Geodesic
A priori estimates of the positive real or imaginary part of a generalized analytic function
Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 4, pp. 50-57 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     title = {A priori estimates of the positive real or imaginary part of a generalized analytic function},
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S. B. Klimentov. A priori estimates of the positive real or imaginary part of a generalized analytic function. Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 4, pp. 50-57. http://geodesic.mathdoc.fr/item/VMJ_2023_25_4_a4/

[1] Bojarsky, B. V., “Weak Solutions to a System of First-Order Didderential Equations of Elliptic Type with Discontinuous Coefficients”, Matematichesky Sbornik, 43:1 (1957), 451–503 (in Russian)

[2] Vekua, I. N., Generalized Analytic Functions, Pergamon Press, Oxford–London–New York–Paris, 1962, xxix+668 pp. | MR | Zbl

[3] Klimentov, S. B., “On a Priori Estimates of the Derivatives of the Radius-Vector of a Surface of Positive Curvature”, Journal of Soviet Mathematics, 44:2 (1989), 215–235 | DOI | MR

[4] Klimentov, S. B., “On a Method of Constructing the Solutions of Boundary-Value Problems of the Theory of Bendings of Surfaces of Positive Curvature”, Journal of Mathematical Sciences, 51:2 (1990), 2230–2248 | DOI | MR | Zbl

[5] Klimentov, S. B., “The Riemann–Hilbert Problem in Hardy Classes for General First-Order Elliptic Systems”, Russian Mathematics, 60:6 (2016), 29–39 | DOI | MR | Zbl