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1.
On positivity of solutions of degenerate boundary value problems for second-order elliptic equations
In this paper we study thedegenerate mixed boundary value problem:Pu=f in Ω,B
u
=gon Ω∂Г where ω is a domain in ℝ
n
,P is a second order linear elliptic operator with real coefficients, Γ⊆∂Ω is a relatively closed set, andB is an oblique boundary operator defined only on ∂Ω/Γ which is assumed to be a smooth part of the boundary.
The aim of this research is to establish some basic results concerning positive solutions. In particular, we study the solvability
of the above boundary value problem in the class of nonnegative functions, and properties of the generalized principal eigenvalue,
the ground state, and the Green function associated with this problem. The notion of criticality and subcriticality for this
problem is introduced, and a criticality theory for this problem is established. The analogs for the generalized Dirichlet
boundary value problem, where Γ=∂Ω, were examined intensively by many authors. 相似文献
2.
Let u be the Newtonian potential of a real analytic distribution in an open set Ω. In this paper we assume u is analytically
continuable from the complement of Ω into some neighborhood of a point x0 ∈ ∂Ω, and we study conditions under which the analytic continuation implies that ∂Ω is a real analytic hypersurface in some
neighborhood of x0. 相似文献
3.
V. I. Burenkov M. Lanza de Cristoforis 《Proceedings of the Steklov Institute of Mathematics》2008,260(1):68-89
We consider the Robin Laplacian in two bounded regions Ω1 and Ω2 of ℝ
N
with Lipschitz boundaries and such that Ω2 ⊂ Ω1, and we obtain two-sided estimates for the eigenvalues λ
n,2 of the Robin Laplacian in Ω2 via the eigenvalues λ
n, 1 of the Robin Laplacian in Ω1. Our estimates depend on the measure of the set difference Ω\Ω2 and on suitably defined characteristics of vicinity of the boundaries ∂Ω1 and ∂Ω2, and of the functions defined on ∂Ω1 and on ∂Ω2 that enter the Robin boundary conditions. 相似文献
4.
Kazuhiro Takimoto 《Calculus of Variations and Partial Differential Equations》2006,26(3):357-377
We consider the boundary blowup problem for k-curvature equation, i.e., H
k
[u] = f(u) g(|Du|) in an n-dimensional domain Ω, with the boundary condition u(x) → ∞ as dist (x,∂Ω) → 0. We prove the existence result under some hypotheses. We also establish the asymptotic behavior of a solution near the boundary ∂Ω.
Mathematics Subject Classification (2000) 35J65, 35B40, 53C21 相似文献
5.
Anders Björn 《Journal d'Analyse Mathématique》2010,112(1):49-77
In this paper, we study cluster sets and essential cluster sets for Sobolev functions and quasiharmonic functions (i.e., continuous
quasiminimizers). We develop their basic theory with a particular emphasis on when they coincide and when they are connected.
As a main result, we obtain that if a Sobolev function u on an open set Ω has boundary values f in Sobolev sense and f |∂Ω is continuous at x
0 ∈ ∂Ω, then the essential cluster set (u, x
0,Ω) is connected. We characterize precisely in which metric spaces this result holds. Further, we provide some new boundary
regularity results for quasiharmonic functions. Most of the results are new also in the Euclidean case. 相似文献
6.
Consider a compact manifold M with boundary ∂
M endowed with a Riemannian metric g and a magnetic field Ω. Given a point and direction of entry at the boundary, the scattering relation Σ determines the point
and direction of exit of a particle of unit charge, mass, and energy. In this paper we show that a magnetic system (M,∂
M,g,Ω) that is known to be real-analytic and that satisfies some mild restrictions on conjugate points is uniquely determined
up to a natural equivalence by Σ. In the case that the magnetic field Ω is taken to be zero, this gives a new rigidity result
in Riemannian geometry that is more general than related results in the literature. 相似文献
7.
In this paper, we study the asymptotic behavior of the solutionsu
ε (ε is a small parameter) of boundaryvalue problems for the heat equation in the domain Ωε=Ω−∪Ω
ε
+
∪γ one part of which (Ω
ε
+
) contains ε-periodically situated channels with diameters of order ε and the other part of which (Ω+) is a homogeneous medium; γ=∂Ω
ε
+
∩∂Ω+. On the boundary of the channels the Neumann boundary condition is posed, and on ∂Ωε∩∂Ω the Dirichlet boundary condition is prescribed. The homogenized problem is the Dirichlet problem in Ω with the transmission
condition on γ. The estimates for the difference betweenu
ε and the solution of the homogenized problem are obtained. Bibliography: 14 titles.
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 20, pp. 27–47, 1997. 相似文献
8.
For any multiply connected domain Ω in R2, let S be the boundary of the convex hull in H3 of R2\Ω which faces Ω. Suppose in addition that there exists a lower bound l > 0 of the hyperbolic lengths of closed geodesics in Ω. Then there is always a K-quasiconformal mapping from S to Ω, which extends continuously to the identity on S = Ω, where K depends only on l. We also give a numerical estimate of K by using the parameter l. 相似文献
9.
Jiří Neustupa 《Annali dell'Universita di Ferrara》2009,55(2):353-365
We prove the existence of a weak solution to the steady Navier–Stokes problem in a 2D domain Ω, whose boundary ∂Ω consists of two unbounded components Γ
− and Γ
+. We impose an inhomogeneous Dirichlet—type boundary condition on ∂Ω. The condition implies no restriction on fluxes of the solution through the components Γ
− and Γ
+. 相似文献
10.
We study the existence and the properties of reduced measures for the parabolic equations ∂
t
u − Δu + g(u) = 0 in Ω × (0, ∞) subject to the conditions (P): u = 0 on ∂Ω × (0, ∞), u(x, 0) = μ and (P′): u = μ′ on ∂Ω × (0, ∞), u(x, 0) = 0, where μ and μ′ are positive Radon measures and g is a continuous nondecreasing function. 相似文献
11.
David Kalaj 《Mathematische Zeitschrift》2008,260(2):237-252
Let Ω and Ω1 be Jordan domains, let μ ∈ (0, 1], and let be a harmonic homeomorphism. The object of the paper is to prove the following results: (a) If f is q.c. and ∂Ω, ∂Ω1 ∈ C
1,μ
, then f is Lipschitz; (b) if f is q.c., ∂Ω, ∂Ω1 ∈ C
1,μ
and Ω1 is convex, then f is bi-Lipschitz; and (c) if Ω is the unit disk, Ω1 is convex, and ∂Ω1 ∈ C
1,μ
, then f is quasiconformal if and only if its boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in
L
∞. These extend the results of Pavlović (Ann. Acad. Sci. Fenn. 27:365–372, 2002).
相似文献
12.
N. A. Shirokov 《Journal of Mathematical Sciences》1997,87(5):3925-3940
Denote by Kω(z, ζ) the Bergman kernel of a pseudoconvex domain Ω. For some classes of domains Ω, a relationship is found between the rate
of increase of Kω(z, z) as z tends to ∂Ω, and a purely geometric property of Ω. Bibliography: 5 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 222–245. 相似文献
13.
Exact propagators are obtained for the degenerate second order hyperbolic operators ∂2
t
-t
2l
Δ
x
, l=1,2,..., by analytic continuation from the degenerate elliptic operators ∂2
t
+t
2l
Δ
x
. The partial Fourier transforms are also obtained in closed form, leading to integral transform formulas for certain combinations
of Bessel functions and modified Bessel functions. 相似文献
14.
Kaouther Ammar 《Central European Journal of Mathematics》2010,8(3):548-568
The paper addresses the existence and uniqueness of entropy solutions for the degenerate triply nonlinear problem: b(v)
t
− div α(v, ▽g(v)) = f on Q:= (0, T) × Ω with the initial condition b(v(0, ·)) = b(v
0) on Ω and the nonhomogeneous boundary condition “v = u” on some part of the boundary (0, T) × ∂Ω”. The function g is continuous locally Lipschitz continuous and has a flat region [A
1, A
2,] with A
1 ≤ 0 ≤ A
2 so that the problem is of parabolic-hyperbolic type. 相似文献
15.
Paolo Albano 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(2):273-281
In an open bounded set Ω, we consider the distance function from ∂Ω associated to a Riemannian metric with C
1,1 coefficients. Assuming that Ω is convex near a boundary point x
0, we show that the distance function is differentiable at x
0 if and only if there exists the tangent space to ∂Ω at x
0. Furthermore, if the distance function is not differentiable at x
0 then there exists a Lipschitz continuous curve, with initial point at x
0, such that the distance function is not differentiable along such a curve.
相似文献
16.
Dian K. Palagachev 《Journal of Global Optimization》2008,40(1-3):305-318
We derive W
2,p
(Ω)-a priori estimates with arbitrary
p ∈(1, ∞), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular
coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field ℓ that is tangent
to ∂Ω at the points of a non-empty set ε ⊂ ∂Ω and is of emergent type on ∂Ω.
相似文献
17.
The equation-δu = χ uo(-1/uΒ + λf (x, u)) in Ω with Dirichlet boundary condition on ∂Ω has a maximal solution uλ ≥0 for every λ 0. For λ less than a constant λ*,
the solution vanishes inside the domain; and for λ λ*, the solution is positive. We obtain optimal regularity ofuλ even in the presence of the free boundary.
Supported in part by H. J. Sussmann’s NSF Grant DMS01-03901.
Supported by FAPESP. He also thanks Rutgers University. 相似文献
18.
Martin Fuchs 《manuscripta mathematica》1991,72(1):131-140
Given a smooth domain Ω in ℝ
m+1 with compact closure and a smooth integrable functionh: ℝ
m+1→ℝ satisfyingh(x)≥H
∂Ω
(x) on ∂Ω whereH
∂ω denotes the mean curvature of ∂Ω calculated w.r.t. the interior unit normal we show that there is a setA⊂ℝ
m+1 with the properties
andH
∂A=h on ∂A. 相似文献
19.
Burglind Jöricke 《Journal of Geometric Analysis》1999,9(2):257-300
Let Ω be a bounded strictly pseudoconvex domain in ℂn, n ≥ 3, with boundary ∂Ω, of class C2. A compact subset K is called removable if any analytic function in a suitable small neighborhood of ∂Ω K extends to an analytic
function in Ω. We obtain sufficient conditions for removability in geometric terms under the condition that K is contained
in a generic C2 -submanifold M of co-dimension one in ∂Ω. The result uses information on the global geometry of the decomposition of a CR-manifold
into CR-orbits, which may be of some independent interest. The minimal obstructions for removability contained in M are compact
sets K of two kinds. Either K is the boundary of a complex variety of co-dimension one in Ω or it is an exceptional minimal
CR-invariant subset of M, which is a certain analog of exceptional minimal sets in co-dimension one foliations. It is shown
by an example that the latter possibility may occur as a nonremovable singularity set.
Further examples show that the germ of envelopes of holomorphy of neighborhoods of ∞Ω K for K ⊂ M may be multisheeted. A couple
of open problems are discussed. 相似文献
20.
Let Ω and Π be two finitely connected hyperbolic domains in the complex plane
\Bbb C{\Bbb C}
and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities
Cn(W,P) := supf ? A(W,P)supz ? W\frac|f(n)(z)| R(f(z),P)n! (R(z,W))nC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n}
are finite for all
n ? \Bbb N{n \in {\Bbb N}}
if and only if ∂Ω and ∂Π do not contain isolated points. 相似文献