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 ${\rm{Re}}\,\omega(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\in D,\zeta\in \partial D,$ with continuous boundary data $\varphi :\partial D\to\mathbb R$ for Beltrami equations $\omega_{\bar{z}}=\mu(z) \omega_z+\sigma (z)$, $|\mu(z)|<1$ a.e., with sources $\sigma :D\to\mathbb C$ 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 $D$ of the complex plane $\mathbb C$ with no boun\-da\-ry component degenerated to a single point for sources $\sigma$ in $L_p(D)$, $p>2$, with compact support in $D$. Moreover, we prove in such domains existence, representation and regularity of weak solutions of the Dirichlet problem for the Poisson type equation ${\rm div} [A(z)\nabla\,u(z)] = g(z)$ whose source $g\in L_p(D)$, $p>1$, has compact support in $D$ and whose mat\-rix valued coefficient $A(z)$ guarantees its locally uniform ellipticity.Comment: 31 pages. arXiv admin note: substantial text overlap with arXiv:2111.1037

Heat-Resistant Intermetallic Compounds and Ceramic Dispersion Alloys for Additive Manufacturing: A Review

Many industries such as aerospace, power generation, and ground transportation demand structural materials with high specific strength at elevated temperatures. Up until now, many types of heat-resistant materials including Ni-based superalloys, intermetallic compounds, and dispersion-strengthened alloys have been developed for specific applications in these industries. Moreover, with the recent development of additive manufacturing techniques, these industries can now benefit from the rapid prototyping abilities, geometric freedom, and increased mechanical properties that can be achieved through various additive manufacturing processes. With this in mind, the progress made in additive manufacturing of heat-resistant intermetallic compounds and ceramic dispersion alloys is herein examined. A brief introduction is provided on the target industries, applications, and the compositions of heat-resistant alloys of current research interest. Then, recent research on heat-resistant intermetallic compounds and ceramic dispersion alloys fabricated by additive manufacturing processes such as laser powder bed fusion, laser direct energy deposition, and electron-beam powder bed fusion are reviewed with information provided on microstructure, processing parameters, strengthening mechanisms, and mechanical properties. Finally, an outlook is provided with future research suggestions