151 research outputs found
On equilibrium shapes of charged flat drops
Equilibrium shapes of two-dimensional charged, perfectly conducting liquid
drops are governed by a geometric variational problem that involves a perimeter
term modeling line tension and a capacitary term modeling Coulombic repulsion.
Here we give a complete explicit solution to this variational problem. Namely,
we show that at fixed total charge a ball of a particular radius is the unique
global minimizer among all sufficiently regular sets in the plane. For sets
whose area is also fixed, we show that balls are the only minimizers if the
charge is less than or equal to a critical charge, while for larger charge
minimizers do not exist. Analogous results hold for drops whose potential,
rather than charge, is fixed
Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains
We study spectral properties for , the Krein--von Neumann
extension of the perturbed Laplacian defined on
, where is measurable, bounded and nonnegative, in a
bounded open set belonging to a class of nonsmooth
domains which contains all convex domains, along with all domains of class
, . In particular, in the aforementioned context we establish
the Weyl asymptotic formula #\{j\in\mathbb{N} |
\lambda_{K,\Omega,j}\leq\lambda\} = (2\pi)^{-n} v_n |\Omega|
\lambda^{n/2}+O\big(\lambda^{(n-(1/2))/2}\big) {as} \lambda\to\infty, where
denotes the volume of the unit ball in
, and , , are the non-zero
eigenvalues of , listed in increasing order according to their
multiplicities. We prove this formula by showing that the perturbed Krein
Laplacian (i.e., the Krein--von Neumann extension of defined on
) is spectrally equivalent to the buckling of a clamped
plate problem, and using an abstract result of Kozlov from the mid 1980's. Our
work builds on that of Grubb in the early 1980's, who has considered similar
issues for elliptic operators in smooth domains, and shows that the question
posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl
asymptotic formula continues to have an affirmative answer in this nonsmooth
setting.Comment: 60 page
On maximal parabolic regularity for non-autonomous parabolic operators
We consider linear inhomogeneous non-autonomous parabolic problems associated
to sesquilinear forms, with discontinuous dependence of time. We show that for
these problems, the property of maximal parabolic regularity can be
extrapolated to time integrability exponents . This allows us to prove
maximal parabolic -regularity for discontinuous non-autonomous
second-order divergence form operators in very general geometric settings and
to prove existence results for related quasilinear equations
Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach
Five types of blow-up patterns that can occur for the 4th-order semilinear
parabolic equation of reaction-diffusion type
u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1,
\quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For
the semilinear heat equation , various blow-up patterns
were under scrutiny since 1980s, while the case of higher-order diffusion was
studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure
Instability of the anomalies in the one-dimensional Anderson model at weak disorder
We study the asymptotic behavior of the invariant measure, the Lyapunov exponent, and the density of states in the weak disorder limit in the case where the single-site potential distribution μ is not centered and for the special energies E =cos( πp/q ). We also prove that in general the above quantities can be continuously extended to zero disorder as continuous functions in the disorder parameter for all energies E ∈(−1, 1).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45159/1/10955_2005_Article_BF01029199.pd
Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems
Asymptotic Formulas for Eigenvalues of Elliptic Operators of Second Order in unbounded Domains of the Maz'ja Class
Singularities of Rotationally Symmetric Solutions of Boundary Value Problems for the Lamé Equations
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