A Holder Continuous Nowhere Improvable Function with Derivative Singular Distribution

We present a class of functions $\mathcal{K}$ in $C^0(\R)$ which is variant of the Knopp class of nowhere differentiable functions. We derive estimates which establish \mathcal{K} \sub C^{0,\al}(\R) for 0<\al<1 but no $K \in \mathcal{K}$ is pointwise anywhere improvable to C^{0,\be} for any \be>\al. In particular, all $K$'s are nowhere differentiable with derivatives singular distributions. $\mathcal{K}$ furnishes explicit realizations of the functional analytic result of Berezhnoi. Recently, the author and simulteously others laid the foundations of Vector-Valued Calculus of Variations in $L^\infty$ (Katzourakis), of $L^\infty$-Extremal Quasiconformal maps (Capogna and Raich, Katzourakis) and of Optimal Lipschitz Extensions of maps (Sheffield and Smart). The "Euler-Lagrange PDE" of Calculus of Variations in $L^\infty$ is the nonlinear nondivergence form Aronsson PDE with as special case the $\infty$-Laplacian. Using $\mathcal{K}$, we construct singular solutions for these PDEs. In the scalar case, we partially answered the open $C^1$ regularity problem of Viscosity Solutions to Aronsson's PDE (Katzourakis). In the vector case, the solutions can not be rigorously interpreted by existing PDE theories and justify our new theory of Contact solutions for fully nonlinear systems (Katzourakis). Validity of arguments of our new theory and failure of classical approaches both rely on the properties of $\mathcal{K}$.Comment: 5 figures, accepted to SeMA Journal (2012), to appea

Existence and uniqueness of global solutions to fully nonlinear second order elliptic systems

We consider the problem of existence and uniqueness of strong a.e. solutions u:Rn⟶RNu:Rn⟶RN to the fully nonlinear PDE system F(⋅,D2u)=f, a.e. on Rn,(1) F(⋅,D2u)=f, a.e. on Rn,(1) when f∈L2(Rn)Nf∈L2(Rn)N and F is a Carathéodory map. (1) has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanato’s ellipticity condition on F. By introducing a new much weaker notion of ellipticity, we prove solvability of (1) in a tailored Sobolev “energy” space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a “perturbation device” which allows to use Campanato’s near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form (1)

Bulk-sensitive Photoemission of Mn5Si3

We have carried out a bulk-sensitive high-resolution photoemission experiment on Mn5Si3. The measurements are performed for both core level and valence band states. The Mn core level spectra are deconvoluted into two components corresponding to different crystallographic sites. The asymmetry of each component is of noticeable magnitude. In contrast, the Si 2p spectrum shows a simple Lorentzian shape with low asymmetry. The peaks of the valence band spectrum correspond well to the peak positions predicted by the former band calculation.Comment: To be published in: Solid State Communication

The eigenvalue problem for the ∞-Bilaplacian

We consider the problem of finding and describing minimisers of the Rayleigh quotient Λ∞:=infu∈W2,∞(Ω)∖{0}∥Δu∥L∞(Ω)∥u∥L∞(Ω), Λ∞:=infu∈W2,∞(Ω)∖{0}‖Δu‖L∞(Ω)‖u‖L∞(Ω), where Ω⊆RnΩ⊆Rn is a bounded C1,1C1,1 domain and W2,∞(Ω)W2,∞(Ω) is a class of weakly twice differentiable functions satisfying either u=0u=0 on ∂Ω∂Ω or u=|Du|=0u=|Du|=0 on ∂Ω∂Ω . Our first main result, obtained through approximation by LpLp -problems as p→∞p→∞ , is the existence of a minimiser u∞∈W2,∞(Ω)u∞∈W2,∞(Ω) satisfying {Δu∞∈Λ∞Sgn(f∞)Δf∞=μ∞ a.e. in Ω, in D′(Ω), {Δu∞∈Λ∞Sgn(f∞) a.e. in Ω,Δf∞=μ∞ in D′(Ω), for some f∞∈L1(Ω)∩BVloc(Ω)f∞∈L1(Ω)∩BVloc(Ω) and a measure μ∞∈M(Ω)μ∞∈M(Ω) , for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue Λ∞Λ∞ on the domain, establishing the validity of a Faber–Krahn type inequality: among all C1,1C1,1 domains with fixed measure, the ball is a strict minimiser of Ω↦Λ∞(Ω)Ω↦Λ∞(Ω) . This result is shown to hold true for either choice of boundary conditions and in every dimension

Nonlinear Dynamics of Aeolian Sand Ripples

We study the initial instability of flat sand surface and further nonlinear dynamics of wind ripples. The proposed continuous model of ripple formation allowed us to simulate the development of a typical asymmetric ripple shape and the evolution of sand ripple pattern. We suggest that this evolution occurs via ripple merger preceded by several soliton-like interaction of ripples.Comment: 6 pages, 3 figures, corrected 2 typo

Quasivariational solutions for first order quasilinear equations with gradient constraint

We prove the existence of solutions for an evolution quasi-variational inequality with a first order quasilinear operator and a variable convex set, which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable {\em a priori} estimates. We obtain also the existence of stationary solutions, by studying the asymptotic behaviour in time. In the variational case, corresponding to a constraint independent of the solution, we also give uniqueness results

Microsecond folding dynamics of the F13W G29A mutant of the B domain of staphylococcal protein A by laser-induced temperature jump

The small size (58 residues) and simple structure of the B domain of staphylococcal protein A (BdpA) have led to this domain being a paradigm for theoretical studies of folding. Experimental studies of the folding of BdpA have been limited by the rapidity of its folding kinetics. We report the folding kinetics of a fluorescent mutant of BdpA (G29A F13W), named F13W*, using nanosecond laser-induced temperature jump experiments. Automation of the apparatus has permitted large data sets to be acquired that provide excellent signal-to-noise ratio over a wide range of experimental conditions. By measuring the temperature and denaturant dependence of equilibrium and kinetic data for F13W*, we show that thermodynamic modeling of multidimensional equilibrium and kinetic surfaces is a robust method that allows reliable extrapolation of rate constants to regions of the folding landscape not directly accessible experimentally. The results reveal that F13W* is the fastest-folding protein of its size studied to date, with a maximum folding rate constant at 0 M guanidinium chloride and 45°C of 249,000 (s-1). Assuming the single-exponential kinetics represent barrier-limited folding, these data limit the value for the preexponential factor for folding of this protein to at least ≈2 x 10(6) s(-1)