BMO and Dirichlet problem for semi-linear equations in the plane - Gutlyanskiĭ, Vladimir; Nesmelova, Olga; Ryazanov, Vladimir; Yakubov, Eduard (Topological Methods in Nonlinear Analysis, 2024)
First we study the Dirichlet problem${\rm{Re}}\omega(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\inD,\zeta\in \partial D,$ 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 thecomplex plane $\mathbb C$, which is either simply connected or hasno boundary components degenerated to a single point, and thatthe equations are locally uniform elliptic with possiblesingularities at the boundary. For $\sigma\in L_p(D)$, $p> 2$, withcompact support, and continuous $q\colon \mathbb R\to\mathbb C$,$q(t)/t\to 0$ as $t\to\infty$, we establish a series of effectivecriteria for existence of solutions of the Dirichlet problem interms of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz integralmeans. We also establish representation and regularity ofthese solutions. Then, we prove existence, representation andregularity 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 ${\rmdiv} [A(z) ablau(z)] = g(z)Q(u(z))$ for $g\in L_p(D)$,$p> 1$, with compact support, and continuous $Q:\mathbb R\to\mathbbR$, $Q(t)/t\to 0$ as $t\to\infty$. We also assume here conditions onthe matrix coefficients $A(z)$ guaranteing locally uniformellipticity of these equations. Finally, we give examples ofpossible applications of the obtained results to various semi-linearequations of the mathematical physics in anisotropic andinhomogeneous media.