The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions

Arendt, Wolfgang; Metafune, Giorgio; Pallara, Diego; Romanell, S.

doi:10.1007/s00233-002-0010-8

Cited by 66 publications

(87 citation statements)

References 22 publications

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“…Alternatively, using the equation u t = ∆u in the boundary condition, problem (1) can be written as heat equation with Wentzell condition, which again possess a good existence theory. See [2], [3], [7], [10], [13], [14], [15], [16], [1], [9], [20], [18], [19], [27], and the references therein. The corresponding solutions generate a semigroup S t in H 0 = L 2 (Ω) by the standard formula…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Heat Equation with Dynamical Boundary Conditions of Reactive Type

Vázquez

Vitillaro

2008

Communications in Partial Differential Equations

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Abstract. The aim of this paper is to study the initial and boundary value problemwhere Ω is a bounded regular open domain in R N (N ≥ 1), Γ = ∂Ω, ν is the outward normal to Ω, and k < 0. In particular we prove that the problem is ill-posed when N ≥ 2, while is well-posed in dimension N = 1. Moreover we carefully study the case when Ω is a ball in R N . As a byproduct we give several results on the elliptic eigenvalue problem −∆u = λu, in Ω, ∆u = ku ν on Γ.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Heat Equation with Dynamical Boundary Conditions of Reactive Type

Vázquez

Vitillaro

2008

Communications in Partial Differential Equations

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“…)u + ∂u ∂n + ∆u = 0 on ∂Ω. In order to consider the Laplacian with Wentzell boundary conditions it is convenient to work on H := L 2 (Ω) ⊕ L 2 (∂Ω) (see [3] or [10]). Set V = {(u, Tr(u)), u ∈ H 1 (Ω)} and define the form We apply again Theorem 1.3 and obtain maximal L p -regularity on L 2 (Ω)⊕L 2 (∂Ω) for all p ∈ (1, ∞) and u(0) ∈ H 1 (Ω) under the sole condition that α > 0.…”

Section: Elliptic Operators With Wentzell Boundary Conditionsmentioning

confidence: 99%

Maximal regularity for non-autonomous evolution equations governed by forms having less regularity

Ouhabaz

2015

Arch. Math.

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Abstract. We consider the maximal regularity problem for non-autonomous evolution equationsEach operator A(t) is associated with a sesquilinear form a(t) on a Hilbert space H. We assume that these forms all have the same domain V . It is proved in [11] that if the forms have some regularity with respect to t (e.g., piecewise α-Hölder continuous for some α > 1 /2) then the above problem has maximal Lp-regularity for all u 0 in the real-interpolation space (H, D(A(0))) 1−1/p,p . In this paper we prove that the regularity required there can be improved for a class of sesquilinear forms. The forms considered here are such that the difference a(t; ·, ·) − a(s; ·, ·) is continuous on a larger space than the common domain V . We give three examples which illustrate our results.

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“…The result is based on non-linear semigroup theory. Following the ideas in [2], we are able to apply our methods also to equations with Wentzell-Robin boundary conditions. We do not have to assume that the L 2 -realization of the operator is a subdifferential, i.e., we do not assume that the corresponding elliptic problem has a variational formulation.…”

Section: P−2 ∂U(tx) ∂νmentioning

confidence: 99%

Quasilinear elliptic and parabolic Robin problems on Lipschitz domains

Nittka

2012

Nonlinear Differ. Equ. Appl.

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Abstract. We prove Hölder continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This includes the p-Laplace operator for all p ∈ (1, ∞), but also operators with unbounded coefficients. Based on the elliptic result we show that the corresponding parabolic problem is wellposed in the space C(Ω) provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in C(Ω) is m-accretive and densely defined. Thus it generates a non-linear strongly continuous contraction semigroup on C(Ω). Mathematics Subject Classification (1991

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The Laplacian with Wentzell-Robin boundary conditions on spaces of continuous functions

Cited by 66 publications

References 22 publications

Heat Equation with Dynamical Boundary Conditions of Reactive Type

Heat Equation with Dynamical Boundary Conditions of Reactive Type

Maximal regularity for non-autonomous evolution equations governed by forms having less regularity

Quasilinear elliptic and parabolic Robin problems on Lipschitz domains

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