We present a simple Bellman function proof of a bilinear estimate for elliptic operators in divergence form with real coefficients and with nonnegative potentials. The constants are dimension-free. The ▫$p$▫-range of applicability of this estimate is ▫$(1,\infty)$▫ for any real accretive (nonsymmetric) matrix ▫$A$▫ of coefficients.
COBISS.SI-ID: 16051545
We study a system of conserved quantities of the periodic Klein-Gordon equation. We obtain these quantities by means of a perturbation construction from a Lax pair of the periodic sine-Gordon equation. We show that for a suitable choice of values of the spectral parameter, our conserved quantities have simple expressions in terms of the Fourier coefficients of the initial data. Moreover, they turn out to be the action variables. This provides an interesting illustration of the role of the spectral parameter. Our perturbation construction also provides a new Lax pair for the Klein-Gordon equation, and our action variables arise from this Lax pair. This turns out to be a special case ▫$(k = 2)$▫ of a more general Lax pair for a certain ▫$k$▫-jet system of the sine-Gordon equation. The structure algebra of this Lax pair is the algebra ▫${\mathcal{TA}}_k$▫ of upper triangular ▫$k \times k$▫ block Toeplitz matrices whose blocks are elements of ▫$\mathfrak{su}(2)$▫. The Ad-invariant functions on the Lie group ▫${\mathcal{TG}}_k$▫ corresponding to the Lie algebra ▫${\mathcal{TA}}_k$▫ are needed for the construction of the integrals. These functions are not given by the spectra of the matrices. They have to be constructed by other means.
COBISS.SI-ID: 15909465
From Preface: This book is an attempt to present a coherent account of Oka theory, from the classical Oka-Grauert theory originating in the works of Kiyoshi Oka and Hans Grauert to the contemporary developments initiated by Mikhael Gromov. Chapter 1 contains some preparatory material, and Chapter 2 is a brief survey of Stein space theory. In Chapter 3 we construct open Stein neighborhoods of certain types of sets in complex spaces that are used in Oka theory. Chapter 4 contains an exposition of the theory of holomorphic automorphisms of Euclidean spaces and of the density property, a subject closely intertwined with our main theme. In Chapter 5 we develop Oka theory for stratified fiber bundles with Oka fibers (this includes the classical Oka-Grauert theory), and in Chapter 6 we treat Oka-Gromov theory for stratified subelliptic submersions over Stein spaces. Chapters 7 and 8 contain applications ranging from classical to the recent ones. In Chapter 8 we present results on regular holomorphic maps of Stein manifolds; highlights include the optimal embedding theorems for Stein manifolds and Stein spaces, proper holomorphic embeddings of some bordered Riemann surfaces into ▫${\mathbb C}^2$▫, and the construction of noncritical holomorphic functions, submersions and foliations on Stein manifolds. In Chapter 9 we explore implications of Seiberg-Witten theory to the geometry of Stein surfaces, and we present the Eliashberg-Gompf construction of Stein structures on manifolds with suitable handlebody decomposition.
COBISS.SI-ID: 16008025