338 research outputs found
On integral conditions in the mapping theory
It is established interconnections between various integral conditions that
play an important role in the theory of space mappings and in the theory of
degenerate Beltrami equations in the plane.Comment: 15 pages, changes related to Corollary 3.2, see (3.28
Dirichlet problem for Poisson equations in Jordan domains
We study the Dirichlet problem for the Poisson equations △u(z) = g(z) with g ∈ Lp, p > 1, and continuous boundary data φ : ∂D → ℝ in arbitrary Jordan domains D in ℂ and prove the existence of continuous solutions u of the problem.Мы изучаем задачу Дирихле для уравнений Пуассона △u(z) = g(z) с g ∈ Lp, p > 1, и непрерывными граничными данными φ : ∂D → ℝ в произвольных жордановых областях D ⊂ ℂ и доказываем существование непрерывных решений u этой задачи.Ми вивчаємо задачу Дiрихле для рiвнянь Пуасона △u(z) = g(z) с g ∈ Lp, p > 1, та неперервними граничними даними φ : ∂D → ℝ в довiльних жорданових областях D ⊂ ℂ та доводимо iснування неперервних рiшень u цiєї задач
To the theory of semi-linear Beltrami equations
The present paper is devoted to the study of semi-linear Beltrami equations
which are closely relevant to the corresponding semi-linear Poisson type
equations of mathematical physics on the plane in anisotropic and inhomogeneous
media.
In its first part, applying completely continuous ope\-ra\-tors by
Ahlfors-Bers and Leray--Schauder approach, we prove existence of regular
solutions of the semi-linear Beltrami equations with no boundary conditions.
Moreover, here we derive their representation through solutions of the Vekua
type equations and generalized analytic functions with sources.
As consequences, it is given a series of applications of these results to
semi-linear Poisson type equations and to the corresponding equations of
mathematical physics describing such phenomena as diffusion with physical and
chemical absorption, plasma states and stationary burning in anisotropic and
inhomogeneous media.
The second part of the paper contains existence, representation and
regularity results for nonclassical solutions to the Hilbert (Dirichlet)
boundary value problem for semi-linear Beltrami equations and to the Poincare
(Neumann) boundary value problem for semi-linear Poisson type equations with
arbitrary boundary data that are measurable with respect to logarithmic
capacity.Comment: 28 pages. arXiv admin note: text overlap with arXiv:2107.1066
On Dirichlet problem for degenerate Beltrami equations with sources
The present paper is devoted to the study of the Dirichlet problem
as with continuous boundary data
for Beltrami equations ,
a.e., with sources in the case of locally
uniform ellipticity. In this case, we establish a series of effective integral
criteria of the type of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz on
singularities of the equations at the boundary for existence, representation
and regularity of solutions in arbitrary bounded domains of the complex
plane with no boun\-da\-ry component degenerated to a single point
for sources in , , with compact support in . Moreover,
we prove in such domains existence, representation and regularity of weak
solutions of the Dirichlet problem for the Poisson type equation whose source , , has compact
support in and whose mat\-rix valued coefficient guarantees its
locally uniform ellipticity.Comment: 31 pages. arXiv admin note: substantial text overlap with
arXiv:2111.1037
Representations of -semigroups by multiplace functions
We describe the representations of -semigroups, i.e. groupoids with
binary associative operations, by partial -place functions and prove
that any such representation is a union of some family of representations
induced by Schein's determining pairs.Comment: 17 page
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