Limit behaviour of optimal control problems governed by elliptic boundary value problems with equivalued surface

In this paper, we discuss the limit behaviour of optimal control problems governed by elliptic boundary value problems with equivalued surface when the equivalued surface boundary shrinks to a fixed point on the outer boundary of a bounded domain.     

Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of isoperimetric regions: for instance, local convexity of the cone at some boundary point.

We also characterize which are the stable regions in a convex cone, i.e., second order minima of perimeter under a volume constraint. From this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone.

     

Total variation and Cheeger sets in Gauss space

The aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the Gauss space, in both the finite and infinite dimensional case. We first study the finite dimensional case, proving the existence of a maximal Cheeger set which is convex inside any bounded convex set. We also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set. Then we extend these results in the context of the abstract Wiener space, and for that we study the total variation denoising problem in this context.     

向量值测度与线性系统最优控制问题的必要条件

The vector-valued measure defined by the well-posed linear boundary value problems is discussed. The maximum principle of the optimal control problem with non-convex constraintis proved by using the vector-valued measure. Especially, the necessary conditions of the optimal control of elliptic systems is derived without the convexity of the control domain and the cost function. optimal control, maximum principle, distributed parameter system, linear system,vector-valued measure.     

Some homogenization problems for the system of elasticity with nonlinear boundary conditions in perforated domains

The system of linear elasticity is considered in a perforated domain with an ε-periodic structure. External forces nonlinearly depending on the displacements are applied to the surface of the cavities (or channels), while the body is fixed along the outer portion of its boundary. We investigate the asymptotic behavior of solutions to such boundary value problems asε→0 and construct the limit problem, according to the external surface forces and their dependence on the parameter ε. In some cases, this dependence results in the homogenized problem having the form of a variational inequality over a certain closed convex cone in a Sobolev space. This cone is described in terms of the functions involved in the nonlinear boundary conditions on the perforated boundary. A homogenization theorem is also proved for some unilateral problems with boundary conditions of Signorini type for the system of elasticity in a perforated domain. We discuss some cases when the homogenized tensor may depend on the functions specifying the boundary conditions.     

Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2

Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so‐called Ma‐Trudinger‐Wang condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport map. Moreover, our new condition, again combined with the strict convexity of the nonfocal domains, allows us to prove that all injectivity domains are strictly convex too. These results apply, for instance, on any small C⁴‐deformation of the 2‐sphere. © 2009 Wiley Periodicals, Inc.     

Lower semicontinuity in domain optimization problems

In the present paper, the lower semicontinuity of certain classes of functionals is studied when the domain of integration, which defines the functionals, is not fixed. For this purpose, a certain class of domains introduced by Chenais is employed. For this class of domains, a basic lemma is proved that plays an essential role in the derivations of the lower-semicontinuity theorems. These theorems are applied to the study of the existence of the optimal domain in domain optimization problems; a boundary-value problem of Neumann type or Dirichlet type is the main constraint in these optimization problems.The author wishes to express his sincere thanks to the reviewer for his valuable comments, which made the paper more readable; the reviewer also pointed out that Lemma 2.1 in the text is a direct corollary to a lemma by Chenais (Ref. 9). He thanks Prof. Y. Sakawa of Osaka University for encouragement.     

A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary

We present a discontinuous Galerkin method, based on the classical method of Nitsche, for elliptic problems with an immersed boundary representation on a structured grid. In such methods very small elements typically occur at the boundary, leading to breakdown of the discrete coercivity as well as numerical instabilities. In this work we propose a method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain. This construction allows us to prove the crucial inverse inequality that leads to a coercive bilinear form and as a consequence we obtain optimal order a priori error estimates. Furthermore, we prove a bound of the condition number of the stiffness matrix. All the results are valid for polynomials of arbitrary order. We also discuss the implementation of the method and present numerical examples in three dimensions.     

The first Robin eigenvalue with negative boundary parameter

We give a counterexample to the long standing conjecture that the ball maximises the first eigenvalue of the Robin eigenvalue problem with negative parameter among domains of the same volume. Furthermore, we show that the conjecture holds in two dimensions provided that the boundary parameter is small. This is the first known example within the class of isoperimetric spectral problems for the first eigenvalue of the Laplacian where the ball is not an optimiser.     

An integral equation in conformal geometry

Motivated by Carleman's proof of the isoperimetric inequality in the plane, we study the problem of finding a metric with zero scalar curvature maximizing the isoperimetric ratio among all zero scalar curvature metrics in a fixed conformal class on a compact manifold with boundary. We derive a criterion for the existence and make a related conjecture.     

求解二维多区域声波传播问题的一种边界元方法

针对多区域中声波的传播问题,其中每个散射区域的介质是相同的,将散射区域内的声波用一种单双层混合位势的形式来表示,再应用Green定理表示出外部介质区域中的声波,并形成相应的边界积分方程.如果区域个数为M时,传统的边界元方法最终将形成2M个边界积分方程并对应2M个未知函数,而本文的边界元方法最终只形成M个边界积分方程以及对应M个未知函数,从而使得求解的方程和未知数的个数都减少了一倍.最后,通过对数值算例的求解,验证了该方法的可行性及精确性.     

Partially overdetermined elliptic boundary value problems

We consider semilinear elliptic Dirichlet problems in bounded domains, overdetermined with a Neumann condition on a proper part of the boundary. Under different kinds of assumptions, we show that these problems admit a solution only if the domain is a ball. When these assumptions are not fulfilled, we discuss possible counterexamples to symmetry. We also consider Neumann problems overdetermined with a Dirichlet condition on a proper part of the boundary, and the case of partially overdetermined problems on exterior domains.     

An algorithm to solve boundary value problems for differential iinclusions and applications in optimal control

Convergence results for simplicia1 fixed point algorithms applied to problems in Banach-spaces enable constructive proofs of the existence of fixed points for set valued operators [14]. Boundary value problems for differential inclusions will be interpreted in this context. The resulting new algorithm allows numerical treatment of boundary value problems with Peano-typedynamics. The necessary conditions of the Pontryagin Maximum Principle are discussed in this framework, leading to a new indirect method for the computation of optimal trajectories with its focus on global convergence conditions for compact control domains.     

A Bernoulli problem with non-constant gradient boundary constraint

In this article, we present a result about the existence and convexity of solutions to a free boundary problem of Bernoulli type, with non-constant gradient boundary constraint depending on the outer unit normal. In particular, we prove that, in the convex case, the existence of a subsolution guarantees the existence of a classical solution, which is proved to be convex.     

Linear elliptic boundary value problems in varying domains

The paper is devoted to the study of solutions to linear elliptic boundary value problems in domains depending smoothly on a small perturbation parameter. To this end we transform the boundary value problem onto a fixed reference domain and obtain a problem in a fixed domain but with differential operators depending on the perturbation parameter. Using the Fredholm property of the underlying operator we show the differentiability of the transformed solution under the assumption that the dimension of the kernel does not depend on the perturbation parameter. Furthermore, we obtain an explicit representation for the corresponding derivative.     

Dimension splitting for evolution equations

In this paper, we are concerned with splitting methods for the time integration of abstract evolution equations. We introduce an analytic framework which allows us to prove optimal convergence orders for various splitting methods, including the Lie and Peaceman–Rachford splittings. Our setting is applicable for a wide variety of linear equations and their dimension splittings. In particular, we analyze parabolic problems with Dirichlet boundary conditions, as well as degenerate equations on bounded domains. We further illustrate our theoretical results with a set of numerical experiments. This work was supported by the Austrian Science Fund under grant M961-N13.     

Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems

We study nonlocal equations from the area of peridynamics, an instance of nonlocal wave equation, and nonlocal diffusion on bounded domains whose governing equations contain a convolution operator based on integrals. We generalize the notion of convolution to accommodate local boundary conditions. On a bounded domain, the classical operator with local boundary conditions has a purely discrete spectrum, and hence, provides a Hilbert basis. We define an abstract convolution operator using this Hilbert basis, thereby automatically satisfying local boundary conditions. The main goal in this paper is twofold: apply the concept of abstract convolution operator to nonlocal problems and carry out a numerical study of the resulting operators. We study the corresponding initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. To connect to the standard convolution, we give an integral representation of the abstract convolution operator. For discretization, we use a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition.     

New isoperimetric estimates for solutions to Monge–Ampère equations

We prove some sharp estimates for solutions to Dirichlet problems relative to Monge–Ampère equations. Among them we show that the eigenvalue of the Dirichlet problem, when computed on convex domains with fixed measure, is maximal on ellipsoids. This result falls in the class of affine isoperimetric inequalities and shows that the eigenvalue of the Monge–Ampère operator behaves just the contrary of the first eigenvalue of the Laplace operator.     

Isoperimetric properties of Euclidean boundary moments of a simply connected domain

We consider integral functionals of a simply connected domain which depend on the distance to the domain boundary. We prove an isoperimetric inequality generalizing theorems derived by the Schwarz symmetrization method. For L ^p -norms of the distance function we prove an analog of the Payne inequality for the torsional rigidity of the domain. In compare with the Payne inequality we find new extremal domains different from a disk.     

An embedding domain approach for a class of 2-d shape optimization problems: mathematical analysis

This contribution combines a shape optimization approach to free boundary value problems of Bernoulli type with an embedding domain technique. A theoretical framework is developed which allows to prove continuous dependence of the primal and dual variables in the resulting saddle point problems with respect to the domain. This ensures the existence of a solution of a related shape optimization problem in a sufficiently large class of admissible domains.