A priori estimates of the positive real or imaginary part of a generalized analytic function

S. B. Klimentov

Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 4, pp. 50-57

Résumé

We denote by $D=D_z=\{z : |z|1\}$ the unit disk in the complex $z$-plane, $\Gamma= \partial D$. The following property of harmonic functions is well-known. If a real valued function $U(z)\in C(\overline D)$ is harmonic in $D$, $U(z) |_{z\in \Gamma} \geq K = {\rm const}>0$, then $U(z) \geq K$ for all $ z \in \overline D$. The subject of this work is the generalization of this property to the real (imaginary) part of the solution to the elliptic system on $D$: $\partial_{\bar z} w-q_1(z) \partial_z w - q_2(z) \partial_{\bar z} \overline w +A(z)w+B(z) \overline w=0,$ where $w=w(z)=u(z)+iv(z)$ is a desired complex function. $\partial _{\bar z}=\frac12 \big(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\big)$, $\partial _{z}=\frac12 \big(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y}\big)$, are derivatives in Sobolev sense; $q_1(z)$ and $q_2(z)$ are given measurable complex functions satisfying the uniform ellipticity condition of the system $|q_1(z)| + |q_2(z)| \leq q_0 = {\rm const}1$, $ z\in \overline D$; $A(z), B(z)\in L_p(\overline D)$, $p>2$, also are given complex functions.