A Priori Estimates of the Positive Real or Imaginary Part of a Generalized Analytic Function

Klimentov, S. B.
Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 4.

Abstract:
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.

Keywords: first-order elliptic system, generalized analytic function

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