Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces

We consider a transmission problem consisting of two semilinear parabolic equations involving fractional diffusion operators of different orders in a general non-smooth setting with emphasis on Lipschitz interfaces and transmission conditions along the interface. We give a unified framework for the existence and uniqueness of strong and mild solutions, and their global regularity properties.     

Analyticity on L 1 of the heat semigroup with Wentzell boundary conditions

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)}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 L¹(W) ?L¹(?W){L^1(\Omega) \oplus L^1(\partial \Omega)} .     

Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator

In the very influential paper [4 Caffarelli, L.A., Silvestre, L. (2007). An extension problem related to the fractional Laplacian. Commun. Partial Differential Equations 32:1245–1260.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] Caffarelli and Silvestre studied regularity of (?Δ)^s, 0<s<1, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea [15 Stinga, P.R., Torrea, J. (2010). Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differential Equations 35:2092–2122.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and Galé et al. [7 Galé, J., Miana, P., Stinga, P.R. (2013). Extension problem and fractional operators: semigroups and wave equations. J. Evol. Eqn. 13:343–368.[Crossref], [Web of Science ®] , [Google Scholar]] gave several more abstract versions of this extension procedure. The purpose of this paper is to study precise regularity properties of the Dirichlet and the Neumann problem in Hilbert spaces. Then the Dirichlet-to-Neumann operator becomes an isomorphism between interpolation spaces and its part in the underlying Hilbert space is exactly the fractional power.     

Dirichlet and Neumann boundary conditions: What is in between?

Given an admissible measure μ on where is an open set, we define a realization of the Laplacian in with general Robin boundary conditions and we show that generates a holomorphic C ₀ -semigroup on which is sandwiched by the Dirichlet Laplacian and the Neumann Laplacian semigroups. Moreover, under a locality and a regularity assumption, the generator of each sandwiched semigroup is of the form . We also show that if contains smooth functions, then μ is of the form (where σ is the (n-1)-dimensional Hausdorff measure and β a positive measurable bounded function on ); i.e. we have the classical Robin boundary conditions. RID="h1" ID="h1"Dédié à Philippe Bénilan RID="*" ID="h1"This work is part of the DGF-Project: "Regularit?t und Asymptotik für elliptische und parabolische Probleme".     

Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

We prove representation results for solutions of a time‐fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. Our focus is in the problem , where 0<β ≤ 2, 0<α ≤ 1, , (?Δ_d)^α is the discrete fractional Laplacian, and is the Caputo fractional derivative of order β. We discuss important special cases as consequences of the representations obtained.     

The Wave Equation in Lp-Spaces

In the first part of the paper, Gaussian estimates are used to study $L^p$-summability of the solution of the wave equation in $L^p(\Omega)$ associated with a general operator in divergence form with bounded coefficients. Secondly, we prove that if $\Omega$ is a cube in $\RR^N$, then the Laplacian with Dirichlet or Neumann boundary conditions generates an $\al$-times integrated cosine function on $L^p(\Omega),\;1\le p <\infty$ for any $\al\ge (N-1)|\frac{1}{2}-\frac{1}{p}|$.     

The one-dimensional wave equation with general boundary conditions

We show that a realization of the Laplace operator Au := u′′ with general nonlocal Robin boundary conditions α _j u′(j) + β _j u(j) + γ _1–j u(1 ? j) = 0, (j = 0, 1) generates a cosine family on L ^p (0, 1) for every \({p\,{\in}\,[1,\infty)}\). Here α _j , β _j and γ _j are complex numbers satisfying α ₀, α ₁ ≠ 0. We also obtain an explicit representation of local solutions to the associated wave equation by using the classical d’Alembert’s formula.     

The fractional Neumann and Robin type boundary conditions for the regional fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian

Let \({p \in (1,\infty)}\), \({s \in (0,1)}\) and \({\Omega \subset {\mathbb{R}^{N}}}\) a bounded open set with boundary \({\partial\Omega}\) of class C ^1,1. In the first part of the article we prove an integration by parts formula for the fractional p-Laplace operator \({(-\Delta)_{p}^{s}}\) defined on \({\Omega \subset {\mathbb{R}^{N}}}\) and acting on functions that do not necessarily vanish at the boundary \({\partial\Omega}\). In the second part of the article we use the above mentioned integration by parts formula to clarify the fractional Neumann and Robin boundary conditions associated with the fractional p-Laplacian on open sets.     

The Laplacian with Robin Boundary Conditions on Arbitrary Domains

Using a capacity approach, we prove in this article that it is always possible to define a realization of the Laplacian on L ²() with generalized Robin boundary conditions where is an arbitrary open subset of R ⁿ and is a Borel measure on the boundary of . This operator generates a sub-Markovian C ₀-semigroup on L ²(). If d=d where is a strictly positive bounded Borel measurable function defined on the boundary and the (n–1)-dimensional Hausdorff measure on , we show that the semigroup generated by the Laplacian with Robin boundary conditions has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L ^p () is independent of p[1,). Our approach constitutes an alternative way to Daners who considers the (n–1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture.