Description

• Title:
  BMO and Dirichlet problem for semi-linear equations in the plane
• Creator:
  Gutlyanskiĭ, Vladimir;
  Nesmelova, Olga;
  Ryazanov, Vladimir;
  Yakubov, Eduard
• Description:
  First we study 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 $\varphi \colon \partial D\to\mathbb R$ for semi-linear Beltrami equations $\omega_{\overline{z}}-\mu(z) \omega_z=\sigma (z)q({\rm Re}\omega(z))$. We assume here that $D$ is an arbitrary bounded domain of the complex plane $\mathbb C$, which is either simply connected or has no boundary components degenerated to a single point, and that the equations are locally uniform elliptic with possible singularities at the boundary. For $\sigma\in L_p(D)$, $p> 2$, with compact support, and continuous $q\colon \mathbb R\to\mathbb C$, $q(t)/t\to 0$ as $t\to\infty$, we establish a series of effective criteria for existence of solutions of the Dirichlet problem in terms of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz integral means. We also establish representation and regularity of these solutions. Then, we prove existence, representation and regularity of weak solutions of the Dirichlet problem $u(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\in D,\zeta\in \partial D,$ to semi-linear Poisson type equations ${\rm div} [A(z)\nablau(z)] = g(z)Q(u(z))$ for $g\in L_p(D)$, $p> 1$, with compact support, and continuous $Q:\mathbb R\to\mathbb R$, $Q(t)/t\to 0$ as $t\to\infty$. We also assume here conditions on the matrix coefficients $A(z)$ guaranteing locally uniform ellipticity of these equations. Finally, we give examples of possible applications of the obtained results to various semi-linear equations of the mathematical physics in anisotropic and inhomogeneous media.
• Object availability:
  free access
• Rights: https://creativecommons.org/licenses/by-nd/4.0 ;
  Prawa autorskie (c) 2024 Vladimir Gutlyanskiĭ, Olga Nesmelova, Vladimir Ryazanov, Eduard Yakubov
• Date:
  2024.09.15
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 64, No 1 (September 2024); 163 - 199;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/55208/39685 ;https://apcz.umk.pl/TMNA/article/view/55208/39686
• Subject:
  BMO;
  bounded mean oscillation;
  FMO;
  finite mean oscillation;
  Dirichlet problem;
  semi-linear Beltrami equations;
  semi-linear Poisson type equations;
  generalized analytic functions with sources;
  generalized harmonic functions with sources;
  Primary: 30C62, 30C65, 30E25;
  Secondary: 30G30, 35F45, 35J25
• Identifier: https://apcz.umk.pl/TMNA/article/view/55208
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other