On the convergence of solutions of variational problems with bilateral obstacles in variable domains

We establish sufficient conditions for the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral obstacles in variable domains. The given obstacles are elements of the corresponding Sobolev space, and the degeneracy on a set of measure zero is admitted for the difference between the upper and lower obstacles. We show that a weakening of the condition of positivity of this difference on a set of full measure may lead to a certain violation of the established convergence result.     

Connection Between Smoothing Problems with Obstacles and Weights

We study how to reduce the smoothing problems with obstacles to the solution of smoothing problems with weights. We prove that, for problems with obstacles in Hilbert spaces, and in the classical case, especially, in several variable problems, the associated Lagrangian has a saddle point. This implies the existence of equivalent problems with weights.     

Mean curvature flow with obstacles

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss an application of this result to the positive mean curvature flow.     

INVERSE SCATTERING PROBLEMS BY SINGULAR SOURCE METHODS

The inverse scattering problems are to detect the property of obstacles from the measurements outside the obstacles. One of important research areas in this topic is the recovery of boundary property for impenetrable obstacles. In this paper, we would like to give a brief review about the recently developed singular source methods. There are three different methods in this category, namely, linear sampling method, pointsource method and probe method. We also present some recent new results about the probe method.     

Perturbative Analysis of Wave Interaction in Nonlinear Systems

We propose a new way to handle obstacles to asymptotic integrability in perturbed nonlinear PDEs in the method of normal forms (NFs) in the case of multiwave solutions. Instead of including the whole obstacle in the NF, we include only its resonant part (if it exists) in the NF and assign the remainder to the homological equation. This leaves the NF integrable, and its solutions retain the character of the solutions of the unperturbed equation. We use the freedom in the expansion to construct canonical obstacles that are confined to the interaction region of the waves. For soliton solutions (e. g., of the KdV equation), the interaction region is a finite domain around the origin; the canonical obstacles then do not generate secular terms in the homological equation. When the interaction region is infinite (or semi-infinite, e.g., in wave-front solutions of the Burgers equation), the obstacles may contain resonant terms. The obstacles generate waves of a new type that cannot be written as functionals of the solutions of the NF. When the obstacle contributes a resonant term to the NF, this leads to a nonstandard update of the wave velocity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 410–422, August, 2005.     

Solving the two-dimensional findpath problem using a line-triangle representation of the robot

We have developed a new approach to computing the collision boundary of a collection of obstacles grown by a convex robot. The essential idea of this approach involves first representing the robot as a set sum of line segments and triangles. The process of growing the obstacles by the robot can then be viewed as a sequence of steps where each step involves growing the partial grown collection of obstacles by a line segment or a triangle. A fast algorithm has been presented to solve this problem.     

Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

In this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non-homogeneous obstacles using convolution quadrature (CQ) techniques for the time variable and coupled boundary element method/finite element method for the space variable. The applicability of CQ to waves requires polynomial type bounds for operators related to the operator Δ − s ² in the right half complex plane. We propose a new systematic way of dealing with this problem, both at the continuous and semidiscrete-in-space cases. We apply the technique to three different situations: scattering by a group of sound-soft and -hard obstacles, by homogeneous and non-homogeneous obstacles.     

Cauchy Problem for Semilinear Parabolic Equation with Time-Dependent Obstacles: A BSDEs Approach

In the paper we apply methods of the theory of backward stochastic differential equations to prove existence, uniqueness and stochastic representation of solutions of the Cauchy problem for semilinear parabolic equation in divergence form with two time-dependent obstacles. We consider two quite different cases: problems with distinct quasi-continuous obstacles and with irregular obstacles satisfying the so called Mokobodzki condition. As an application we also generalize the Lewy-Stampacchia inequality to non-Radon measures and give new existence result for the Dynkin game problem.     

Computing simple paths among obstacles

Given a set X of points in the plane, two distinguished points s,tX, and a set Φ of obstacles represented by line segments, we wish to compute a simple polygonal path from s to t that uses only points in X as vertices and avoids the obstacles in Φ. We present two results: (1) we show that finding such simple paths among arbitrary obstacles is NP-complete, and (2) we give a polynomial-time algorithm that computes simple paths when the obstacles form a simple polygon P and X is inside P. Our algorithm runs in time O(m²n²), where m is the number of vertices of P and n is the number of points in X.     

Escape Probabilities for Branching Brownian Motion Among Soft Obstacles

We derive asymptotics for the quenched probability that a critical branching Brownian motion killed at a small rate ε in Poissonian obstacles exits from a large domain. Results are formulated in terms of the solution to a semilinear partial differential equation with singular boundary conditions. The proofs depend on a quenched homogenization theorem for branching Brownian motion among soft obstacles.     

Variational problems with obstacles and integral constraints

Various problems in mathematics and physics can be formulated in terms of a variational problem with obstacles and integral constraints, e.g. finding a surface of minimal area with prescribed volume in a bounded region.We are concerned with the regularity of solutions of variational problems: We show that the minima of a variational integral under all Sobolewfunctions with prescribed boundary values, lying between two obstacles, and fulfilling some integral constraints, are bounded and Hölder-continuous. We do not assume any differentiability or convexity of the integrand, but only a Caratheodory and a growth condition.This research has been supported by the Sonderforschungsbereich 72 of the Deutsche Forschungsgemeinschaft.     

On the Invertible Boundary Integral Operator in the Exterior Impedance Problem for the Helmholtz Equation

We propose an approach for reduction of the impedance problem for propagative Helmholtz equation outside several obstacles to the uniquely solvable Fredholm integral equation of the second kind and index zero. The integral equation in this approach is derived by introducing auxiliary boundaries with an appropriate boundary conditions inside obstacles.     

Resonances in scattering by two magnetic fields at large separation and a complex scaling method

We study the quantum resonances in magnetic scattering in two dimensions. The scattering system consists of two obstacles by which the magnetic fields are completely shielded. The trajectories trapped between the two obstacles are shown to generate the resonances near the positive real axis, when the distance between the obstacles goes to infinity. The location is described in terms of the backward amplitudes for scattering by each obstacle. A difficulty arises from the fact that even if the supports of the magnetic fields are largely separated from each other, the corresponding vector potentials are not expected to be well separated. To overcome this, we make use of a gauge transformation and develop a new type of complex scaling method. We can cover the scattering by two solenoids at large separation as a special case. The obtained result heavily depends on the magnetic fluxes of the solenoids. This indicates that the Aharonov–Bohm effect influences the location of resonances.     

Brownian asymptotics in a Poissonian environment

Summary We consider a Brownian motion moving in a random potential obtained by translating a given fixed non negative shape function at the points of a Poisson cloud. We derive the almost sure principal long time behavior of the expectation of the natural Feynman Kac functional, which is insensitive to the detail of the shape function. We also study the situation of hard obstacles where Brownian motion is killed once it comes within distancea of a point of the cloud. The nature of the results then changes between the case whena is small or large in connection with the presence, or absence of an infinite component in the complement of the obstacles.     

The complexity of the free space for a robot moving amidst fat obstacles

We propose a new definition of fatness of geometric objects and compare it with alternative definitions. We show that, under some realistic assumptions, the complexity of the free space for a robot, with any fixed number of degrees of freedom moving in a d-dimensional Euclidean workspace with fat obstacles, is linear in the number of obstacles. The complexity of motion planning algorithms depends on the complexity of the robot's free space, and theoretically, the complexity of the free space can be very high. Thus, our result opens the way to devising provable efficient motion planning algorithms in certain realistic settings.     

BSDEs with two RCLL reflecting obstacles driven by Brownian motion and Poisson measure and a related mixed zero-sum game

In this paper we study Backward Stochastic Differential Equations with two reflecting right continuous with left limit obstacles (or barriers) when the noise is given by Brownian motion and a mutually independent Poisson random measure. The jumps of the obstacle processes could be either predictable or inaccessible. We show the existence and uniqueness of the solution when the barriers are completely separated and the generator uniformly Lipschitz. We do not assume the existence of a difference of supermartingales between the obstacles. As an application, we show that the related mixed zero-sum differential–integral game problem has a value.     

Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options

In this paper we prove optimal interior regularity for solutions to the obstacle problem for a class of second order differential operators of Kolmogorov type. We treat smooth obstacles as well as non-smooth obstacles. All our proofs follow the same line of thought and are based on blow-ups, compactness, barriers and arguments by contradiction. The problem considered arises in financial mathematics, when considering path-dependent derivative contracts with early exercise feature.     

On the piano movers' problem: V. The case of a rod moving in three-dimensional space amidst polyhedral obstacles

This paper, a fifth in a series, solves some additional 3-D special cases of the ?piano movers”? problem, which arises in robotics. The main problem solved in this paper is that of planning the motion of a rod moving amidst polyhedral obstacles. We present polynomial-time motion-planning algorithms for this case, using the connectivity-graph technique described in the preceding papers. We also study certain more general polyhedral problems, which arise in the motion planning problem considered here but have application to other similar problems. Application of these techniques to the problem of planning the motion of a general polyhedral body moving in 3-space amidst polyhedral obstacles is also described.     

Decaying states of perturbed wave equations

We study the solutions of perturbed wave equations that represent free wave motion outside some ball. When there are no trapped rays, it is shown that every solution whose total energy decays to zero must be smooth. This extends results of Rauch to the even-dimensional case and to systems having more than one sound speed. In these results, obstacles are not considered. We show that, even allowing obstacles, waves with compact spatial support cannot decay, assuming a unique continuation hypothesis. An example with obstacle is given where nonsmooth, compactly supported, decaying waves exist.