The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients

The present paper discusses relations between regularity, Dirichlet, and Neumann problems. We investigate the boundary problems for block operators and prove, in particular, that the solvability of the regularity problem does not imply the solvability of the dual Dirichlet problem for general elliptic operators with complex bounded measurable coefficients. This is strikingly different from the case of real operators, for which such an implication was established in 1993 by C. Kenig, J. Pipher [Invent. Math. 113 (3) (1993) 447-509] and since then has served as an integral part of many results.     

The Dirichlet problem for second order parabolic operators in non‐cylindrical domains

In this paper we develope a perturbation theory for second order parabolic operators in non‐divergence form. In particular we study the solvability of the Dirichlet problem in non cylindrical domains with L^p ‐data on the parabolic boundary (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

REGULARITY THEORY FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS WITHN EUMANN BOUNDARY CONDITIONS

Inroductlonu consider here a system     

Spectral properties of the Laplacian on bond-percolation graphs

Bond-percolation graphs are random subgraphs of the d-dimensional integer lattice generated by a standard bond-percolation process. The associated graph Laplacians, subject to Dirichlet or Neumann conditions at cluster boundaries, represent bounded, self-adjoint, ergodic random operators with off-diagonal disorder. They possess almost surely the non-random spectrum [0, 4d] and a self-averaging integrated density of states. The integrated density of states is shown to exhibit Lifshits tails at both spectral edges in the non-percolating phase. While the characteristic exponent of the Lifshits tail for the Dirichlet (Neumann) Laplacian at the lower (upper) spectral edge equals d/2, and thus depends on the spatial dimension, this is not the case at the upper (lower) spectral edge,where the exponent equals 1/2.     

On the mixed problem for the semilinear Darcy‐Forchheimer‐Brinkman PDE system in Besov spaces on creased Lipschitz domains

The purpose of this paper is to study the mixed Dirichlet‐Neumann boundary value problem for the semilinear Darcy‐Forchheimer‐Brinkman system in L _p ‐based Besov spaces on a bounded Lipschitz domain in ${\mathbb{R}}^3$, with p in a neighborhood of 2. This system is obtained by adding the semilinear term | u | u to the linear Brinkman equation. First, we provide some results about equivalence between the Gagliardo and nontangential traces, as well as between the weak canonical conormal derivatives and the nontangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well‐posedness results for the Dirichlet and Neumann problems in L _p ‐based Besov spaces on bounded Lipschitz domains in ${\mathbb{R}}^n$(n ≥3) are also presented. Then, using integral potential operators, we show the well‐posedness in L ₂‐based Sobolev spaces for the mixed problem of Dirichlet‐Neumann type for the linear Brinkman system on a bounded Lipschitz domain in ${\mathbb{R}}^n$(n ≥3). Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann‐to‐Dirichlet operator, we extend the well‐posedness property to some L _p ‐based Sobolev spaces. Next, we use the well‐posedness result in the linear case combined with a fixed point theorem to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy‐Forchheimer‐Brinkman system in L _p ‐based Besov spaces, with p ∈(2?ε ,2+ε ) and some parameter ε >0.     

Well‐conditioned boundary integral formulations for high‐frequency elastic scattering problems in three dimensions

We construct and analyze a family of well‐conditioned boundary integral equations for the Krylov iterative solution of three‐dimensional elastic scattering problems by a bounded rigid obstacle. We develop a new potential theory using a rewriting of the Somigliana integral representation formula. From these results, we generalize to linear elasticity the well‐known Brakhage–Werner and combined field integral equation formulations. We use a suitable approximation of the Dirichlet‐to‐Neumann map as a regularizing operator in the proposed boundary integral equations. The construction of the approximate Dirichlet‐to‐Neumann map is inspired by the on‐surface radiation conditions method. We prove that the associated integral equations are uniquely solvable and possess very interesting spectral properties. Promising analytical and numerical investigations, in terms of spherical harmonics, with the elastic sphere are provided. Copyright © 2014 John Wiley & Sons, Ltd.     

Approximating Spectral Invariants of Harper Operators on Graphs

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada (Sun). A main result in this paper is that the spectral density function of DMLs associated to rational weight functions on graphs with a free action of an amenable discrete group can be approximated by the average spectral density function of the DMLs on a regular exhaustion, with either Dirichlet or Neumann boundary conditions. This then gives a criterion for the existence of gaps in the spectrum of the DML, as well as other interesting spectral properties of such DMLs. The technique used incorporates some results of algebraic number theory.     

Study of the initial value problems appearing in a method of factorization of second-order elliptic boundary value problems

In [J. Henry, A.M. Ramos, Factorization of second order elliptic boundary value problems by dynamic programming, Nonlinear Analysis. Theory, Methods & Applications 59 (2004) 629–647] we presented a method for factorizing a second-order boundary value problem into a system of uncoupled first-order initial value problems, together with a nonlinear Riccati type equation for functional operators. A weak sense was given to that system but we did not perform a direct study of those equations. This factorization utilizes either the Neumann to Dirichlet (NtD) operator or the Dirichlet to Neumann (DtN) operator, which satisfy a Riccati equation. Here we consider the framework of Hilbert–Schmidt operators, which provides tools for a direct study of this Riccati type equation. Once we have solved the system of Cauchy problems, we show that its solution solves the original second-order boundary value problem. Finally, we indicate how this techniques can be used to find suitable transparent conditions.     

Analysis of united boundary‐domain integro‐differential and integral equations for a mixed BVP with variable coefficient

The mixed (Dirichlet–Neumann) boundary‐value problem for the ‘Laplace’ linear differential equation with variable coefficient is reduced to boundary‐domain integro‐differential or integral equations (BDIDEs or BDIEs) based on a specially constructed parametrix. The BDIDEs/BDIEs contain integral operators defined on the domain under consideration as well as potential‐type operators defined on open sub‐manifolds of the boundary and acting on the trace and/or co‐normal derivative of the unknown solution or on an auxiliary function. Some of the considered BDIDEs are to be supplemented by the original boundary conditions, thus constituting boundary‐domain integro‐differential problems (BDIDPs). Solvability, solution uniqueness, and equivalence of the BDIEs/BDIDEs/BDIDPs to the original BVP, as well as invertibility of the associated operators are investigated in appropriate Sobolev spaces. Copyright © 2005 John Wiley & Sons, Ltd.     

Unbounded quantum graphs with unbounded boundary conditions

We consider metric graphs with a uniform lower bound on the edge lengths but no further restrictions. We discuss how to describe every local self‐adjoint Laplace operator on such graphs by boundary conditions in the vertices given by projections and self‐adjoint operators. We then characterize the lower bounded self‐adjoint Laplacians and determine their associated quadratic form in terms of the operator families encoding the boundary conditions.     

Semi‐analytical solution of MHD flow through boundary integrals on the pipe wall

A mathematical model is given for the magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross section of the pipe, coupled with an outer Dirichlet or Neumann magnetic problem. Inner Dirichlet problem is given as the coupled convection‐diffusion equations for the velocity and the induced current of the fluid coupling also to the outer problem, which is defined with the Laplace equation for the induced magnetic field of the exterior region with either Dirichlet or Neumann boundary condition. Unique solution of inner Dirichlet problem is obtained theoretically reducing it into two boundary integral equations defined on the boundary by using the corresponding fundamental solutions. Exterior solution is also given theoretically on the pipe wall with Poisson integral, and it is unique with Dirichlet boundary condition but exists with an additive constant obtained through coupled boundary and solvability conditions in Neumann wall condition. The collocation method is used to discretize these boundary integrals on the pipe wall. Thus, the proposed procedure is an improved theoretical analysis for combining the solution methods for the interior and exterior regions, which are consolidated numerically showing the flow behavior. The solution is simulated for several values of problem parameters, and the well‐known MHD characteristics are observed inside the pipe for increasing values of Hartmann number maintaining the continuity of induced currents on the pipe wall.     

A probabilistic approach to a non‐local quadratic form and its connection to the Neumann boundary condition problem

In this paper we look at a probabilistic approach to a non‐local quadratic form that has lately attracted some interest. This form is related to a recently introduced non‐local normal derivative. The goal is to construct two Markov processes: one corresponding to that form and the other which is related to a probabilistic interpretation of the Neumann problem. We also study the Dirichlet‐to‐Neumann operator for non‐local operators.     

The Dirichlet boundary value problems for p-Schrödinger operators on finite networks

In this paper, we define the discrete p-Schrödinger operators on finite networks and discuss the existence of the Dirichlet eigenvalues and their eigenfunctions for the operators. We also provide various equivalent conditions for the existence of positive solutions for Dirichlet boundary value problems for the operators.     

Determination of a time‐ and space‐dependent heat flux relaxation function by means of a restricted Dirichlet‐to‐Neumann operator

We consider an inverse problem to recover a space‐ and time‐dependent relaxation function of heat flux in a three‐dimensional body on the basis of the restriction of the Dirichlet‐to‐Neumann operator of the related equation of heat flow onto a set of Dirichlet data of the form of a product of a fixed time‐dependent coefficient and a free space‐dependent function. Uniqueness of the solution of this inverse problem is proved. Copyright © 2004 John Wiley & Sons, Ltd.     

On homogenization of solutions of the poisson equation in a perforated domain with different types of boundary conditions on different cavities

In this paper we consider boundary value problems in perforated domains with periodic structures and cavities of different scales, with the Neumann condition on some of them and mixed boundary conditions on others. We take a case when cavities with mixed boundary conditions have so called critical size (see [1]) and cavities with the Neumann conditions have the scale of the cell. In the same way other cases can be studied, when we have the Neumann and the Dirichlet boundary conditions or the Dirichlet condition and the mixed boundary condition on the boundary of cavities.There is a large literature where homogenization problems in perforated domains were studied [2];-[7];     

Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

We prove that Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half-space are well posed in L₂ for small complex L_∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for k-forms are well posed for small perturbations of block matrices.     

Square roots of elliptic second order divergence operators on strongly Lipschitz domains: $L^p$ theory

We study estimates for square roots of second order elliptic non necessarily selfadjoint operators in divergence form on Lipschitz domains subject to Dirichlet or to Neumann boundary conditions, pursuing our work [4] where we considered operators on . We obtain among other things for all if L is real symmetric and the domain bounded, which is new for . We also obtain similar results for perturbations of constant coefficients operators. Our methods rely on a singular integral representation, Calderón-Zygmund theory and quadratic estimates. A feature of this study is the use of a commutator between the resolvent of the Laplacian (Dirichlet and Neumann) and partial derivatives which carries the geometry of the boundary. Received: 12 January 2000 / Published online: 4 May 2001     

Boundary Problems for Harmonic Functions and Norm Estimates for Inverses of Singular Integrals in Two Dimensions

In this paper, we establish sharp well-posedness results for tangential derivative problems for the Laplacian with data in L ^p , 1 < p < ∞, on curvilinear polygons. Furthermore, we produce norm estimates/formulas for inverses of singular integral operators relevant for the Dirichlet, Neumann, tangential derivative, and transmission boundary value problems associated with the Laplacian in a distinguished subclass of Lipschitz domains in two dimensions. Our approach relies on Calderón-Zygmund theory and Mellin transform techniques.     

Approximating spectral invariants of Harper operators on graphs II

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in a paper by Mathai and Yates. The result is generalized to other self adjoint operators with finite propagation speed.

     

Operators with Wentzell boundary conditions and the Dirichlet‐to‐Neumann operator

In this paper we relate the generator property of an operator A with (abstract) generalized Wentzell boundary conditions on a Banach space X and its associated (abstract) Dirichlet‐to‐Neumann operator N acting on a “boundary” space . Our approach is based on similarity transformations and perturbation arguments and allows to split A into an operator A₀₀ with Dirichlet‐type boundary conditions on a space X₀ of states having “zero trace” and the operator N. If A₀₀ generates an analytic semigroup, we obtain under a weak Hille–Yosida type condition that A generates an analytic semigroup on X if and only if N does so on . Here we assume that the (abstract) “trace” operator is bounded that is typically satisfied if X is a space of continuous functions. Concrete applications are made to various second order differential operators.