General Wentzell Boundary Conditions Research Articles

Fourth-order differential operators with interior degeneracy and generalized Wentzell boundary conditions

In this article we consider the fourth-order operators \(A_1u:=(au'')''\) and \(A_2u:=au''''\) in divergence and non divergence form, where \(a:[0,1]\to\mathbb{R}_+\) degenerates in an interior point of the interval. Using the semigroup technique, under suitable assumptions on \(a\), we study the generation property of these operators associated to generalized Wentzell boundary conditions. We prove the well posedness of the corresponding parabolic problems.

Correction to: Strictly elliptic operators with generalized Wentzell boundary conditions on continuous functions on manifolds with boundary

Correction to: Strictly elliptic operators with generalized Wentzell boundary conditions on continuous functions on manifolds with boundary

Strictly elliptic operators with generalized Wentzell boundary conditions on continuous functions on manifolds with boundary

We prove that strictly elliptic operators with generalized Wentzell boundary conditions generate analytic semigroups of angle frac{pi }{2} on the space of continuous functions on a compact manifold with boundary.

The heat conduction model involving two temperatures on the segment with Wentzell boundary conditions

According to the theory of relatively p-bounded operators, we study the Heat Conduction model involving two temperatures for isotropic material, which describes, the rate of change of internal energy due to the movement of the heat flux form one medium to its complement with general Wentzell boundary conditions. In particular, we consider spectrum of one-dimensional Laplace operator on the segment [0,1] with general Wentzell boundary conditions. We examine the relative spectrum in one-dimensional Heat Conduction equation involving two temperatures, and construct the resolving group in the Cauchy-Wentzell problem with general Wentzell boundary conditions. In the paper, these problems are solved under the assumption that the initial space is a contraction of the space L2(0,1).

Maximal regularity, analytic semigroups, and dynamic and general Wentzell boundary conditions with a diffusion term on the boundary

We show maximal regularity results concerning parabolic systems with dynamic boundary conditions and a diffusion theorem on the boundary in the framework of $$L^p$$ spaces, $$1<p<\infty $$. Analyticity results can be derived for the semigroups generated by suitable classes of uniformly elliptic operators with general Wentzell boundary conditions having diffusion terms on the boundary.

Numerical Research of the Barenblatt - Zheltov - Kochina Model on the Interval with Wentzell Boundary Conditions

In terms of numerical research, we study the Barenblatt--Zheltov--Kochina model, which describes dynamics of pressure of a filtered fluid in a fractured-porous medium with the general Wentzell boundary conditions. Based on the theoretical results associated with Galerkin method, we develop an algorithm and implement the numerical solution of the Cauchy--Wentzell problem on the interval $ [0,1] $. In particular, we examine the asymptotic approximation of the spectrum of the one-dimensional Laplace operator and present result of a computational experiment. In the paper, these problems are solved under the assumption that the initial space is a contraction of the space $L^2(0,1)$.

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

AbstractIn 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.

Mode solutions for a Klein-Gordon field in anti–de Sitter spacetime with dynamical boundary conditions of Wentzell type

We study a real, massive Klein-Gordon field in the Poincar\'e fundamental domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime, subject to a particular choice of dynamical boundary conditions of generalized Wentzell type, whereby the boundary data solves a non-homogeneous, boundary Klein-Gordon equation, with the source term fixed by the normal derivative of the scalar field at the boundary. This naturally defines a field in the conformal boundary of the Poincar\'e fundamental domain of AdS. We completely solve the equations for the bulk and boundary fields and investigate the existence of bound state solutions, motivated by the analogous problem with Robin boundary conditions, which are recovered as a limiting case. Finally, we argue that both Robin and generalized Wentzell boundary conditions are distinguished in the sense that they are invariant under the action of the isometry group of the AdS conformal boundary, a condition which ensures in addition that the total flux of energy across the boundary vanishes.

Time asymptotics of structured populations with diffusion and dynamic boundary conditions

This work revisits and extends in various directions a work by J.Z. Farkas and P. Hinow (Math. Biosc and Eng, 8 (2011) 503-513) on structured populations models (with bounded sizes) with diffusion and generalized Wentzell boundary conditions. In particular, we provide first a self-contained $L^{1}$ generation theory making explicit the domain of the generator. By using Hopf maximum principle, we show that the semigroup is always irreducible regardless of the reproduction function. By using weak compactness arguments, we show first a stability result of the essential type and then deduce that the semigroup has a spectral gap and consequently the asynchronous exponential growth property. Finally, we show how to extend this theory to models with arbitrary sizes and point out an open problem pertaining to this extension.

Generalized Wentzell Boundary Conditions and Quantum Field Theory

We discuss a free scalar field subject to generalized Wentzell boundary conditions. On the classical level, we prove well-posedness of the Cauchy problem and in particular causality. Upon quantization, we obtain a field that may naturally be restricted to the boundary. We discuss the holographic relation between this boundary field and the bulk field.

Perturbation of analytic semigroups and applications to partial differential equations

In a recent paper we presented a general perturbation result for generators of \(C_0\)-semigroups, c.f. Theorem 2.1 below. The aim of the present work is to replace, in case the unperturbed semigroup is analytic, the various admissibility conditions appearing in this result by simpler inclusion assumptions on the domain and the range of the perturbation. This is done in Theorem 2.4 and allows to apply our results also in situations which are only in part governed by analytic semigroups. The power of our approach to treat in a unified and systematic way wide classes of PDE’s is illustrated by a generic example, a degenerate differential operator with generalized Wentzell boundary conditions, a reaction diffusion equation with unbounded delay and a perturbed Laplacian.

Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary

We show a result of maximal regularity in spaces of Holder continuous function, concerning linear parabolic systems, with dynamic or Wentzell boundary conditions, with an elliptic diffusion term on the boundary.

Generalized Wentzell boundary conditions for second order operators with interior degeneracy

We consider operators in divergence form, $A_1u=(au')'$, and in nondivergence form, $A_2u=au''$, provided that the coefficient $a$ vanishes in an interior point of the space domain. Characterizing the domain of the operators, we prove that, under suitable assumptions, the operators $A_1$ and $A_2$, equipped with general Wentzell boundary conditions, are nonpositive and selfadjoint on spaces of $L^2$ type.

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

Let p∈C0,1(Ω¯) be such that 1<p⁎⩽p⁎<∞, let Ω⊆RN be a bounded W1,p(⋅)-extension domain, and let μ be an upper d-Ahlfors measure supported on ∂Ω with d∈(N−p⁎,N). We investigate the solvability of a class of quasi-linear boundary value problems involving the p(⋅)-Laplace and p(⋅)-Laplace–Beltrami operators, and either classical Wentzell–Robin boundary conditionsΔp(⋅)udμ+|∇u|p(⋅)−2∂u∂νdHN−1+β|u|p(⋅)−2udμ=0on ∂Ω, or general fully Wentzell-type boundary conditions of the formΔp(⋅)udμ−Δp(⋅),∂Ωudμ+|∇u|p(⋅)−2∂u∂νdHN−1+β|u|p(⋅)−2udμ=0on∂Ω, where β∈L∞(∂Ω,dμ) is such that infx∈∂Ωβ(x)⩾β0 for some constant β0>0, and ∂u∂νdHN−1 denotes the generalized p(⋅)-normal derivative on ∂Ω (in the interpretative sense). We prove that the realization of the p(⋅)-Laplace operator with both of the above boundary conditions generate (nonlinear) ultracontractive submarkovian C0-semigroups on L2(Ω,dx)×L2(∂Ω,dμ), and hence, their associated first order Cauchy problems are both well posed on Lq(⋅)(Ω,dx)×Lq(⋅)(∂Ω,dμ) for all measurable function q with 1⩽q⁎⩽q⁎<∞. In addition, we investigate the associated quasi-linear elliptic problem with general Wentzell boundary conditions, and obtain existence, uniqueness and global regularity of weak solutions to this equation.

Goldstein-wentzell boundary conditions: Recent results with jerry and gisèle goldstein

We present a survey of recent results concerning heat and telegraph equations, equipped with Goldstein-Wentzell boundary conditions (already known as general Wentzell boundary conditions). We focus on the generation of analytic semigroups and continuous dependence of the solutions of the associated Cauchy problems from the boundary conditions.

Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data

Let $Ω\subset\mathbb{R}^N$ ($N\ge 2$) be a bounded domain with a boundary $∂Ω$ of class $C^2$ and let $\alpha,\beta$ be maximal monotone graphs in $\mathbb{R}^2$ satisfying $\alpha(0)\cap\beta(0)\ni 0$. Given $f\in L^1(Ω)$ and $g\in L^1(∂Ω)$, we characterize the existence and uniqueness of weak solutions to the semi-linear elliptic equation $-\Delta u+\alpha(u)\ni f$ in $Ω$ with the nonlinear general Wentzell boundary conditions $-\Delta_{\Gamma} u+\frac{\partial u}{\partial\nu}+\beta(u)\ni g$ on $∂Ω$. We also show the well-posedness of the associated parabolic problem on the Banach space $L^1(Ω)\times L^1(∂Ω)$.

Semi linear parabolic equations with nonlinear general Wentzell boundary conditions

Let $A$ be a uniformly elliptic operator in divergence form with bounded coefficients.We show that on a bounded domain $Ω ⊂ \mathbb{R}^N$ with Lipschitz continuous boundary $∂Ω$, a realization of $Au-\beta_1(x,u)$ in $C(\bar{Ω})$ with the nonlinear general Wentzell boundary conditions $[Au-\beta_1(x,u)]|_{∂Ω}-\Delta_\Gamma u+\partial_\nu^au+\beta_2(x,u)= 0$ on $∂Ω$ generates a strongly continuous nonlinear semigroup on $C(\bar{Ω})$. Here, $\partial_\nu^au$ is the conormal derivative of $u$, and $\beta_1(x,\cdot)$ ($x \in Ω$), $\beta_2(x,\cdot)$ ($x \in ∂Ω$) are continuous on $\mathbb{R}$ satisfying a certain growth condition.

Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains

We show that on a bounded domain $\Omega\subset R^N$ with Lipschitz continuous boundary $\partial \Omega$, weak solutions of the elliptic equation $\lambda u-Au=f$ in $\Omega$ with the boundary conditions $-\gamma\Delta_\Gamma u+\partial_\nu^a u+\beta u=g$ on $\partial \Omega$ are globally Holder continuous on $\bar \Omega$. Here $A$ is a uniformly elliptic operator in divergence form with bounded measurable coefficients, $\Delta_\Gamma$ is the Laplace-Beltrami operator on $\partial \Omega$, $\partial_\nu^a u$ denotes the conormal derivative of $u$, $\lambda,\gamma>0$ are real numbers and $\beta$ is a bounded measurable function on $\partial Omega$. We also obtain that a realization of the operator $A$ in $C(\bar \Omega)$ with the general Wentzell boundary conditions $(Au)|_{\partial \Omega}-\gamma\Delta_\Gamma u+\partial_\nu^a u+\beta u=g$ on $\partial \Omega$ generates a strongly continuous compact semigroup. Some analyticity results of the semigroup are also discussed.

Regularity and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains

We show that on a bounded domain Ω⊂RN with Lipschitz continuous boundary ∂Ω, if 2N/(N+2)<p≤N−1, then weak solutions of the quasi-linear elliptic equation −Δpu+|u|p−2u+α1(x,u)=f in Ω with the general Wentzell boundary conditions −Δp,Γu+|∇u|p−2∂νu+|u|p−2u+α2(x,u)=g weakly on ∂Ω, are globally Hölder continuous on Ω¯. Here, Δp and Δp,Γ denote the p-Laplace operator, and the p-Laplace–Beltrami operator on the boundary, respectively. The functions α1(x,⋅) (for x∈Ω) and α2(x,⋅) (for x∈∂Ω) are continuous on R and satisfy a certain growth condition. We also obtain that a realization of the operator Δpu−α1(x,u) in C(Ω¯) with the boundary conditions [Δpu−α1(x,u)]|∂Ω−Δp,Γu+|∇u|p−2∂νu+α2(x,u)=0 on ∂Ω generates a strongly continuous semigroup of nonlinear operators on C(Ω¯).

Nonautonomous semilinear second order evolution equations with generalized Wentzell boundary conditions

Of concern are the initial-boundary value problems for nonautonomous semilinear second order evolution equations with generalized Wentzell boundary conditions. We succeed in establishing a global wellposedness theorem (in a classical sense) for these problems via a specifically designed operator theoretic approach. Moreover, we obtain sharp estimations for the evolution of the solution along the characteristic curves, which enable us to derive the uniform exponential decay of the associated energies. The results obtained in this paper are still new even for the autonomous case.