What Is a Quantum Stochastic Differential Equation from the Point of View of Functional Analysis?

We prove that a quantum stochastic differential equation is the interaction representation of the Cauchy problem for the Schrödinger equation with Hamiltonian given by a certain operator restricted by a boundary condition. If the deficiency index of the boundary-value problem is trivial, then the corresponding quantum stochastic differential equation has a unique unitary solution. Therefore, by the deficiency index of a quantum stochastic differential equation we mean the deficiency index of the related symmetric boundary-value problem.In this paper, conditions sufficient for the essential self-adjointness of the symmetric boundary-value problem are obtained. These conditions are closely related to nonexplosion conditions for the pair of master Markov equations that we canonically assign to the quantum stochastic differential equation.     

Some Results on Risk-Sensitive Control with Full Observation

The Bellman equation of the risk-sensitive control problem with full observation is considered. It appears as an example of a quasi-linear parabolic equation in the whole space, and fairly general growth assumptions with respect to the space variable x are permitted. The stochastic control problem is then solved, making use of the analytic results. The case of large deviation with small noises is then treated, and the limit corresponds to a differential game. Accepted 25 March 1996     

Mixed nonlocal problem for a nonlinear singular hyperbolic equation

In this paper, we study a nonlocal mixed problem for a nonlinear hyperbolic equation. Based on some a priori estimates and some density arguments, we prove the well posedness of the associated linear problem. The existence and uniqueness of the weak solution of the nonlinear problem are then established by applying an iterative process based on the obtained results for the linear problem. Copyright © 2009 John Wiley & Sons, Ltd.     

On the transmission problem of the helmholtz equation for quadrants

The plane transmission problem of the Helmholtz equation for quadrants is characterized by a one-dimensional singular integral equation, which refers to the Fourier transform of the normal derivative of the solution along the x-axis. It is derived by solving the transmission problem for the upper and the lower half-plane involving a Neumann condition at y = 0. This is done by a two-dimensional Laplace transform technique. The inverse Laplace transform with respect to the second cartesian coordinate and the restriction of this one to y = 0 then lead to the integral equation. Thereby the transmission conditions of the original problem at y = 0 have to be taken into account. The resulting integral equation is of generalized Wiener-Hopf-type. It is solved via the contraction theorem imposing restricting conditions on the wave numbers.     

Conservation laws with a random source

We study the scalar conservation law with a noisy nonlinear source, namely,u _l + f(u)_x = h(u, x, t) + g(u)W(t), whereW(t) is the white noise in the time variable, and we analyse the Cauchy problem for this equation where the initial data are assumed to be deterministic. A method is proposed to construct approximate weak solutions, and we then show that this yields a convergent sequence. This sequence converges to a (pathwise) solution of the Cauchy problem. The equation can be considered as a model of deterministic driven phase transitions with a random perturbation in a system of two constituents. Finally we show some numerical results motivated by two-phase flow in porous media. This research has been supported by VISTA (a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap, Statoil) and NAVF (the Norwegian Research Council for Science and the Humanities).     

On holomorphic solutions of some inhomogeneous linear differential equations in a banach space over a non-archimedean field

Let A be a closed linear operator on a Banach space $ \mathfrak{B} $ \mathfrak{B} over the field Ω of complex p-adic numbers having an inverse operator defined on the whole $ \mathfrak{B} $ \mathfrak{B} , and f be a locally holomorphic at 0 $ \mathfrak{B} $ \mathfrak{B} -valued vector function. The problem of existence and uniqueness of a locally holomorphic at 0 solution of the differential equation y ^(m) − Ay = f is considered in this paper. In particular, it is shown that this problem is solvable under the condition $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} = 0. It is proved also that if the vector-function f is entire, then there exists a unique entire solution of this equation. Moreover, the necessary and sufficient conditions for the Cauchy problem for such an equation to be correctly posed in the class of locally holomorphic functions are presented.     

Global existence to a reaction–diffusion equation through a self‐similar‐like upper solution

A demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation ?u_t+Δu+u^p=0 in ?ⁿ×[0, ∞), for exponents p?3 and space dimensions n?3. This method does not require the initial value to have a specific uniform smallness condition, but rather to satisfy a bell‐like form. The method is based on a specific upper solution, which models the diffusion process of the heat equation. The upper solution is not self‐similar, but does have a self‐similar‐like form. After transforming the reaction–diffusion problem into an equivalent one, whose initial value is uniformly very small, a local solution is obtained in the time interval [0, 1] by the use of this upper solution. This local solution is then extended to [0, ∞) through an infinite sequence of extensions. At each step, an appropriate change of variables will transform the extension into a problem nearly identical to the local problem in [0, 1]. These transformations exploit the diffusive and self‐similar‐like nature of the upper solution. Copyright © 2009 John Wiley & Sons, Ltd.     

On a boundary value problem for a p‐Dirac equation

The p‐Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p‐Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p‐Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p‐Laplace equation into the p‐Dirac equation. This equation will be solved iteratively by using a fixed‐point theorem. Applying operator‐theoretical methods for the p‐Dirac equation and p‐Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd.     

Integral representation for the solution of the stationary Schrödinger equation in a cone

In this paper, we consider the Dirichlet problem for the stationary Schrödinger equation in a cone with continuous boundary data. For a solution u of the stationary Schrödinger equation in a cone, we prove that if its positive part u⁺ satisfying a slowly growing condition, then its negative part u^? can also be dominated by a similar slowly growing condition. Meanwhile, u can be represented by its integral in the boundary of the cone.     

Two Dimensional Incompressible Ideal Flow Around a Small Obstacle

Abstract

In this article we study the asymptotic behavior of incompressible, ideal, time-dependent two dimensional flow in the exterior of a single smooth obstacle when the size of the obstacle becomes very small. Our main purpose is to identify the equation satisfied by the limit flow. We will see that the asymptotic behavior depends on γ, the circulation around the obstacle. For smooth flow around a single obstacle, γ is a conserved quantity which is determined by the initial data. We will show that if γ = 0, the limit flow satisfies the standard incompressible Euler equations in the full plane but, if γ≠ 0, the limit equation acquires an additional forcing term. We treat this problem by first constructing a sequence of approximate solutions to the incompressible 2D Euler equation in the full plane from the exact solutions obtained when solving the equation on the exterior of each obstacle and then passing to the limit on the weak formulation of the equation. We use an explicit treatment of the Green's function of the exterior domain based on conformal maps, a priori estimates obtained by carefully examining the limiting process and the Div-Curl Lemma, together with a standard weak convergence treatment of the nonlinearity for the passage to the limit.     

Boundary integral equations for the Helmholtz equation: The third boundary value problem

In this paper we study the application of boundary integral equation methods for the solution of the third, or Robin, boundary value problem for the exterior Helmholtz equation. In contrast to earlier work, the boundary value problem is interpreted here in a weak sense which allows data to be specified in L_∞ (?D), ?D being the boundary of the exterior domain which we assume to be Lyapunov of index 1. For this exterior boundary value problem, we employ Green's theorem to derive a pair of boundary integral equations which have a unique simultaneous solution. We then show that this solution yields a solution of the original exterior boundary value problem.     

Reduction of Time-Dependent Systems Admitting a Superposition Principle

The problem of differential equation systems admitting a nonlinear superposition principle is analyzed from a geometric perspective. We show how it is possible to reduce the problem of finding the general solution of such a differential equation system defined by a Lie group G to a pair of simpler problems, one in a subgroup H and the other on a homogeneous space. The theory is illustrated with several examples and applications.     

Elastic wave diffraction by a wedge-shaped inclusion

By means of Sommerfeld integrals techniques the problem of diffraction is reduced to a functional equation in a complex domain. Such an equation is known in the literature as an equation with non-Carleman translation. The theory of pseudodifferential operators reduces to the theory of functional equations and, consequently, the problem of diffraction, to a Fredholm integral equation. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 50–70, 1990. Translated by B. V. Budaev.     

General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients

In this article, we consider a linear-quadratic optimal control problem (LQ problem) for a controlled linear stochastic differential equation driven by a multidimensional Browinan motion and a Poisson random martingale measure in the general case, where the coefficients are allowed to be predictable processes or random matrices. By the duality technique, the dual characterization of the optimal control is derived by the optimality system (so-called stochastic Hamilton system), which turns out to be a linear fully coupled forward-backward stochastic differential equation with jumps. Using a decoupling technique, the connection between the stochastic Hamilton system and the associated Riccati equation is established. As a result, the state feedback representation is obtained for the optimal control. As the coefficients for the LQ problem are random, here, the associated Riccati equation is a highly nonlinear backward stochastic differential equation (BSDE) with jumps, where the generator depends on the unknown variables K, L, and H in a quadratic way (see (5.9) herein). For the case where the generator is bounded and is linearly dependent on the unknown martingale terms L and H, the existence and uniqueness of the solution for the associated Riccati equation are established by Bellman's principle of quasi-linearization.     

An integral equation arising in a convective heat (mass) transfer problem through a boundary layer

A problem of Lighthill and Levich, where convective heat (mass) transfer occurs through a boundary layer, is considered. This gives rise to a certain integral equation. The unique L₂ solution of a generalized version of this integral equation is obtained providing the solution, in closed form, of the original problem.     

On a nonlinear elliptic equation arising in a free boundary problem

 Let p ^* =n/(n−2) and n≥3. In this paper, we first classify all non-constant solutions of

Two inverse problems of finding a coefficient in a parabolic equation

For a parabolic equation, we consider inverse problems of reconstructing a coefficient that depends on the space variables alone. The first problem is to find a lower-order coefficient c(x) multiplying u(x, t), and the second problem is to find the coefficient a(x) multiplying Δu. As additional information, the integral of the solution with respect to time with some weight function is given. The coefficients of the equation depend both on time and on the space variables. We obtain sufficient conditions for the existence of generalized solutions of our problems; moreover, for the first problem, we also prove uniqueness and construct an iterative sequence that converges to the desired coefficient almost everywhere in the domain. We present examples of input data of these problems for which the assumptions of our theorems are necessarily true.     

The first initial boundary value problem for a compound type equation in a three-dimensional multiply connected region

The method of boundary integral equations is used for solving the first initial boundary value problem for a compound type equation in a three-dimensional multiply connected region. The problem is reduced to a uniquely solvable integral equation. The solution of the problem is obtained in the form of dynamic potentials whose density satisfies this integral equation. Thus the existence theorem is proved. Moreover, the uniqueness of the solution is also studied. All the results are valid for both interior and exterior regions provided that the corresponding conditions at infinity are taken into account. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 249–265, August, 2000.     

When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable?

We consider solutions of the initial-Neumann problem for the heat equation on bounded Lipschitz domains in ℝ ^N and classify the solutions whose spatial level surfaces are invariant with respect to the time variable. (Of course, the values of each solution on its spatial level surfaces vary with time.) The prototype of such classification is a result of Alessandrini, which proved a conjecture of Klamkin. He considered the initial-Dirichlet problem for the heat equation on bounded domains and showed that if all the spatial level surfaces of the solution are invariant with respect to the time variable under the homogeneous Dirichlet boundary condition, then either the initial data is an eigenfunction or the domain is a ball and the solution is radially symmetric with respect to the space variable. His proof is restricted to the initial-Dirichlet problem for the heat equation. In the present paper, in order to deal with the initial-Neumann problem, we overcome this obstruction by using the invariance condition of spatial level surfaces more intensively with the help of the classification theorem ofisoparametric hypersurfaces in Euclidean space of Levi-Civita and Segre. Furthermore, we can deal with nonlinear diffusion equations, such as the porous medium equation.     

On a resonant-superlinear elliptic problem

We start by discussing the solvability of the following superlinear problem where , is a smooth bounded domain and f satisfies a one-sided Landesman-Lazer condition. We also consider systems of semilinear elliptic equations with nonlinearities of the above form, so exhibiting superlinearity as and resonance as . A priori bounds for the solutions of the equation and the system are obtained by using Hardy-type inequalities . Existence of solutions is then obtained using topological degree arguments. Received: 18 March 2002 / Accepted: 8 May 2002 / Published online: 5 September 2002