Probabilistic approach to the neumann problem

The basic problem considered in this paper is to solve the following Neumann boundary value problem probabilistically: where we assume that q is in a certain functional class to be specified below, and φ is a bounded measurable function on the boundary. We give a martingale formulation of the Neumann problem and show that this formulation is essentially equivalent to the classical formulation. The paper culminates in an explicit formula for the solution of this problem in terms of reflecting Brownian motion and its boundary local time.     

Neumann boundary control of a coupled transport-diffusion system with boundary observation

We study the Neumann boundary stabilization problem of a coupled transport-diffusion system in the case where the observation is done at the boundary. In the recent paper of Sano and Nakagiri [H. Sano, S. Nakagiri, Stabilization of a coupled transport-diffusion system with boundary input, J. Math. Anal. Appl. 363 (2010) 57-72], we treated the stabilization problem for the case with Neumann boundary control and distributed observation. The novelty of this paper is the formulation of the boundary observation equation in a Hilbert space. We have an interesting result of its being expressed by using an -bounded operator with . Moreover, it is shown that a reduced-order model with a finite-dimensional state variable is controllable and observable. This means that one can always construct a finite-dimensional stabilizing controller for the original infinite-dimensional system by using a residual mode filter (RMF) approach.     

Solvability of the classical initial-boundary-value problems for the heat-conduction equation in a dihedral angle

One proves the unique solvability of the fundamental initial-boundary-value problems for the heat-conduction equation in an -dimensional infinite dihedral angle and one obtains coercive estimates of their solutions in weighted Hölder norms. The Neumann problem is investigated in detail; for the Dirichlet problem and for the problem with mixed boundary conditions, one gives the formulation of the basic result.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 146–180, 1984.     

Steady flows of compressible fluids in a rigid container with upper free boundary

We consider fluid in a smooth rigid container whose lateral boundary is a piece of vertical cylinder, bounded above by a free upper surface. As basic flow we consider the non homogeneous rest state in the presence of gravity, and of a surface tension. Under these assumptions, we study the existence of a steady free boundary and a steady motion in of an isothermal viscous gas, resulting as perturbation to the rest state in correspondence of small non potential perturbations to the (large potential) gravitational force. We linearize the problem by prescribing the unknown domain , then we make use of the iterative scheme introduced by Heywood and Padula. Our method is based on an iteration between the Neumann problem for a non homogeneous Stokes system for the velocity, the Neumann problem for an elliptic problem on for height, and a steady transport equation for the perturbation to the density. The difference of boundary condition between lateral boundary and free upper surfaces causes a singularity at the intersection (contact line). To avoid singularities on the contact line, we adopt weighted Sobolev spaces.     

A radial basis function method for solving PDE-constrained optimization problems

In this article, we apply the theory of meshfree methods to the problem of PDE-constrained optimization. We derive new collocation-type methods to solve the distributed control problem with Dirichlet boundary conditions and also discuss the Neumann boundary control problem, both involving Poisson’s equation. We prove results concerning invertibility of the matrix systems we generate, and discuss a modification to guarantee invertibility. We implement these methods using Matlab, and produce numerical results to demonstrate the methods’ capability. We also comment on the methods’ effectiveness in comparison to the widely-used finite element formulation of the problem, and make some recommendations as to how this work may be extended.     

Multiple traces boundary integral formulation for Helmholtz transmission problems

We present a novel boundary integral formulation of the Helmholtz transmission problem for bounded composite scatterers (that is, piecewise constant material parameters in ??subdomains??) that directly lends itself to operator preconditioning via Calderón projectors. The method relies on local traces on subdomains and weak enforcement of transmission conditions. The variational formulation is set in Cartesian products of standard Dirichlet and special Neumann trace spaces for which restriction and extension by zero are well defined. In particular, the Neumann trace spaces over each subdomain boundary are built as piecewise $\widetilde{H}^{-1/2}$ -distributions over each associated interface. Through the use of interior Calderón projectors, the problem is cast in variational Galerkin form with an operator matrix whose diagonal is composed of block boundary integral operators associated with the subdomains. We show existence and uniqueness of solutions based on an extension of Lions?? projection lemma for non-closed subspaces. We also investigate asymptotic quasi-optimality of conforming boundary element Galerkin discretization. Numerical experiments in 2-D confirm the efficacy of the method and a performance matching that of another widely used boundary element discretization. They also demonstrate its amenability to different types of preconditioning.     

Existence of positive solutions of superlinear second-order Neumann boundary value problem

In this paper, we consider the Neumann boundary value problem

     

Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem

We consider the stability in an inverse problem of determining the potential q entering the wave equation in a bounded smooth domain of Rd from boundary observations. The observation is given by a hyperbolic (dynamic) Dirichlet to Neumann map associated to a wave equation. We prove a log-type stability estimate in determining q from a partial Dirichlet to Neumann map provided that q is a priori known in a neighbourhood of the boundary of the spatial domain and satisfies an additional condition. Next, we use this result to establish a stability estimate related to the multidimensional Borg-Levinson theorem.     

Existence of multiple positive solutions for inhomogeneous Neumann problem

In this paper, we study the existence and nonexistence of multiple positive solutions for the inhomogeneous Neumann boundary value problem

(∗)     

Liouville type equation with exponential Neumann boundary condition and with singular data

In this paper we will analyze the blow-up behaviors of solutions to the singular Liouville type equation with exponential Neumann boundary condition. We generalize the Brezis–Merle type concentration-compactness theorem to this Neumann problem. Then along the line of the Li–Shafrir type quantization property we show that the blow-up value \(m(0) \in 2\pi \mathbb N\cup \{ 2\pi (1+\alpha )+2\pi (\mathbb N\cup \{0\})\}\) if the singular point 0 is a blow-up point. In the end, when the boundary value of solutions has an additional condition, we can obtain the precise blow-up value \(m(0)=2\pi (1+\alpha )\).     

Laplace–Beltrami equation on hypersurfaces and Γ‐convergence

A mixed boundary value problem for the stationary heat transfer equation in a thin layer around a surface with the boundary is investigated. The main objective is to trace what happens in Γ‐limit when the thickness of the layer converges to zero. The limit Dirichlet BVP for the Laplace–Beltrami equation on the surface is described explicitly, and we show how the Neumann boundary conditions in the initial BVP transform in the Γ‐limit. For this, we apply the variational formulation and the calculus of Günter's tangential differential operators on a hypersurface and layers, which allow global representation of basic differential operators and of corresponding boundary value problems in terms of the standard Euclidean coordinates of the ambient space . Copyright © 2017 John Wiley & Sons, Ltd.     

Existence and multiplicity results for some nonlinear problems with singular -Laplacian

Using Leray–Schauder degree theory we obtain various existence and multiplicity results for nonlinear boundary value problems

where l(u,u^′)=0 denotes the Dirichlet, periodic or Neumann boundary conditions on [0,T], is an increasing homeomorphism, (0)=0. The Dirichlet problem is always solvable. For Neumann or periodic boundary conditions, we obtain in particular existence conditions for nonlinearities which satisfy some sign conditions, upper and lower solutions theorems, Ambrosetti–Prodi type results. We prove Lazer–Solimini type results for singular nonlinearities and periodic boundary conditions.     

Construction of branching diffusion processes and their optimal stochastic control

In this work we construct a branching diffusion process whose individuals are interdependent, as the unique solution of a martingale problem. As an application we propose and solve a closed loop, finite horizon optimal control problem.     

Construction of N-Particle Langevin Dynamics for <Emphasis Type="Italic">H</Emphasis><Superscript>1,∞</Superscript>-Potentials via Generalized Dirichlet Forms

We construct an N-particle Langevin dynamics on a cuboid region in with periodic boundary condition, i.e., a diffusion process solving the Langevin equation with periodic boundary condition in the sense of the corresponding martingale problem. Our approach works for general H ^1,∞ potentials allowing N-particle interactions and external forces. Of course, the corresponding forces are not necessarily continuous. Since the generator of the dynamics is non-sectorial, for the construction we use the theory of generalized Dirichlet forms.      

Non-existence and uniqueness results for boundary value problems for Yang-Mills connections

We show uniqueness results for the Dirichlet problem for Yang-Mills connections defined in -dimensional () star-shaped domains with flat boundary values. This result also shows the non-existence result for the Dirichlet problem in dimension 4, since in 4-dimension, there exist countably many connected components of connections with prescribed Dirichlet boundary value. We also show non-existence results for the Neumann problem. Examples of non-minimal Yang-Mills connections for the Dirichlet and the Neumann problems are also given.

     

The Neumann problem for the planar Stokes system

The Neumann problem for the Stokes system is studied on bounded and unbounded domains with Ljapunov boundary (i.e. of class ${{\mathcal C}^{1,\alpha }}$ ) in the plane. We construct a solution of this problem in the form of appropriate potentials and reduce the problem to an integral equation systems on the boundary of the domain. We determine a necessary and sufficient condition for the solvability of the problem. Then we study the direct integral equation method and prove that a solution of the corresponding integral equation can be obtained by the successive approximation.     

The Heat Equaton with General Periodic Boundary Conditions

In this paper we address the well posedness of the linear heat equation under general periodic boundary conditions in several settings depending on the properties of the initial data. We develop an L ^q theory not based on separation of variables and use techniques based on uniform spaces. We also allow less directions of periodicity than the dimension of the problem. We obtain smoothing estimates on the solutions. Also, based on symmetry arguments, we handle Dirichlet or Neumann boundary conditions in some faces of the unit cell.     

Recovery of An Unknown Flux in Parabolic Problems with Nonstandard Boundary Conditions: Error Estimates

In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain ^N, with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant (t), accompanied with a nonlocal (integral) Dirichlet side condition.We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution u and also of the unknown function .