Feynman formulas and functional integrals for diffusion with drift in a domain on a manifold

We obtain representations for the solution of the Cauchy-Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as limits of integrals over the Cartesian powers of the domain; the integrands are elementary functions depending on the geometric characteristics of the manifold, the coefficients of the equation, and the initial data. It is natural to call such representations Feynman formulas. Besides, we obtain representations for the solution of the Cauchy-Dirichlet problem for the diffusion equation with drift in a domain on a compact Riemannian manifold as functional integrals with respect to Weizsäcker-Smolyanov surface measures and the restriction of the Wiener measure to the set of trajectories in the domain; such a restriction of the measure corresponds to Brownian motion in a domain with absorbing boundary. In the proof, we use Chernoff’s theorem and asymptotic estimates obtained in the papers of Smolyanov, Weizsäcker, and their coauthors.     

On the regularity of the heat equation solution in non-cylindrical domains: Two approaches

In this work, we investigate the behavior of the solution of the Cauchy-Dirichlet problem for a parabolic equation set in a three-dimensional domain with edges. We also give new regularity results for the weak solution of this equation in terms of the regularity of the initial data.     

Functional Integrals for the Schrodinger Equation on Compact Riemannian Manifolds

In this paper, we represent the solution of the Cauchy problem for the Schrodinger equation on compact Riemannian manifolds in terms of functional integrals with respect to the Wiener measure corresponding to the Brownian motion in a manifold and with respect to the Smolyanov surface measures constructed from the Wiener measure on trajectories in the underlying space. The representation of the solution is obtained for the case of analytic (on some sets) potential and analytic initial condition under certain assumptions on the geometric characteristics of the manifold. In the proof, we use a method due to Doss and the representations via functional integrals of the solution to the Cauchy problem for the heat equation in a compact Riemannian manifold.     

EXISTENCE，UNIQUENESS AND PROPERTIES OF THE SOLUTIONS OF A DEGENERATE PARABOLIC EQUATION WITH DIFFUSION－ADVECTION－ABSORPTION

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On the Cauchy-Dirichlet Problem in the Half Space for Parabolic SPDEs in Weighted Hoelder Spaces

We study the Cauchy-Dirichlet problem for a second order linear parabolic stochastic differential equation (SPDE) with constant coefficients in a half-space. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Hölder classes with weights.     

A priori estimates for solutions of the first initial boundary-value problem for systems of fully nonlinear partial differential equations

We prove a priori estimates for a solution of the first initial boundary-value problem for a system of fully nonlinear partial differential equations (PDE) in a bounded domain. In the proof, we reduce the initial boundary-value problem to a problem on a manifold without boundary and then reduce the resulting system on the manifold to a scalar equation on the total space of the corresponding bundle over the manifold. St. Petersburg Architecture Building University, St. Petersburg. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 3, pp. 338–363, March, 1997.     

A priori estimates for a solution to the first initial-boundary-value problem for a system of fully nonlinear parabolic equations

A priori estimates for a solution to a system of fully nonlinear parabolic equations are obtained in a bounded domain under the condition that the solution vanishes on the boundary of the domain. The method of obtaining a priori estimates is based on the possibility of reducing the problem under consideration to the Cauchy problem for a scalar equation on a manifold without boundary in some linear space. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 46–71.     

Nonlinear potentials of the cauchy-dirichlet problem for the integrodifferential Bellman equation

The Cauchy-Dirichlet problem for the integrodifferential Bellman equation, arising in the theory of contolled Ito processes, is investigated. Sufficient conditions are given, under which this problem has a viscosity solution Lipschitz continuous inx and Hölder continuous int with the exponent 1/2. The proof is based on the method of nonlinear potentials.     

THE PERMUTATION FORMULA OF SINGULAR INTEGRALS WITH BOCHNER-MARTINELLI KERNEL ON STEIN MANIFOLDS

Using the method of localization, the authors obtain the permutation formula of singular integrals with Bochner-Martinelli kernel for a relative compact domain with C(1) smooth boundary on a Stein manifold. As an application the authors discuss the regularization problem for linear singular integral equations with Bochner-Martinelli kernel and variable coefficients; using permutation formula, the singular integral equation can be reduced to a fredholm equation.     

A contact problem for a viscoelastic plate and an elastic beam

Under consideration is the problem of contact of a viscoelastic plate with an elastic beam. To characterize the viscoelastic deformation of the plate, the hereditary integrals are used. The differential formulation of the problem with the conditions in the form of a system of equalities and inequalities in the domain of possible contact is presented, and its equivalence to a variational inequality is proved. The unique solvability of the problem is proved as well as the existence of the time derivative of the solution. A limit problem is also considered as the bending rigidity of the plate tends to infinity.     

On the asymptotics of a solution to an equation with a small parameter at some of the highest derivatives

We study the asymptotic behavior of a solution of the first boundary value problem for a second-order elliptic equation in a nonconvex domain with smooth boundary in the case where a small parameter is a factor at only some of the highest derivatives and the limit equation is an ordinary differential equation. Although the limit equation has the same order as the initial equation, the problem is singulary perturbed. The asymptotic behavior of its solution is studied by the method of matched asymptotic expansions.     

A hybrid method for numerical solution of Poisson’s equation in a domain with a dielectric corner

An electrostatic problem of determining a potential in a domain containing an incoming dielectric corner, which reduces to solving Poisson’s equation in this domain, is considered. A specific feature of the solution of this problem is that it is bounded in a neighborhood of the dielectric corner but its gradient increases without limit. An efficient hybrid algorithm for the numerical solution of the problem, based on the finite element method and taking into account the known asymptotic representation of the solution in the neighborhood of the dielectric corner, is proposed.     

Existence and uniqueness of a regular solution of the Cauchy-Dirichlet problem for the equation of turbulent filtration

The existence and uniqueness of a regular solution of the Cauchy-Dirichlet problem for doubly nonlinear parabolic equations are proved, and inner and boundary weighted gradient estimates for these solutions are established. The equation of turbulent filtration is admissible for the results obtained. Bibliography: 19 titles. Dedicated to O. A. Ladyzhenskaya on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997, pp. 153–198. Translated by A. V. Ivanov.     

On the Global-in-Time Existence of a Generalized Solution to a Free-Boundary Problem

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.

     

Model oblique derivative problem for the heat equation with a discontinuous boundary function

The oblique derivative problem for the heat equation is considered in a model formulation with a boundary function that can be discontinuous and with the boundary condition understood as the limit in the normal direction almost everywhere on the lateral boundary of the domain. An example is given showing that the solution is not unique in this formulation. A solution is sought in the parabolic Zygmund space H ₁, which is an analogue of the parabolic Hölder space for an integer smoothness exponent. A subspace of H ₁ is introduced in which the existence and uniqueness of the solution is proved under suitable assumptions about the data of the problem.     

A generalized Cauchy problem for the linear differential equations of coupled physical - mechanical fields

A generalized Cauchy problem for a partial differential equation with constant coefficients, which is encountered in the study of physical processes in continuous media with widened physical - mathematical fields (see /1/) (generalized coupled thermoeleasticity /2/, coupled thermoeleasticity, porous media saturated with a viscous fluid /5/, mass and heat transfer /6/, linearized magnetoelasticity /7/, etc.) is considered. The characteristic properties of the solution of the problem, under certain constraints imposed on an equation by the stability condition, are studied. The presence of waves of higher and lower order is characteristic for the solution; in the course of time the lower-order waves are maintained and take a characteristic form. In the general case, the solution is represented in the form of integrals over the segments which link the singular points of Fourier - Laplace transforms with respect to time of the solution under consideration. The methods proposed enable an exact investigation to be made of the processes described by the equation for any time constants, and they also enable one to isolate the singularities at the fronts of propagating perturbations. As an application, the dynamic processes taking place in a thermoelastic subsapce (2) as a result of applying a mechanical and a thermal input at the boundary is studied. It is shown that in the case of unit perturbation of the boundary, the stress and temperature waves in the course of time assume a bell-shaped form and propagate with adiabatic velocity. A numerical analysis of the process which occurs due to sudden application of the force and of the thermal shock at the boundary is given.     

Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach

We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ?Ω of the domain as the square root of the distance to ?Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.     

The inverse problem of determining the source in the stationary transport equation on a riemannian manifold

The transport equation with an unknown right-hand side is considered on a compact Riemannian manifold. The right-hand side of this equation is recovered from values of the outcoming flow. Assumptions under which the solution of the inverse problem is unique are formulated.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 236–242.     

Formulation of a Ritz-Galerkin type procedure for the approximate solution of the neutron transport equation

The one-group neutron transport equation is commonly given as an integrodifferential equation for the neutron density ψ(x, ω) over a domain G × S in the five-dimensional phase space $\text{E}{}_3\text{× S(¦ ω ¦ = 1)}$. In this paper we show how, by decomposing the domain of the transport operator into a complementary pair of manifolds by means of a projection operator, any transport problem can be formulated, on either manifold, in terms of a symmetric positive definite operator. We use Friedrichs' method to extend the operator to a selfadjoint operator and look for a generalized solution by minimizing a certain functional over the appropriate Hilbert space. A Ritz-Galerkin type approximation procedure is formulated, and an estimate for the difference between the exact and approximate solution is given. The procedure is illustrated for a special choice of finite dimensional subspace.     

Coefficient inverse problem for Poisson’s equation in a cylinder

The inverse problem of determining the coefficient on the right-hand side of Poisson’s equation in a cylindrical domain is considered. The Dirichlet boundary value problem is studied. Two types of additional information (overdetermination) can be specified: (i) the trace of the solution to the boundary value problem on a manifold of lower dimension inside the domain and (ii) the normal derivative on a portion of the boundary. (Global) existence and uniqueness theorems are proved for the problems. The study is performed in the class of continuous functions whose derivatives satisfy a Hölder condition.