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1
Content available remote

A counterexample to the $L^{p}$-Hodge decomposition

100%
Hajłasz P.
Banach Center Publications
|
1996
|
tom 33
|
nr 1
79-83
EN
We construct a bounded domain $Ω ⊂ ℝ^2$ with the cone property and a harmonic function on Ω which belongs to $W_0^{1,p}(Ω)$ for all 1 ≤ p < 4/3. As a corollary we deduce that there is no $L^p$-Hodge decomposition in $L^{p}(Ω,ℝ^2)$ for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in $W^{1,p}(Ω)$ for all p > 4.
2
Content available remote

A generalization of the saddle point method with applications

100%
Schechter M.
Annales Polonici Mathematici
|
1992
|
tom 57
|
nr 3
269-281
EN
We show that one can drop an important hypothesis of the saddle point theorem without affecting the result. We then show how this leads to stronger results in applications.
3
Content available remote

Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations

100%
Gorban Y.
Open Mathematics
|
2017
|
tom 15
|
nr 1
768-786
EN
In the present article we deal with the Dirichlet problem for a class of degenerate anisotropic elliptic second-order equations with L1-right-hand sides in a bounded domain of ℝn(n ⩾ 2) . This class is described by the presence of a set of exponents q1,…, qn and a set of weighted functions ν1,…, νn in growth and coercitivity conditions on coefficients of the equations. The exponents qi characterize the rates of growth of the coefficients with respect to the corresponding derivatives of unknown function, and the functions νi characterize degeneration or singularity of the coefficients with respect to independent variables. Our aim is to investigate the existence of entropy solutions of the problem under consideration.
4
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Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type

100%
Brzychczy S.
Annales Polonici Mathematici
|
1993
|
tom 58
|
nr 2
139-146
EN
Consider a nonlinear differential-functional equation (1) Au + f(x,u(x),u) = 0 where $Au := ∑_{i,j=1}^m a_{ij}(x) (∂²u)/(∂x_i ∂x_j)$, $x=(x_1,...,x_m) ∈ G ⊂ ℝ^m$, G is a bounded domain with $C^{2+α}$ (0 < α < 1) boundary, the operator A is strongly uniformly elliptic in G and u is a real $L^p(G̅)$ function. For the equation (1) we consider the Dirichlet problem with the boundary condition (2) u(x) = h(x) for x∈ ∂G. We use Chaplygin's method [5] to prove that problem (1), (2) has at least one regular solution in a suitable class of functions. Using the method of upper and lower functions, coupled with the monotone iterative technique, H. Amman [3], D. H. Sattinger [13] (see also O. Diekmann and N. M. Temme [6], G. S. Ladde, V. Lakshmikantham, A. S. Vatsala [8], J. Smoller [15]) and I. P. Mysovskikh [11] obtained similar results for nonlinear differential equations of elliptic type. A special case of (1) is the integro-differential equation $Au + f(x,u(x), ∫_G u(x)dx) = 0$. Interesting results about existence and uniqueness of solutions for this equation were obtained by H. Ugowski [17].
5
Content available remote

Infinite dimension of solutions of the Dirichlet problem

100%
Ryazanov V.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
It is proved that the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. has the infinite dimension.
6
Content available remote

On some elliptic transmission problems

88%
Athanasiadis C., Stratis I.
Annales Polonici Mathematici
|
1996
|
tom 63
|
nr 2
137-154
EN
Boundary value problems for second order linear elliptic equations with coefficients having discontinuities of the first kind on an infinite number of smooth surfaces are studied. Existence, uniqueness and regularity results are furnished for the diffraction problem in such a bounded domain, and for the corresponding transmission problem in all of $ℝ^N$. The transmission problem corresponding to the scattering of acoustic plane waves by an infinitely stratified scatterer, consisting of layers with physically different materials, is also studied.
7
Content available remote

On variational impulsive boundary value problems

88%
Galewski M.
Open Mathematics
|
2012
|
tom 10
|
nr 6
1969-1980
EN
Using the variational approach, we investigate the existence of solutions and their dependence on functional parameters for classical solutions to the second order impulsive boundary value Dirichlet problems with L1 right hand side.
8
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An integro-differential inequality related to the smallest positive eigenvalue ofp(x)-Laplacian Dirichlet problem

88%
Wiśniewski D., Bodzioch M.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2016
|
tom 15
EN
We consider the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem.
9
Content available remote

Vector-valued holomorphic and harmonic functions

88%
Arendt W.
Concrete Operators
|
2016
|
tom 3
|
nr 1
68-76
EN
Holomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions with values in a Banach space.
10
Content available remote

Second order BVPs with state dependent impulses via lower and upper functions

75%
Rachůnková I., Tomeček J.
Open Mathematics
|
2014
|
tom 12
|
nr 1
128-140
EN
The paper deals with the following second order Dirichlet boundary value problem with p ∈ ℕ state-dependent impulses: z″(t) = f (t,z(t)) for a.e. t ∈ [0, T], z(0) = z(T) = 0, z′(τ i+) − z′(τ i−) = I i(τ i, z(τ i)), τ i = γ i(z(τ i)), i = 1,..., p. Solvability of this problem is proved under the assumption that there exists a well-ordered couple of lower and upper functions to the corresponding Dirichlet problem without impulses.
11
Content available remote

On bifurcation intervals for nonlinear eigenvalue problems

75%
Przybycin J.
Annales Polonici Mathematici
|
1999
|
tom 71
|
nr 1
39-46
EN
We give a sufficient condition for [μ-M, μ+M] × {0} to be a bifurcation interval of the equation u = L(λu + F(u)), where L is a linear symmetric operator in a Hilbert space, μ ∈ r(L) is of odd multiplicity, and F is a nonlinear operator. This abstract result provides an elementary proof of the existence of bifurcation intervals for some eigenvalue problems with nondifferentiable nonlinearities. All the results obtained may be easily transferred to the case of bifurcation from infinity.
12
Content available remote

Stability and Continuity of Functions of Least Gradient

63%
Hakkarainen H., Korte R., Lahti P., Shanmugalingam N.
Analysis and Geometry in Metric Spaces
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2015
|
tom 3
|
nr 1
EN
In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions. We also apply these tools to prove a maximum principle for functions of least gradient that arise as solutions to a Dirichlet problem.
13
Content available remote

An existence result for a quadrature surface free boundary problem

63%
Barkatou M., Seck D., Ly I.
Open Mathematics
|
2005
|
tom 3
|
nr 1
39-57
EN
The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existences is given in the radial case.
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