We prove an optimal regularity result for elliptic operators −∇ • µ∇ : W 1,q 0 → W −1,q for a q > 3 in the case when the coefficient function µ has a jump across a C 1 interface and is continuous elsewhere. A counterexample shows that the C 1 condition cannot be relaxed in general. Finally, we draw some conclusions for corresponding parabolic operators.

We show that elliptic second order operators A of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of A are discontinuous and A is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.

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We show theoretically, that the detuning between the resonance frequencies of differently pumped DFB sections gives rise to two different pulsation mechanisms, 1) dispersive self Q-switching of a single-mode and 2) beating oscillations between two modes of nearly equal threshold gain. Our analysis is based on the dynamic coupled wave equations accomplished with carrier rate equations. We demonstrate the existence of certain isolated values of the detuning between both sections, at which two longitudinal eigenmodes become degenerate. In the degeneration point, the longitudinal excess factor of spontaneous emission has a singularity and the system of eigenmodes becomes incomplete. We derive reduced equations governing the dynamics in the vicinity of degeneration points. For an example device, the numerical integration of these equations clearly demonstrates the two different self pulsations with repetition rates of more than 100 GHz.

Abstract. We show that, under very general conditions on the domain Ω and the Dirichlet part D of the boundary, the operator −∇·µ∇+1 1/2 with mixed boundary conditions provides a topological isomorphism between W 1,p D (Ω) and L p (Ω), if p ∈ ]1, 2]. IntroductionThe main purpose of this paper is to identify the domain of the square root of a divergence form operator(The subscript D indicates the subspace of W 1,p (Ω) whose elements vanish on the boundary part D.) Our focus lies on non-smooth geometric situations in R d for d ≥ 2. So, we allow for mixed boundary conditions and, additionally, deviate from the Lipschitz property of the domain Ω in the following spirit: the boundary ∂Ω decomposes into a closed subset D (the Dirichlet part) and its complement, which may share a common frontier within ∂Ω. Concerning D, we only demand that it satisfies the well-known Ahlfors-David condition (equivalently: is a (d − 1)-set in the sense of Jonsson/Wallin [37, II.1]), and only for points from the complement we demand bi-Lipschitzian charts around. As special cases, the pure Dirichlet (D = ∂Ω) and pure Neumann case (D = ∅) are also included in our considerations. Finally the coefficient function µ is just supposed to be real, measurable, bounded and elliptic in general, cf. Assumption 4.2. Together, this setting should cover nearly everything that occurs in real-world problems -as long as the domain does not have irregularities like cracks meeting the Neumann boundary part ∂Ω \ D.The identification of the domain for fractional powers of elliptic operators, in particular that of square roots, has a long history. Concerning Kato's square root problem -in the Hilbert space L 2 -see e.g. [10], [24], [6] (here only the non-selfadjoint case is of interest). Early efforts, devoted to the determination of domains for fractional powers in the non-Hilbert space case seem to culminate in [48]. In recent years the problem has been investigated in the case of L p (p = 2) for instance in [5], [8], [35], [36], [33], [9]; but only the last three are dedicated to the case of a nonsmooth Ω = R d . In [9] the domain is a strong Lipschitz domain and the boundary conditions are either pure Dirichlet or pure Neumann. Our result generalizes this to a large extent and, at the same time, gives a new proof for these special cases, using more 'global' arguments. Since, in the case of a non-symmetric coefficient function µ, for the nonsmooth constellations described above no general condition is known that assures (−∇ · µ∇ + 1) 1/2 : W 1,2 D (Ω) → L 2 (Ω) to be an isomorphism, this is supposed as one of our assumptions. This serves then as our starting point to show the corresponding isomorphism property of (−∇ · µ∇ + 1)