We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality. We propose a notion of domain with boundary of positive mean curvature and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here least gradient is defined as minimizing total variation (in the sense of BV functions) and boundary conditions are satisfied in the sense that the boundary trace of the solution exists and agrees with the given boundary data. This extends the result of Sternberg, Williams and Ziemer [27] to the non-smooth setting. Via counterexamples we also show that uniqueness of solutions and existence of continuous solutions can fail, even in the weighted Euclidean setting with Lipschitz weights. * The authors thank Estibalitz Durand-Cartagena, Marie Snipes and Manuel Ritoré for fruitful discussions about the subject of the paper.

A. Trace classes of Sobolev-type functions in metric spaces are subject of this paper. In particular, functions on domains whose boundary has an upper codimension-θ bound are considered. Based on a Poincaré inequality, existence of a Borel measurable trace is proven whenever the power of integrability of the "gradient" exceeds θ. The trace T is shown to be a compact operator mapping a Sobolev-type space on a domain into a Besov space on the boundary. Su cient conditions for T to be surjective are found and counterexamples showing that surjectivity may fail are also provided. The case when the exponent of integrability of the "gradient" is equal to θ, i.e., the codimension of the boundary, is also discussed. Under some additional assumptions, the trace lies in L θ on the boundary then. Essential sharpness of these extra assumptions is illustrated by an example.

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In this paper, first-order Sobolev-type spaces on abstract metric measure spaces are defined using the notion of (weak) upper gradients, where the summability of a function and its upper gradient is measured by the "norm" of a quasi-Banach function lattice. This approach gives rise to so-called Newtonian spaces. Tools such as moduli of curve families and Sobolev capacity are developed, which allows us to study basic properties of these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are established.

Two new isoquinoline alkaloids, named fumaranine (2) and fumarostrejdine (10), along with 18 known alkaloids were isolated from aerial parts of Fumaria officinalis. The structures of the isolated compounds were elucidated on the basis of spectroscopic analyses and by comparison with literature data. The absolute configuration of the new compound 2 was determined by comparing its circular dichroism spectra with those of known analogs. Compounds isolated in sufficient amounts were evaluated for their acetylcholinesterase, butyrylcholinesterase, prolyl oligopeptidase (POP), and glycogen synthase kinase-3β inhibitory activities. Parfumidine (8) and sinactine (15) exhibited potent POP inhibition activities (IC50 99±5 and 53±2 μM, resp.).