arXiv subjects
Publications and source records attributed to Peter L. Polyakov.
We present an application of the Faddeev-Henkin exponential ansatz and of the d-to-d-bar map on the boundary to inverse conductivity problem on a bordered Riemann surface in CP2. In our approach we use integral formulas for operator d-bar developed in [HP1]-[HP4] and integral formulas for holomorphic functions on Riemann surfaces from [P].
We construct a boundary integral formula for harmonic functions on open, smoothly-bordered subdomains of Riemann surfaces embeddable into $\C¶^2$. The formula may be considered as an analogue of the Green's formula for domains in $\C$.
We construct a Cauchy type formula on open subdomains of Riemann surfaces
We present a construction of the explicit Hodge decomposition for $\bar\partial$-equation on Riemann surfaces.
We construct an explicit homotopy formula for the d-bar complex on a complete intersection subvariety V in CP^n. This formula can be interpreted as a Hodge-type decomposition for residual currents on V.
In the article we address two issues related to the perturbation method introduced by Zhang and Lu, and applied to solving linear stochastic parabolic PDE. Those issues are: the construction of the perturbation series, and its convergence.
We show that the complex Radon transform realizes an isomorphism between the space of residual $\bar\partial$-cohomologies of a locally complete intersection subvariety in a linearly concave domain of ${\C}P^n$ and the space of holomorphic solutions of the associated homogeneous system of differential equations withconstant coefficients in the dual domain in $({\C}P^n)^*$.
Let $G\subset \C P^n$ be a linearly convex compact with smooth boundary, $D={\C}P^n\setminus G$, and let $D^* \subset (\C P^n)^*$ be the dual domain. Then for an algebraic, not necessarily reduced, complete intersection subvariety $V$ of dimension $d$ we construct an explicit inversion formula for the complex Radon transform $R_V:\ H^{d,d-1}(V\cap D)\to H^{1,0}(D^*)$, and explicit formulas for solutions of an appropriate boundary value problem for the corresponding system of differential equations with constant coefficients on $D^*$.
We study solvability of the {\it generalized Possio integral equation} - a tool in analysis of a boundary value problem in 2D subsonic aeroelasticity with the Kutta-Joukowski condition - {\it "zero pressure discontinuity"} - $ψ(x,0,t)=0$ on the complement of a finite interval in the whole real line $\R$. The corresponding problem with boundary condition on finite intervals adjacent to the "chord" was considered in \cite{P}.
We study a boundary value problem in subsonic aeroelasticity.
We prove that the integral operators $R_r$ and $H_r$ constructed in \cite{P} and such that $$f = \bar\partial_{\bold M} R_r(f) + R_{r+1}(\bar\partial_{\bold M} f) + H_r(f),$$ for a differential form $f \in C_{(0,r)}^{\infty}({\bold M})$ on a regular q-concave CR manifold ${\bold M}$ admit sharp estimates in the Lipschitz scale.