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1
Content available remote Solution of plane Dirichlet problem using compactly supported 2D wavelet scaling functions
100%
Katunin A.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2012
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tom Vol. 11, nr 1
31-40
EN
The paper presents the solution of the homogeneous plane Dirichlet problem using the wavelet-Galerkin method with various 2D compactly supported wavelet scaling functions. An analysis of approximation accuracy was performed with respect to the orders of investigated wavelet scaling functions and the level of approximation. The most effective scaling functions for solving the Dirichlet problem were indicated and discussed.
2
Content available remote The semicoercive case of the Dirichlet problem of even order
80%
Sidorczyk M.
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1998
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tom Vol. 31, nr 3
615-625
3
Content available Pseudo-differential equations and conical potentials: 2-dimensional case
80%
Vasilyev V. B.
Opuscula Mathematica
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2019
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tom Vol. 39, no. 1
109--124
EN
We consider two-dimensional elliptic pseudo-differential equation in a plane sector. Using a special representation for an elliptic symbol and the formula for a general solution we study the Dirichlet problem for such equation. This problem was reduced to a system of linear integral equations and then after some transformations to a system of linear algebraic equations. The unique solvability for the Dirichlet problem was proved in Sobolev-Slobodetskii spaces and a priori estimate for the solution is given.
4
Content available remote A generalization of the saddle point method with applications
80%
Schechter M.
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1992
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tom 57
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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.
5
Content available remote Infinite dimension of solutions of the Dirichlet problem
80%
Ryazanov V.
Open Mathematics
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2015
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tom 13
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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 Existence of solution of the nonlinear Dirichlet problem for differential-functional equations of elliptic type
80%
Brzychczy S.
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1993
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tom 58
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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].
7
Content available remote Existence of bounded solutions for fourth order Dirichlet problems with convex-concave nonlinearity
80%
Galewski M.
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2009
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tom Vol. 42, nr 1
75-81
EN
We consider the Dirichlet boundary value problem for higher order O. D. E. with nonlinearity being the sum of a derivative of a convex and of a concave function in case when no growth condition is imposed on the concave part.
8
Content available remote Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
80%
Gorban Y.
Open Mathematics
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2017
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tom 15
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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.
9
Content available Positive solutions with specific asymptotic behavior for a polyharmonic problem on Rn
80%
Dhifli A.
Opuscula Mathematica
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2015
|
tom Vol. 35, no. 1
5--19
EN
This paper is concerned with positive solutions of the semilinear polyharmonic equation [formula] on Rn, where m and n are positive integers with n > 2m, α ∈ e (—1,1). The coefncient a is assumed to satisfy[formula], where Λ ∈ (2m,∞) and [formula]is positive with [formula], one also assumes that [formula]. We prove the existence of a positive solution u such that [formula], with [formula] and a function L, given explicitly in terms of L and satisfying the same condition as infinity. (Given positive functions ∫ and g on Rn, ∫≈ g means that [formula]for some constant c > 1.)
10
Content available remote Dirichlet problems with variable boundary data for nonlinear partial differential equations
80%
Bors D.
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2000
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tom Vol. 33, nr 2
295-312
EN
In this paper we consider some systems of partial differential equations with variable boundary data. Some sufficient conditions under which solutions of these systems continuously depend on boundary data are given. The proofs of the main result of this work are based on some variational methods.
11
Content available remote On some elliptic transmission problems
70%
Athanasiadis C. , Stratis I.
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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.
12
Content available An integro-differential inequality related to the smallest positive eigenvalue ofp(x)-Laplacian Dirichlet problem
70%
Wiśniewski D. , Bodzioch M.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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2016
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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.
13
Content available Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in the annulus
70%
Dridi S. , Khamessi B.
Opuscula Mathematica
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2015
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tom Vol. 35, no. 1
21--36
EN
In this paper, we establish existence and asymptotic behavior of a positive clas­sical solution to the following semilinear boundary value problem: [formula] Here O is an annulus in [formula] and q is a positive function in [formula], satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub- and supersolutions with Karamata regular variation theory.
14
Content available remote Second order BVPs with state dependent impulses via lower and upper functions
60%
Rachůnková I. , Tomeček J.
|
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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.
15
Content available remote On bifurcation intervals for nonlinear eigenvalue problems
60%
Przybycin J.
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1999
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tom 71
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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.
16
Content available Numerical Implementation of the Fictitious Domain Method for Elliptic Equations
60%
Temirbekov A. N. , Wójcik W.
International Journal of Electronics and Telecommunications
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2014
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tom Vol. 60, No. 3
219-223
EN
In this paper, we consider an elliptic equation with strongly varying coefficients. Interest in the study of these equations is connected with the fact that this type of equation is obtained when using the fictitious domain method. In this paper, we propose a special method for the numerical solution of elliptic equations with strongly varying coefficients. A theorem is proved for the rate of convergence of the iterative process developed. A computational algorithm and numerical calculations are developed to illustrate the effectiveness of the proposed method.
17
Content available remote Existence and uniqueness of solutions of the dirichlet nonlocal problem with nonlocal initial condition
51%
Byszewski L. , Winiarska T.
|
|
tom R. 111, z. 2-NP
3--7
EN
The aim of this paper is to prove the existence and uniqueness of solutions of the Dirichlet nonlocal problem with nonlocal initial condition. The considerations are extensions of results by E. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero.
PL
W artykule udowodniono istnienie i jednoznaczność rozwiązań nielokalnego zagadnienia Dirichleta z nielokalnym warunkiem początkowym. Rozważania są rozszerzeniami rezultatów otrzymanych przez E. Andreu-Vaillo, J. M. Mazóna, J. D. Rossi i J. J. Toledo-Melero.
18
Content available remote Hemivariational inequalities governed by the p-Laplacian -Dirichlet problem
51%
Naniewicz Z.
Control and Cybernetics
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2004
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tom Vol. 33, no 2
181-210
EN
A hemivariational inequality involving p-Laplacian is studied under the hypothesis that the nonlinear part fulfills the unilateral growth condition (Naniewicz, 1994). The existence of solutions for problems with Dirichlet boundary conditions is established by making use of Chang's version of the critical point theory for non-smooth locally Lipschitz functionals (Chang, 1981), combined with the Galerkin method. A class of problems with nonlinear potentials fulfilling the classical growth hypothesis without Ainbrosetti-Rabinowitz type assumption is discussed. The approach is based on the recession technique introduced in Naniewicz (2003).
19
On some nonlinear hyperbolic p(x,t)-Laplacian equations
51%
Ahmedatt T. , Touzani A. , Aberqi A. , Yazough C.
Journal of Applied Analysis
|
2018
|
tom Vol. 24, nr 1
55--69
EN
This paper is devoted to study the global existence of solutions of the hyperbolic Dirichlet equation Utt=Lu+f(x,t) in ΩT=Ω×(0,T), where L is a nonlinear operator and ϕ(x,t,⋅), f(x,t) and the exponents of the nonlinearities p(x,t) and μ(x,t) are given functions.
20
Content available remote Stability and Continuity of Functions of Least Gradient
51%
Hakkarainen H. , Korte R. , Lahti P. , Shanmugalingam N.
|
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.
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