellowship at Cornell University and by NSF at Mathematical Sciences Research Institute, Berkeley and at Stanford 相似文献
17.
We formulate a modified nodal cubic spline collocation scheme for the solution of the biharmonic Dirichlet problem on the
unit square. We prove existence and uniqueness of a solution of the scheme and show how the scheme can be solved on an N × N uniform partition of the square at a cost O(N
2 log2 N + mN
2) using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. We demonstrate numerically that m proportional to log2
N guarantees the desired convergence rates. Numerical results indicate the fourth order accuracy of the approximations in the
global maximum norm and the fourth order accuracy of the approximations to the first order partial derivatives at the partition
nodes.
相似文献
18.
Salim A. Messaoudi 《Mathematische Nachrichten》2001,231(1):105-111
In this paper we consider the nonlinearly damped semilinear wave equation utt – Δu + aut |ut|m – 2 = bu|u|p – 2 associated with initial and Dirichlet boundary conditions. We prove that any strong solution, with negative initial energy, blows up in finite time if p > m. This result improves an earlier one in [2]. 相似文献
19.
The concept of complex Dirichlet forms
c
resp. operators L
c
in complex weighted L
2-spaces is introduced. Perturbations of classical Dirichlet forms by forms associated with complex first-order differential operators provide examples of complex Dirichlet forms.Complex Dirichlet operators L
c
are unitarily equivalent with (a family of) Schrödinger operators with electromagnetic potentials.To
c
there is associated a pair of real-valued non symmetric Dirichlet forms on the corresponding real weighted L
2-spaces, which in turn are associated with (non-symmetric) diffusion processes.Results by Stannat on non symmetric Dirichlet forms and their perturbations can be used for discussing the essential self-adjointness of L
c
.New closability criteria for (perturbation of) non symmetric Dirichlet forms are obtained. 相似文献
20.
We study the unique solvability of the Dirichlet problem for the biharmonic equation in the exterior of a compact set under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight |x|a. Depending on the value of the parameter a,a we prove uniqueness theorems or present exact formulas for the dimension of the solution space of the Dirichlet problem. 相似文献