1,570 research outputs found
A Holder Continuous Nowhere Improvable Function with Derivative Singular Distribution
We present a class of functions in 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 is pointwise anywhere improvable to C^{0,\be} for any \be>\al.
In particular, all 's are nowhere differentiable with derivatives singular
distributions. 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 (Katzourakis), of
-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 is the nonlinear nondivergence
form Aronsson PDE with as special case the -Laplacian.
Using , we construct singular solutions for these PDEs. In the
scalar case, we partially answered the open 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 .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:Rnâ¶RNu:Rnâ¶RN to the fully nonlinear PDE system
F(â
,D2u)=f, a.e. on Rn,(1)
F(â
,D2u)=f, a.e. on Rn,(1)
when fâL2(Rn)NfâL2(Rn)N and F is a CarathĂ©odory map. (1) has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanatoâs ellipticity condition on F. By introducing a new much weaker notion of ellipticity, we prove solvability of (1) in a tailored Sobolev âenergyâ space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a âperturbation deviceâ which allows to use Campanatoâs near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form (1)
A nonhomogeneous boundary value problem in mass transfer theory
We prove a uniqueness result of solutions for a system of PDEs of
Monge-Kantorovich type arising in problems of mass transfer theory. The results
are obtained under very mild regularity assumptions both on the reference set
, and on the (possibly asymmetric) norm defined in
. In the special case when is endowed with the Euclidean
metric, our results provide a complete description of the stationary solutions
to the tray table problem in granular matter theory.Comment: 22 pages, 2 figure
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 â-Bilaplacian
We consider the problem of finding and describing minimisers of the Rayleigh quotient
Îâ:=infuâW2,â(Ω)â{0}â„Îuâ„Lâ(Ω)â„uâ„Lâ(Ω),
Îâ:=infuâW2,â(Ω)â{0}âÎuâLâ(Ω)âuâLâ(Ω),
where ΩâRnΩâRn is a bounded C1,1C1,1 domain and W2,â(Ω)W2,â(Ω) is a class of weakly twice differentiable functions satisfying either u=0u=0 on âΩâΩ or u=|Du|=0u=|Du|=0 on âΩâΩ . Our first main result, obtained through approximation by LpLp -problems as pââpââ , is the existence of a minimiser uââW2,â(Ω)uââW2,â(Ω) satisfying
{ÎuââÎâSgn(fâ)Îfâ=ÎŒâ a.e. in Ω, in DâČ(Ω),
{ÎuââÎâSgn(fâ) a.e. in Ω,Îfâ=ÎŒâ in DâČ(Ω),
for some fââL1(Ω)â©BVloc(Ω)fââL1(Ω)â©BVloc(Ω) and a measure ÎŒââM(Ω)ÎŒââM(Ω) , for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue ÎâÎâ on the domain, establishing the validity of a FaberâKrahn type inequality: among all C1,1C1,1 domains with fixed measure, the ball is a strict minimiser of ΩâŠÎâ(Ω)ΩâŠÎâ(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
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