ON CORRECT SOLVABILITY OF DIRICHLET PROBLEM IN A HALF-SPACE FOR REGULAR EQUATIONS WITH NON-HOMOGENEOUS BOUNDARY CONDITIONS
DOI:
https://doi.org/10.46991/PYSU:A/2023.57.2.044Keywords:
regular operator, characteristic polyhedron, multianisotropic Sobolev spaceAbstract
In this paper we consider the following Dirichlet problem with non-homogeneous boundary conditions in a multianisotropic Sobolev space $W_2^{\mathfrak{M}}(R^2 \times R_+)$ $$\begin{cases} P(D_x, D_{x_3}) u = f(x, x_3), \quad x_3 > 0, \quad x \in R^2, \\ D_{x_3}^s u \big\rvert_{x_3 = 0} = \varphi_s(x),\quad s = 0, \dots, m-1. \end{cases} $$ It is assumed that $P(D_x, D_{x_3})$ is a multianisotopic regular operator of a special form with a characteristic polyhedron $\mathfrak{M}$. We prove unique solvability of the problem in the space $W_2^{\mathfrak{M}}(R^2 \times R_+)$, assuming additionally, that $f(x, x_3)$ belongs to $L_2(R^2 \times R^+)$ and has a compact support, boundary functions $\varphi_s$ belong to special Sobolev spaces of fractional order and have compact supports.
References
Karapetyan G.A., Petrosyan H.A. Correct Solvability of the Dirichlet Problem in the Half-space for Regular Hypoelliptic Equations. Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) 54 (2019), 45-69. https://doi.org/10.3103/S1068362319040022
Ghazaryan H.G. The Newton Polyhedron, Spaces of Differentiable Functions and General Theory of Differential Equations. Armenian Journal of Mathematics 9 (2017), 102-145.
Karapetyan G.A. Integral Representations of Functions and Embedding Theorems for Multianisotropic Spaces on the Plane with One Anisotropy Vertex. Journal of Contemporary Mathematical Analysis 51 (2016), 269-281. https://doi.org/10.3103/S1068362316060017
Karapetyan G.A. Integral Representation of Functions and Embedding Theorems for Multianisotropic Spaces for the Three-dimensional Case. Eurasian Mathematical Journal 7 (2016), 19-39.
Karapetyan G.A., Arakelyan M.K. Estimation of Multianisotropic Kernels and their Application to the Embedding Theorems. Transactions of A. Razmadze Mathematical Institute 171 (2017), 48-56.
Karapetyan G.A. Integral Representations of Functions and Embedding Theorems for $n$-dimensional Multianisotropic Spaces with One Anisotropy Vertex. Siberian Mathematical Journal 58 (2017), 445-460. https://doi.org/10.1134/S0037446617030089
Karapetyan G.A., Petrosyan H.A. Embedding Theorems for Multianisotropic Spaces with Two Vertices of Anisotropicity. Proc. of the YSU. Physical and Mathematical Sci. 51 (2017), 29-37. https://doi.org/10.46991/PYSU:A/2017.51.1.029
Karapetyan G.A. An Integral Representation and Embedding Theorems in the Plane for Multianisotropic Spaces. Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) 52 (2017), 267-275. https://doi.org/10.3103/S1068362317060024
Karapetyan G.A., Arakelyan M.K. Embedding Theorems for General Multianisotropic Spaces. Matematical Notes 104 (2018), 422-438. https://doi.org/10.1134/S0001434618090092
Khachaturyan M.A., Hakobyan A.R. On Traces of Functions from Multianisotropic Sobolev Spaces. Vestnik RAU. Phys.-Math. Est. Nauki 1 (2021), 56-77 (in Russian).
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Proceedings of the YSU
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.