1,519 research outputs found
Convexity criteria and uniqueness of absolutely minimizing functions
We show that absolutely minimizing functions relative to a convex Hamiltonian
are uniquely determined by their boundary
values under minimal assumptions on Along the way, we extend the known
equivalences between comparison with cones, convexity criteria, and absolutely
minimizing properties, to this generality. These results perfect a long
development in the uniqueness/existence theory of the archetypal problem of the
calculus of variations in Comment: 34 page
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, uniqueness and structure of second order absolute minimisers
Let â Rn be a bounded open C1,1 set. In this paper we prove the existence
of a unique second order absolute minimiser uâ of the functional
Eâ(u, O) := F(·, u)Lâ(O), O â measurable,
with prescribed boundary conditions for u and Du on â and under natural assumptions
on F. We also show that uâ is partially smooth and there exists a harmonic
function fâ â L1() such that
F(x, uâ(x)) = eâ sgn
fâ(x)
for all x â { fâ = 0}, where eâ is the infimum of the global energy
An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions
We present a new, easy, and elementary proof of Jensen's Theorem on the
uniqueness of infinity harmonic functions. The idea is to pass to a finite
difference equation by taking maximums and minimums over small balls.Comment: 4 pages; comments added, proof simplifie
Radiological impact of naturally occurring radionuclides in bottled water
Consumption of bottled water is increasing year after year in Europe. Due to the local geology from where the water is extracted; bottled water could be enhanced with radionuclides. This study focuses on the activity concentrations of 210Po, 210Pb, 226Ra, 228Ra, 234U and 238U in bottled water available in the Swedish market, to assess the radiological impact to different age groups. The results showed that among the 26 brands studied, only three could exceed the threshold value for drinking water: 0.1 mSv/year. For two brands, the dose was mainly due to the activity concentrations of 238U and 234U being up to 714 and 1162 mBq/L, respectively. While for one brand, the dose was mainly due to the activity concentration of both 210Po and 210Pb being around 100 mBq/L. For the remainder brands, 228Ra was the main contributor to the committed effective dose
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
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