共查询到20条相似文献,搜索用时 31 毫秒
1.
Abstract In this paper we deal with the Dirichlet problem for the Laplace equation in a plane exterior domain Ω with a Lipschitz boundary. We prove that, if the boundary datum a is square summable, then the problem admits a solution which tends to a in the sense of nontangential convergence, is unique in a suitable function class and vanishes at infinity as r–k if and only if a satisfies k compatibility conditions, which we are able to explicit when Ω is the exterior of an ellipse.
Keywords: Dirichlet problem, Asymptotic behavior, Potential theory
Mathematics Subject Classification (2000): 31A05, 31A10 相似文献
2.
Andrea Aspri Elena Beretta Corrado Mascia 《Mathematical Methods in the Applied Sciences》2016,39(10):2415-2430
In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half‐space with a cavity C. Zero normal derivative is assumed at the boundary of the half‐space; differently, at ?C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ?C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half‐space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g, which describes a constant pressure at ?C, we recover a simplified representation based on a polarization tensor. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
3.
An inverse problem for a family of two parameters time fractional diffusion equations with nonlocal boundary conditions
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The determination of a space‐dependent source term along with the solution for a 1‐dimensional time fractional diffusion equation with nonlocal boundary conditions involving a parameter β>0 is considered. The fractional derivative is generalization of the Riemann‐Liouville and Caputo fractional derivatives usually known as Hilfer fractional derivative. We proved existence and uniqueness results for the solution of the inverse problem while over‐specified datum at 2 different time is given. The over‐specified datum at 2 time allows us to avoid initial condition in terms of fractional integral associated with Hilfer fractional derivative. 相似文献
4.
A counter-example to the boundary regularity of solutions to elliptic quasilinear systems 总被引:1,自引:0,他引:1
Mariano Giaquinta 《manuscripta mathematica》1978,24(2):217-220
It is shown that solutions to the Dirichlet problem for quasilinear elliptic systems in a domain ofR
n n3 with smooth boundary datum can be singular at the boundary. 相似文献
5.
In this work, we study the Cauchy problem of the nonlinear spatially homogeneous Landau equation with hard potentials in a close-to-equilibrium framework. We prove that the solution to the Cauchy problem with the initial datum in L2 enjoys an analytic regularizing effect, and the evolution of the analytic radius is the same as that of heat equations.
相似文献6.
Netra Khanal Jiahong Wu Juan-Ming Yuan Bing-Yu Zhang 《Journal of Nonlinear Science》2010,20(3):341-360
Spatially periodic complex-valued solutions of the Burgers and KdV–Burgers equations are studied in this paper. It is shown
that for any sufficiently large time T, there exists an explicit initial datum such that its corresponding solution of the Burgers equation blows up at T. In addition, the global convergence and regularity of series solutions is established for initial data satisfying mild conditions. 相似文献
7.
P. de Mottoni A. Schiaffino K. P. Hadeler 《Mathematical Methods in the Applied Sciences》1981,3(1):11-20
We consider the problem where a and f are 1-periodic in t, a is positive, f satisfies appropriate decreasing conditions; smoothness of a, f, ?Ω is also assumed. Denote by λ0 the principal eigenvalue of Δ with zero Dirichlet boundary conditions, and define . We prove: (a) if ε ≤ 0, then no non-negative periodic solution exists but zero, and any solution with continuous non-negative initial datum converges to zero uniformly as t → ∞; (b) if ε > 0, then a unique non trivial non-negative 1-periodic solution u* exists, and any solution with continuous, non-negative not identically zero initial datum approaches uniformly u* as t → ∞. 相似文献
8.
Summary We study existence, uniqueness and regularity of the strict, classical and strong solution u C([0,T],E) of the non-autonomous evolution equation u(t)–A(t)u(t)= f(t), with the initial datum, u(0)=x, in a Banach space E, where {A(t)} is a family of infinitesimal generators of analytic semi-groups whose domains are constant in t and possibly not dense in E. We prove necessary and sufficient conditions for existence and Hölder regularity of the solutions and their first derivative. 相似文献
9.
A collisionless plasma is modelled by the Vlasov–Maxwell system. In the presence of very large velocities, relativistic corrections are meaningful. When magnetic effects are ignored this formally becomes the relativistic Vlasov–Poisson equation. The initial datum for the phase space density ƒ0(x, v) is assumed to be sufficiently smooth, non‐negative and cylindrically symmetric. If the (two‐dimensional) angular momentum is bounded away from zero on the support of ƒ0(x, v), it is shown that a smooth solution to the Cauchy problem exists for all times. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
10.
P. de Mottoni E. Santi K.-P. Hadeler 《Mathematical Methods in the Applied Sciences》1986,8(1):458-477
We present an algorithm for approximating the solution of the degenerate diffusion problem ut = (?(u))xx in (0,1) × R+ (with zero Dirichlet boundary conditions, and nonnegative initial datum u0), where ?(u) = min {ku1} for some ? > 0. The algorithm also provides an approximation for the interface curves which represent the boundary of the Mushy Region ?? = {(x, t): ? (u(x, t)) = 1}. The convergence of the algorithm is proved. 相似文献
11.
Jrgen Bang-Jensen Tibor Jordn 《Journal of Algorithms in Cognition, Informatics and Logic》2000,37(2):326
Splitting off a pair su, sv of edges in a graph G means the operation that deletes su and sv and adds a new edge uv. Given a graph G = (V + s, E) which is k-edge-connected (k ≥ 2) between vertices of V and a specified subset R V, first we consider the problem of finding a longest possible sequence of disjoint pairs of edges sx, sy, (x ,y R) which can be split off preserving k-edge-connectivity in V. If R = V and d(s) is even then a well-known theorem of Lovász asserts that a complete R-splitting exists, that is, all the edges connecting s to R can be split off in pairs. This is not the case in general. We characterize the graphs possessing a complete R-splitting and give a formula for the length of a longest R-splitting sequence. Motivated by the connection between splitting off results and connectivity augmentation problems we also investigate the following problem that we call the split completion problem: given G and R as above, find a smallest set F of new edges incident to s such that G′ = (V + s, E + F) has a complete R-splitting. We give a min-max formula for F as well as a polynomial algorithm to find a smallest F. As a corollary we show a polynomial algorithm which finds a solution of size at most k/2 + 1 more than the optimum for the following augmentation problem, raised in [[2]]: given a graph H = (V, E), an integer k ≥ 2, and a set R V, find a smallest set F′ of new edges for which H′ = (V, E + F′) is k-edge-connected and no edge of F′ crosses R. 相似文献
12.
Pietro Celada Stefania Perrotta 《Calculus of Variations and Partial Differential Equations》2001,12(4):371-398
We consider the problem of minimizing multiple integrals of product type, i.e.
where is a bounded, open set in , is a possibly nonconvex, lower semicontinuous function with p-growth at infinity for some and the boundary datum is in (or simply in if ). Assuming that the convex envelope off is affine on each connected component of the set , we prove attainment for () for every continuous, positively bounded below function g such that (i) every point is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that the class of coefficents g that yield existence to () is dense in the space of continuous, positive functions on . We present examples which show that these conditions for attainment are essentially sharp.
Received April 12, 2000 / Accepted May 9, 2000 / Published online November 9, 2000 相似文献
13.
We study special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u 0 ∈ H 3/4+ (?) whose restriction belongs to H l ((b, ∞)) for some l ∈ ?+ and b ∈ ? we prove that the restriction of the corresponding solution u(·, t) belongs to H l ((β, ∞)) for any β ∈ ? and any t ∈ (0, T). Thus, this type of regularity propagates with infinite speed to its left as time evolves. 相似文献
14.
Q 《European Journal of Combinatorics》2002,23(8):1073
Ito, Tanabe and Terwilliger recently introduced the notion of a tridiagonal pair. We apply their results to distance-regular graphs and obtain the following theorem.Theorem Let Γ denote a distance-regular graph with diameter D ≥ 3. Suppose Γ is Q -polynomial with respect to the orderingE0 , E1, , EDof the primitive idempotents. For 0 ≤ i ≤ D, let midenote the multiplicity ofEi . Then (i)mi − 1 ≤ mi (1 ≤ i ≤ D / 2),(ii)mi ≤ mD − i (0 ≤ i ≤ D / 2).By proving the above theorem we resolve a conjecture of Dennis Stanton. 相似文献
15.
On the long time behavior of the stochastic heat equation 总被引:2,自引:0,他引:2
We consider the stochastic heat equation in one space dimension and compute – for a particular choice of the initial datum
– the exact long time asymptotic. In the Carmona-Molchanov approach to intermittence in non stationary random media this corresponds
to the identification of the sample Lyapunov exponent. Equivalently, by interpreting the solution as the partition function of a directed polymer in a random environment, we obtain
a weak law of large numbers for the quenched free energy. The result agrees with the one obtained in the physical literature
via the replica method. The proof is based on a representation of the solution in terms of the weakly asymmetric exclusion
process.
Received: 11 November 1997 / Revised version: 31 July 1998 相似文献
16.
Rajendra M. Pawale 《Designs, Codes and Cryptography》2011,58(2):111-121
Quasi-symmetric designs with intersection numbers x > 0 and y = x + 2 under the condition λ > 1 are investigated. If D(v, b, r, k, λ; x, y) is a quasi-symmetric design with above conditions then it is shown that either λ = x + 1 or x + 2 or D is a design with the parameters given in the Table 6 or complement of one of these designs. 相似文献
17.
We consider a 2D vorticity configuration where vorticity is highly concentrated around a curve and exponentially decaying away from it: the intensity of the vorticity is O(1/ε) on the curve while it decays on an O(ε) distance from the curve itself. We prove that, if the initial datum is of vortex-layer type, Euler solutions preserve this structure for a time that does not depend on ε . Moreover, the motion of the center of the layer is well approximated by the Birkhoff-Rott equation. © 2020 Wiley Periodicals, Inc. 相似文献
18.
In this paper, we study the initial-boundary value problem of the porous medium equation u
t
= Δu
m
+ V(x)u
p
in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|)
σ
. Let ω
1 denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l
2 + (n − 2)l = ω
1. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u
0 unless u
0 = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u
0 ≥ 0. 相似文献
19.
In this paper, we consider a class of Ginzburg-Landau functionalsE
ε associated with a couple of non-commuting vector fields which yield a “degenerate” energy. We study the asymptotic behavior
of the minimizers, showing that it does not depend on the topological degree of the boundary datum; and we prove uniqueness
and regularity of the minimizer of the limit problem, in spite of the lack of lifting theorems in the natural function spaces
for the limit functional.
The authors were supported by University of Bologna, funds for selected research topics, and by GNAMPA of the INDAM, Italy, project “Analysis in metric spaces and subelliptic equations.” 相似文献
Résumé Dans cet article, nous considérons une classe de fonctionnellesE ε du type Ginzburg-Landau associée a un couple de champs de vecteurs définissant une énergie dégénérée. Nous étudions le comportement asymptotique des minimiseurs. Nous démontrons que ce comportement ne dépend pas du degré topologique de la donnée a la frontiere et nous prouvons l’unicité et la régularité du minimiseur du probléme limite, malgré l’absence d’un théorème de lifting dans les espaces de Sobolev naturels pour la même fonctionnelle.
The authors were supported by University of Bologna, funds for selected research topics, and by GNAMPA of the INDAM, Italy, project “Analysis in metric spaces and subelliptic equations.” 相似文献
20.
Ondřej Kurka 《Acta Mathematica Hungarica》2012,134(3):209-268
In spite of the Lebesgue density theorem, there is a positive δ such that, for every non-trivial measurable set S⊂ℝ, there is a point at which both the lower densities of S and of ℝ∖S are at least δ. The problem of determining the supremum of possible values of this δ was studied in a paper of V. I. Kolyada, as well as in some recent papers. We solve this problem in the present work. 相似文献