A Cheeger inequality for the drift Laplacian with Wentzell boundary condition

In this paper, we derive optimal upper and lower bounds on the dimension of the attractor AW for scalar reaction-diffusion equations with a Wentzell (dynamic) boundary condition. We are also interested in obtaining explicit bounds about the constants involved in our asymptotic estimates, and to compare these bounds to previously known estimates for the dimension of the global attractor AK; K \in {D;N; P}, of reactiondiffusion equations subject to Dirichlet, Neumann and periodic boundary conditions. The explicit estimates we obtain show that the dimension of the global attractor AW is of different order than the dimension of AK; for each K \in {D;N; P} ; in all space dimensions that are greater or equal than three.

We introduce a nonlinear structured population model with diffusion in the state space. Individuals are structured with respect to a continuous variable which represents a pathogen load. The class of uninfected individuals constitutes a special compartment that carries mass, hence the model is equipped with generalized Wentzell (or dynamic) boundary conditions. Our model is intended to describe the spread of infection of a vertically transmitted disease, for example Wolbachia in a mosquito population. Therefore the (infinite dimensional) nonlinearity arises in the recruitment term. First we establish global existence of solutions and the Principle of Linearised Stability for our model. Then, in our main result, we formulate simple conditions, which guarantee the existence of non-trivial steady states of the model. Our method utilizes an operator theoretic framework combined with a fixed point approach. Finally, in the last section we establish a sufficient condition for the local asymptotic stability of the positive steady state.

We study mixed hyperbolic systems with dynamic and Wentzell boundary conditions. The boundary condition contains a tangential operator which is strongly elliptic on the boundary. We prove results of generation of strongly continuous groups and well-posedness.

We investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain Omega subseteq mathbb {R}^d with Lipschitz boundary Gamma . More precisely, using form methods, we show that the associated operator on the ground space L^2(Omega )times L^2(Gamma ) has compact resolvent and generates a holomorphic and strongly continuous real semigroup of self-adjoint operators. Furthermore, we give a full characterization of the domain in terms of Sobolev spaces, also proving Hölder regularity of solutions, allowing classical interpretation of the boundary condition. Finally, we investigate spectrum and asymptotic behavior of the semigroup, as well as eventual positivity.

In this paper we give new derivations of the heat and wave equation which incorporate the boundary conditions into the formulation of the problems. The principle of least action and the inclusion of a kinetic energy contribution on the boundary are used to derive the wave equation together with kinetic boundary conditions. The methods described for both equations admit all of the standard boundary conditions as well as general Wentzell and dynamic boundary conditions; in addition the boundary conditions arise naturally as part of the formulation of the problems. The physical interpretation for general Wentzell boundary conditions is given for both the heat and wave equations.

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 $$C([0,1];\;\mathbb{R})$$ with Wentzell boundary conditions.

Eigenvalue inequalities for the Laplacian with mixed boundary conditions

It is known that a diffusion process on a domain D with smooth boundary is determined by a pair of analytical data (A, L), where A is a second order differential operator of elliptic type and L is a Wentzell's boundary condition which consists of the sum of a second order differential operator and non-local terms. Here we shall afford a concrete example in this framework. We discuss the case where the boundary condition L possesses two non-local terms, one corresponds to the Cauchy process on the boundary and the other to a stable process of order β∈(0,1)∩Q having inward jumps from the boundary. We shall show analytically the existence and uniqueness of the Feller semigroup, and hence the diffusion

Uniform stabilization of wave equation subject to second-order boundary conditions is considered in this article. Both dynamic (Wentzell) and static (with higher derivatives in space only) boundary conditions are discussed. In contrast to the classical wave equation where stabilization can be achieved by applying boundary velocity feedback, for a Wentzell-type problem boundary damping alone does not cause the energy to decay uniformly to zero. This is the case for both dynamic and static second-order conditions. In order to achieve uniform decay rates of the associated energy, it is necessary to dissipate part of the collar near the boundary. It will be shown how a combination of partially localized boundary feedback and partially localized collar feedback leads to uniform decay rates that are described by a nonlinear differential equation. This goal is attained by combining techniques used for stabilization of ‘unobserved’ Neumann conditions with differential geometry techniques effective for stabilization on compact manifolds. These lead to a construction of special non-radial multipliers which are geometry dependent and allow reconstruction of the high-order part of the potential energy from the damping that is supported only in a far-off region of the domain.

This paper is devoted to the computation of certain directional semi-derivatives of eigenvalue functionals of self-adjoint elliptic operators involving a variety of boundary conditions. A uniform treatment of these problems is possible by considering them as a problem of calculating the semi-derivative of a minimum with respect to a parameter. The applicability of this approach, which can be traced back to the works of Danskin [8, 9] and Zolésio [28], to the treatment of eigenvalue problems (where the full shape derivative may not exist, due to multiplicity issues), has been illustrated by Zolésio in [29] (see also [10, Chap. 10] and included references). Despite this, some of the recent literature (see, for example, [1] or [7]) on the shape sensitivity of eigenvalue problems still continue to employ methods such as the material derivative method or Lagrangian methods which seem less adapted to this class of problems. The Delfour–Zolésio approach does not seem to be fully exploited in the existing literature: we aim to recall the importance and the simplicity of the ideas from [8, 28], by applying it to the analysis of the shape sensitivity for eigenvalue functionals for a class of elliptic operators in the scalar setting (Laplacian or diffusion in heterogeneous media), thus recovering known results in the case of Dirichlet or Neumann boundary conditions and obtaining new results in the case of Steklov or Wentzell boundary conditions.

Quasi-linear variable exponent boundary value problems with Wentzell–Robin and Wentzell boundary conditions

A large system of ordinary differential equations is approximated by a parabolic partial differential equation with dynamic boundary condition and a different one with Robin boundary condition. Using the theory of differential operators with Wentzell boundary conditions and similar theories, we give estimates on the order of approximation. The theory is demonstrated on a voter model where the Fourier method applied to the PDE is of great advantage.

We are concerned with dynamical behaviors of solutions to nonlinear damped wave equations with nonlinear dampings and force terms, and subject to Wentzell boundary conditions which can be used to describe, for instance, the boundary behavior of a vibrating elastic body (resp. membrane) coated (resp. edged) with a thin layer (resp. coil) of high rigidity. Here the internal dampings are only assumed to be locally distributed and, especially, may disappear gradually over time. We find a new and effective method to overcome all the difficulties caused by the interplay of vanishing localized dampings, Wentzell boundary conditions, as well as nonlinear force terms. Ideal uniform decay rates of solution energies are obtained in terms of the exponents associated with the time-varying damping. Our result shows that the dynamical behavior of solutions is clearly stable without any bifurcation and chaos. To illustrate our theoretical results, we provide some numerical simulations.

We show that the semigroup generated by the realization of the Laplace operator with Wentzell boundary conditions in a bounded smooth domain is analytic on \({L^1(\Omega) \oplus L^1(\partial \Omega)}\) .

In this paper we consider a Cahn–Hilliard model endowed with the Wentzell boundary condition, which arises from the study of spinodal decomposition in binary mixtures confined to a bounded domain with permeable wall. Under the assumption that the nonlinearity is analytic with respect to the unknown dependent function, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of an extended Łojasiewicz–Simon type inequality with boundary term. Estimates of the convergence rate are also obtained.