Adjoint and self-adjoint boundary value problems on a geometric graph

We consider boundary value problems of arbitrary order for linear differential equations on a geometric graph. Solutions of boundary value problems are coordinated at the interior vertices of the graph and satisfy given conditions at the boundary vertices. For considered boundary value problems, we construct adjoint boundary value problems and obtain a self-adjointness criterion. We describe the structure of the solution set of homogeneous self-adjoint boundary value problems with alternating coefficients of a differential equation and obtain nondegeneracy conditions for these boundary value problems.     

Elliptic boundary problems on manifolds with polycylindrical ends

We investigate general Shapiro-Lopatinsky elliptic boundary value problems on manifolds with polycylindrical ends. This is accomplished by compactifying such a manifold to a manifold with corners of in general higher codimension, and we then deal with boundary value problems for cusp differential operators. We introduce an adapted Boutet de Monvel's calculus of pseudodifferential boundary value problems, and construct parametrices for elliptic cusp operators within this calculus. Fredholm solvability and elliptic regularity up to the boundary and up to infinity for boundary value problems on manifolds with polycylindrical ends follows.     

Singular boundary integral equations of boundary value problems of elastic dynamics in the case of subsonic running loads

We consider boundary value problems for an elastic medium bounded by a cylindrical surface on which a load with a constant-in-time form is moving at a constant subsonic velocity. This class of problems is a model class for the dynamics of underground structures like transport tunnels and ground transport. The method of generalized functions is developed for the solution of boundary value problems. We construct dynamic analogs of Green formulas and use them to derive singular boundary integral equations, which solve the considered boundary value problems.     

Mixed boundary value problems and parametrices in the edge calculus

Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. We develop a new method for computing the interface conditions in terms of the index of boundary value problems in weighted spaces on infinite cones, combined with structures from the calculus of boundary value problems on a manifold with edges. This will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator. The approach itself is completely general.     

The Method of Singular Equations in Boundary Value Problems in Kinetic Theory

We develop a new effective method for solving boundary value problems in kinetic theory. The method permits solving boundary value problems for mirror and diffusive boundary conditions with an arbitrary accuracy and is based on the idea of reducing the original problem to two problems of which one has a diffusion boundary condition for the reflection of molecules from the wall and the other has a mirror boundary condition. We illustrate this method with two classical problems in kinetic theory: the Kramers problem (isothermal slip) and the thermal slip problem. We use the Bhatnagar-Gross-Krook equation (with a constant collision frequency) and the Williams equation (with a collision frequency proportional to the molecular velocity).__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 437–454, June, 2005.     

Boundary value problems on manifolds with fibered boundary

We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah–Patodi–Singer problems in subspaces (it contains both as special cases). The boundary conditions in this theory are taken as elements of the C *‐algebra generated by pseudodifferential operators and families of pseudodifferential operators in the fibers. We prove the Fredholm property for elliptic boundary value problems and compute a topological obstruction (similar to Atiyah–Bott obstruction) to the existence of elliptic boundary conditions for a given elliptic operator. Geometric operators with trivial and nontrivial obstruction are given. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solving Boundary-Value Problems for Systems of Hyperbolic Conservation Laws with Rapidly Varying Coefficients

We study how boundary conditions affect the multiple-scale analysis of hyperbolic conservation laws with rapid spatial fluctuations. The most significant difficulty occurs when one has insufficient boundary conditions to solve consistency conditions. We show how to overcome this missing boundary condition difficulty for both linear and nonlinear problems through the recovery of boundary information. We introduce two methods for this recovery (multiple-scale analysis with a reduced set of scales, and a combination of Laplace transforms and multiple scales) and show that they are roughly equivalent. We also show that the recovered boundary information is likely to contain secular terms if the initial conditions are nonzero. However, for the linear problem, we demonstrate how to avoid these secular terms to construct a solution that is valid for all time. For nonlinear problems, we argue that physically relevant problems do not exhibit the missing boundary condition difficulty.     

Continuity of the temperature in boundary heat control problems

Motivated by the boundary heat control problems formulated in the book of Duvaut and Lions, we study a boundary Stefan problem and a boundary porous media problem. We prove continuity of the solution with the appropriate modulus. We also extend the results to the fractional order case and to the anomalous diffusion problems.     

Conditions for the solvability of boundary value problems for quasi-elliptic systems in a half-space

We consider boundary value problems in a half-space for a class of quasi-elliptic systems with constant coefficients. We assume that the boundary value problems satisfy the Lopatinskii condition. We obtain necessary and sufficient conditions for their unique solvability in Sobolev spaces.     

Sous-solutions d'équations elliptiques dans L1

We study subsolutions for semilinear elliptic boundary value problems in L1. We consider as well nonlinear as linear boundary conditions. The nonlinear functions may be multivated. We characterize in terms of p.d.e. the subsolutions defined by a nonlinear functional analysis argument. Applications are given to obtain existence results for semilinear elliptic boundary value problems and comparison and estimates for nonlinear parabolic boundary value problems.     

First-order three-point boundary value problems at resonance

We consider three-point boundary value problems for a system of first-order equations in perturbed systems of ordinary differential equations at resonance. We obtain new results for the above boundary value problems with nonlinear boundary conditions. The existence of solutions is established by applying a version of Brouwer’s Fixed Point Theorem which is due to Miranda.     

Overdetermined partial boundary value problems on finite networks

In this study, we define a class of non-self-adjoint boundary value problems on finite networks associated with Schrödinger operators. The novel feature of this study is that no data are prescribed on part of the boundary, whereas both the values of the function and of its normal derivative are given on another part of the boundary. We show that overdetermined partial boundary value problems are crucial for solving inverse boundary value problems on finite networks since they provide the theoretical foundations for the recovery algorithm. We analyze the uniqueness and the existence of solution for overdetermined partial boundary value problems based on the nonsingularity of partial Dirichlet-to-Neumann maps. These maps allow us to determine the value of the solution in the part of the boundary where no data was prescribed. We also execute full conductance recovery for spider networks.     

Solutions of the main boundary value problems for a loaded second-order parabolic equation with constant coefficients

We give well-posed statements of the main initial–boundary value problems in a rectangular domain and in a half-strip for a second-order parabolic equation that contains partial Riemann–Liouville fractional derivatives with respect to one of the two independent variables. We construct Green functions and representations of solutions of these problems. We prove existence and uniqueness theorems for the first boundary value problem and the problem in the half-strip with the boundary condition of the first kind.     

Boundary value problems with global projection conditions

Parametrices of elliptic boundary value problems for differential operators belong to an algebra of pseudodifferential operators with the transmission property at the boundary. However, generically, smooth symbols on a manifold with boundary do not have this property, and several interesting applications require a corresponding more general calculus. We introduce here a new algebra of boundary value problems that contains Shapiro-Lopatinskij elliptic as well as global projection conditions; the latter ones are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. We show that every elliptic operator admits (up to a stabilisation) elliptic conditions of that kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. Moreover, we construct parametrices in the calculus.     

Elliptic dilation–contraction problems on manifolds with boundary. <Emphasis Type="Italic">C</Emphasis>*-theory

We study boundary value problems with dilations and contractions on manifolds with boundary. We construct a C*- algebra of such problems generated by zero-order operators. We compute the trajectory symbols of elements of this algebra, obtain an analog of the Shapiro–Lopatinskii condition for such problems, and prove the corresponding finiteness theorem.     

Homogenization of elliptic problems with alternating boundary conditions in a thick two-level junction of type 3:2:2

The asymptotic behavior of solutions to boundary value problems for the Poisson equation is studied in a thick two-level junction of type 3:2:2 with alternating boundary conditions. The thick junction consists of a cylinder with ε-periodically stringed thin disks of variable thickness. The disks are divided into two classes depending on their geometric structure and boundary conditions. We consider problems with alternating Dirichlet and Neumann boundary conditions and also problems with different alternating Fourier (Neumann) conditions. We study the influence of the boundary conditions on the asymptotic behavior of solutions as ε → 0. Convergence theorems, in particular, convergence of energy integrals, are proved. Bibliography: 31 titles. Illustrations: 1 figure.     

On the solvability of initial boundary value problems for nonlinear time-dependent Schrödinger equations

In this paper, we study the initial boundary value problems for a nonlinear time-dependent Schrödinger equation with Dirichlet and Neumann boundary conditions, respectively. We prove the existence and uniqueness of solutions of the initial boundary value problems by using Galerkin’s method.     

A survey of results on nonlinear Venttsel problems

We review the recent results for boundary value problems with boundary conditions given by second-order integral-differential operators. Particular attention has been paid to nonlinear problems (without integral terms in the boundary conditions) for elliptic and parabolic equations. For these problems we formulate some statements concerning a priori estimates and the existence theorems in Sobolev and Hölder spaces.     

Linear boundary value problems for normally solvable operator equations in a Banach space

We consider linear boundary value problems for operator equations with generalized-invertible operator in a Banach or Hilbert space. We obtain solvability conditions for such problems and indicate the structure of their solutions. We construct a generalized Green operator and analyze its properties and the relationship with a generalized inverse operator of the linear boundary value problem. The suggested approach is illustrated in detail by an example.     

Temporal boundary value problems in interfacial fluid dynamics

We study problems in interfacial fluid dynamics which do not have well-posed initial value problems. We prove existence of solutions for these problems by considering instead boundary value problems, where boundary data is specified at two different times. We develop a general framework, for problems on the real line and for problems which are spatially periodic. A variety of boundary conditions are considered, including Dirichlet, Neumann and mixed conditions. The framework is applied to two specific problems from interfacial fluid dynamics: a family of generalizations of the Boussinesq equations developed by Bona, Chen and Saut, and the vortex sheet.