View source for Carleman boundary value problem

A boundary value problem for analytic functions involving a shift which reverses the direction of traversing the boundary. It was first considered by T. Carleman [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c0204001.png" /> be a simple closed Lyapunov curve (cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]) in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c0204002.png" />-plane and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c0204003.png" /> be the finite domain bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c0204004.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c0204005.png" /> be a given complex-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c0204006.png" /> inducing a one-to-one mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c0204007.png" /> onto itself reversing the direction of traversing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c0204008.png" /> and satisfying additionally the Carleman condition:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c0204009.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

(it is further supposed that the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c02040010.png" /> satisfies a Hölder condition). Then the Carleman boundary value problem consists in finding a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c02040011.png" />, analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c02040012.png" /> except for a finite number of poles, continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c02040013.png" /> and subject to the boundary condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c02040014.png" /></td> </tr></table>

where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c02040015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c02040016.png" /> given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c02040017.png" /> satisfy a Hölder condition and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c02040018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c02040019.png" />.

The Carleman boundary value problem has also been studied under the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020400/c02040020.png" /></td> </tr></table>

which is more general than (*), and so has the Carleman boundary value problem for several unknown functions (see [[#References|[2]]], [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T. Carleman,   "Sur la théorie des équations intégrales et ses applications" , ''Verh. Internat. Mathematiker Kongress. Zürich, 1932'' , '''1''' , O. Füssli  (1932)  pp. 138–151</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. Muskhelishvili,   "Singular integral equations" , Wolters-Noordhoff  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.P. Vekua,   "Systems of singular integral equations and some boundary value problems" , Moscow  (1970)  (In Russian)</TD></TR></table>