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Institute
This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces.
In a bounded domain with smooth boundary in R^3 we consider the stationary Maxwell equations
for a function u with values in R^3 subject to a nonhomogeneous condition
(u,v)_x = u_0 on
the boundary, where v is a given vector field and u_0 a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.
We consider the Navier-Stokes equations in the layer R^n x [0,T] over R^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed Hölder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all t in [0,T]. On using the particular properties of the de Rham complex we conclude that the Fréchet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of Hölder spaces.
The aim of this paper is to bring together two areas which are of great importance for the study of overdetermined boundary value problems. The first area is homological algebra which is the main tool in constructing the formal theory of overdetermined problems. And the second area is the global calculus of pseudodifferential operators which allows one to develop explicit analysis.
We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy-Riemann equations. The boundary value problem need not satisfy the Shapiro-Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed
problems, and construct an explicit formula for approximate solutions.
When trying to extend the Hodge theory for elliptic complexes on compact closed manifolds to the case of compact manifolds with boundary one is led to a boundary value problem for
the Laplacian of the complex which is usually referred to as Neumann problem. We study the Neumann problem for a larger class of sequences of differential operators on
a compact manifold with boundary. These are sequences of small curvature, i.e., bearing the property that the composition of any two neighbouring operators has order less than two.
Let A be a nonlinear differential operator on an open set X in R^n and S a closed subset of X. Given a class F of functions in X, the set S is said to be removable for F relative to A if any weak solution of A (u) = 0 in the complement of S of class F satisfies this equation weakly in all of X. For the most extensively studied classes F we show conditions on S which guarantee that S is removable for F relative to A.
We describe a natural construction of deformation quantisation on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.
We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to the study of Lagrangian problems.
We consider a Cauchy problem for the heat equation in a cylinder X x (0,T) over a domain X in the n-dimensional space with data on a strip lying on the lateral surface. The strip is of the form
S x (0,T), where S is an open subset of the boundary of X. The problem is ill-posed. Under natural restrictions on the configuration of S we derive an explicit formula for solutions of this problem.
We continue our study of invariant forms of the classical equations of mathematical physics,
such as the Maxwell equations or the Lamé system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded
in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations.
In this paper we study the convergence of continuous Newton method for solving nonlinear equations with holomorphic mappings in complex Banach spaces. Our contribution is based on a recent progress in the geometric theory of spirallike functions. We prove convergence theorems and illustrate them by numerical simulations.
The paper is devoted to asymptotic analysis of the Dirichlet problem for a second order partial differential equation containing a small parameter multiplying the highest order derivatives. It corresponds to a small perturbation of a dynamical system having a stationary solution in the domain. We focus on the case where the trajectories of the system go into the domain and the stationary solution is a proper node.