Existence of solutions for semilinear elliptic boundary value problems on arbitrary open sets

We show the existence of a weak solution of a semilinear elliptic Dirichlet problem on an arbitrary open set Ω. We make no assumptions about the open set Ω and very mild regularity assumptions on the semilinearity f, plus a coerciveness assumption which depends on the optimal Poincaré–Steklov constant λ₁. The proof is based on Schaefer’s fixed point theorem applied to a sequence of truncated problems. We state a simple uniqueness result. We also generalize the results to Robin boundary conditions [17].     

On-Surface Synthesis of Cumulene-Containing Polymers via Two-Step Dehalogenative Homocoupling of Dibromomethylene-Functionalized Tribenzoazulene

Cumulene compounds are notoriously difficult to prepare and study because their reactivity increases dramatically with the increasing number of consecutive double bonds. In this respect, the emerging field of on-surface synthesis provides exceptional opportunities because it relies on reactions on clean metal substrates under well-controlled ultrahigh-vacuum conditions. Here we report the on-surface synthesis of a polymer linked by cumulene-like bonds on a Au(111) surface via sequential thermally activated dehalogenative C−C coupling of a tribenzoazulene precursor equipped with two dibromomethylene groups. The structure and electronic properties of the resulting polymer with cumulene-like pentagon–pentagon and heptagon–heptagon connections have been investigated by means of scanning probe microscopy and spectroscopy methods and X-ray photoelectron spectroscopy, complemented by density functional theory calculations. Our results provide perspectives for the on-surface synthesis of cumulene-containing compounds, as well as protocols relevant to the stepwise fabrication of carbon–carbon bonds on surfaces.     

Hot Branching Dynamics in a Light‐Harvesting Iron Carbene Complex Revealed by Ultrafast X‐ray Emission Spectroscopy

Iron N‐heterocyclic carbene (NHC) complexes have received a great deal of attention recently because of their growing potential as light sensitizers or photocatalysts. We present a sub‐ps X‐ray spectroscopy study of an Fe^IINHC complex that identifies and quantifies the states involved in the deactivation cascade after light absorption. Excited molecules relax back to the ground state along two pathways: After population of a hot ³MLCT state, from the initially excited ¹MLCT state, 30 % of the molecules undergo ultrafast (150 fs) relaxation to the ³MC state, in competition with vibrational relaxation and cooling to the relaxed ³MLCT state. The relaxed ³MLCT state then decays much more slowly (7.6 ps) to the ³MC state. The ³MC state is rapidly (2.2 ps) deactivated to the ground state. The ⁵MC state is not involved in the deactivation pathway. The ultrafast partial deactivation of the ³MLCT state constitutes a loss channel from the point of view of photochemical efficiency and highlights the necessity to screen transition‐metal complexes for similar ultrafast decays to optimize photochemical performance.     

Collective All‐Carbon Magnetism in Triangulene Dimers

Triangular zigzag nanographenes, such as triangulene and its π‐extended homologues, have received widespread attention as organic nanomagnets for molecular spintronics, and may serve as building blocks for high‐spin networks with long‐range magnetic order, which are of immense fundamental and technological relevance. As a first step towards these lines, we present the on‐surface synthesis and a proof‐of‐principle experimental study of magnetism in covalently bonded triangulene dimers. On‐surface reactions of rationally designed precursor molecules on Au(111) lead to the selective formation of triangulene dimers in which the triangulene units are either directly connected through their minority sublattice atoms, or are separated via a 1,4‐phenylene spacer. The chemical structures of the dimers have been characterized by bond‐resolved scanning tunneling microscopy. Scanning tunneling spectroscopy and inelastic electron tunneling spectroscopy measurements reveal collective singlet–triplet spin excitations in the dimers, demonstrating efficient intertriangulene magnetic coupling.