Abstract quasilinear equations of second order with Wentzell boundary conditions

We introduce a general framework to treat abstract quasilinear equations of second order with Wentzell boundary conditions. As an example we study a wave equation for a second order quasilinear differential operator on with Wentzell boundary conditions.     

The Laplacian on $$ C(\overline \Omega) $$ with generalized Wentzell boundary conditions

In this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain in defined by generates an analytic semigroup of angle on for every > 0 and (for the definition of cf. (1.3)).Received: 13 July 2002     

Abstract wave equations with generalized Wentzell boundary conditions

We introduce a general framework which allows to verify if abstract wave equations with generalized Wentzell boundary conditions are well-posed, i.e., are governed by a cosine family. As an example we study wave equations for second order differential operators on C[0,1] with non-local Wentzell-type boundary conditions. Moreover, in Appendix A we give a perturbation result for sine and cosine families.     

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.     

Elliptic operators with general Wentzell boundary conditions,analytic semigroups and the angle concavity theorem

We prove a very general form of the Angle Concavity Theorem, which says that if (T (t)) defines a one parameter semigroup acting over various L^p spaces (over a fixed measure space), which is analytic in a sector of opening angle θ_p, then the maximal choice for θ_p is a concave function of 1 – 1/p. This and related results are applied to give improved estimates on the optimal L^p angle of ellipticity for a parabolic equation of the form ?u /?t = Au, where A is a uniformly elliptic second order partial differential operator with Wentzell or dynamic boundary conditions. Similar results are obtained for the higher order equation ?u /?t = (–1)^{m +l}A^mu, for all positive integers m.     

Analysis and discretization of the volume penalized Laplace operator with Neumann boundary conditions

We study the properties of an approximation of the Laplace operator with Neumann boundary conditions using volume penalization. For the one-dimensional Poisson equation we compute explicitly the exact solution of the penalized equation and quantify the penalization error. Numerical simulations using finite differences allow then to assess the discretization and penalization errors. The eigenvalue problem of the penalized Laplace operator with Neumann boundary conditions is also studied. As examples in two space dimensions, we consider a Poisson equation with Neumann boundary conditions in rectangular and circular domains.     

Asymptotics for 2‐D wave equations with Wentzell boundary conditions in the square

This paper concerns the asymptotics of the linear wave equation with frictional damping only on Wentzell boundary in the square. After reformulating the model into an abstract Cauchy problem, we show that the spectrum for the corresponding operator matrix has no purely imaginary values. Moreover, by analyzing a family of eigenvalues for the operator matrix, we prove that there exists a solution of the system, whose energy decay rate can be arbitrarily slow.     

Impulsive problems for semilinear differential equations with nonlocal conditions

This paper is concerned with the existence of impulsive semilinear differential equations with nonlocal conditions. Using the technique of fixed point theory, existence results are obtained, for mild solutions, when the nonlocal item is Lipschitz, is not Lipschitz and not compact, respectively.     

Existence of nondensely defined evolution equations with nonlocal conditions

This paper is concerned with the existence for nondensely defined evolution equations with nonlocal conditions. Using the techniques of fixed point theory and approximate solutions, existence results are obtained, for integral solutions, when the nonlocal item is Lipschitz continuous or continuous, respectively. Examples are also given to illustrate our results.     

Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces

In this paper, we establish the existence results for semilinear differential systems with nonlocal initial conditions in Banach spaces. The approaches used are fixed point theorems combined with convex-power condensing operators. The first result obtained will be applied to a class of semilinear parabolic equations.     

Small-amplitude solutions of the sine-Gordon equation on an interval under dirichlet or Neumann boundary conditions

Summary We give a complete classification of the small-amplitude finite-gap solutions of the sine-Gordon (SG) equation on an interval under Dirichlet or Neumann boundary conditions. Our classification is based on an analysis of the finite-gap solutions of the boundary problems for the SG equation by means of the Schottky uniformization approach.On leave from IPPI, Moscow, Russia     

Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models

Let be a nonnegative, smooth function with , supported in , symmetric, , and strictly increasing in . We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation

We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as : they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments.

     

Parabolic systems with coupled boundary conditions

We consider elliptic operators with operator-valued coefficients and discuss the associated parabolic problems. The unknowns are functions with values in a Hilbert space W. The system is equipped with a general class of coupled boundary conditions of the form f|∂Ω∈Y and , where Y is a closed subspace of L²(∂Ω;W). We discuss well-posedness and further qualitative properties, systematically reducing features of the parabolic system to operator-theoretical properties of the orthogonal projection onto Y.     

Dissipative q-Dirac operator with general boundary conditions

In this paper, we consider the symmetric q-Dirac operator. We describe dissipative, accumulative, self-adjoint and the other extensions of such operators with general boundary conditions. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.     

Substochastic semigroups for transport equations with conservative boundary conditions

We consider the free streaming operator associated with conservative boundary conditions. It is known that this operator (with its usual domain) admits an extension A which generates a C₀-semigroup in L¹. With techniques borrowed from the additive perturbation theory of substochastic semigroups, we describe precisely its domain and provide necessary and sufficient conditions ensuring to be stochastic. We apply these results to examples from kinetic theory.     

Spectral analysis of transport equations with bounce‐back boundary conditions

We investigate the spectral properties of the time‐dependent linear transport equation with bounce‐back boundary conditions. A fine analysis of the spectrum of the streaming operator is given and the explicit expression of the strongly continuous streaming semigroup is derived. Next, making use of a recent result from Sbihi (J. Evol. Equations 2007; 7 :689–711), we prove, via a compactness argument, that the essential spectrum of the transport semigroup and that of the streaming semigroup coincide on all L^p‐spaces with 1<p<∞. Application to the linear Boltzmann equation for granular gases is provided. Copyright © 2008 John Wiley & Sons, Ltd.     

Fourth‐order compact schemes of a heat conduction problem with Neumann boundary conditions

In this article, a set of fourth‐order compact finite difference schemes is developed to solve a heat conduction problem with Neumann boundary conditions. It is derived through the compact difference schemes at all interior points, and the combined compact difference schemes at the boundary points. This set of schemes is proved to be globally solvable and unconditionally stable. Numerical examples are provided to verify the accuracy.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Positive solutions for singular elliptic equations with mixed Dirichlet‐Neumann boundary conditions

In this paper, we investigate the existence of positive solutions for singular elliptic equations with mixed Dirichlet‐Neumann boundary conditions involving Sobolev‐Hardy critical exponents and Hardy terms by using the concentration compactness principle, the strong maximum principle and the Mountain Pass lemma. We also prove, under complementary conditions, that there is no nontrivial solution if the domain is star‐shaped with respect to the origin.     

Uniqueness theorems for differential pencils with eigenparameter boundary conditions and transmission conditions

Inverse spectral problems are considered for differential pencils with boundary conditions depending polynomially on the spectral parameter and with a finite number of transmission conditions. We give formulations of the associated inverse problems such as Titchmarsh–Weyl theorem, Hochstadt–Lieberman theorem and Mochizuki–Trooshin theorem, and prove corresponding uniqueness theorems. The obtained results are generalizations of the similar results for the classical Sturm–Liouville operator on a finite interval.