51 research outputs found
Quantitative uniqueness for elliptic equations with singular lower order terms
We use a Carleman type inequality of Koch and Tataru to obtain quantitative
estimates of unique continuation for solutions of second order elliptic
equations with singular lower order terms. First we prove a three sphere
inequality and then describe two methods of propagation of smallness from sets
of positive measure.Comment: 23 pages, v2 small changes are done and some mistakes are correcte
The Volume of a Local Nodal Domain
Let M either be a closed real analytic Riemannian manifold or a closed smooth
Riemannian surface. We estimate from below the volume of a nodal domain
component in an arbitrary ball provided that this component enters the ball
deeply enough.Comment: 21 pages; introduction improved putting the problem in a larger
context
Maximizing Neumann fundamental tones of triangles
We prove sharp isoperimetric inequalities for Neumann eigenvalues of the
Laplacian on triangular domains.
The first nonzero Neumann eigenvalue is shown to be maximal for the
equilateral triangle among all triangles of given perimeter, and hence among
all triangles of given area. Similar results are proved for the harmonic and
arithmetic means of the first two nonzero eigenvalues
The mean curvature of cylindrically bounded submanifolds
We give an estimate of the mean curvature of a complete submanifold lying
inside a closed cylinder in a product Riemannian manifold
. It follows that a complete hypersurface of given
constant mean curvature lying inside a closed circular cylinder in Euclidean
space cannot be proper if the circular base is of sufficiently small radius. In
particular, any possible counterexample to a conjecture of Calabion complete
minimal hypersurfaces cannot be proper. As another application of our method,
we derive a result about the stochastic incompleteness of submanifolds with
sufficiently small mean curvature.Comment: First version (December 2008). Final version, including new title
(February 2009). To appear in Mathematische Annale
Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs
Transport through generalized trees is considered. Trees contain the simple
nodes and supernodes, either well-structured regular subgraphs or those with
many triangles. We observe a superdiffusion for the highly connected nodes
while it is Brownian for the rest of the nodes. Transport within a supernode is
affected by the finite size effects vanishing as For the even
dimensions of space, , the finite size effects break down the
perturbation theory at small scales and can be regularized by using the
heat-kernel expansion.Comment: 21 pages, 2 figures include
The stability for the Cauchy problem for elliptic equations
We discuss the ill-posed Cauchy problem for elliptic equations, which is
pervasive in inverse boundary value problems modeled by elliptic equations. We
provide essentially optimal stability results, in wide generality and under
substantially minimal assumptions. As a general scheme in our arguments, we
show that all such stability results can be derived by the use of a single
building brick, the three-spheres inequality.Comment: 57 pages, review articl
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
The hybrid spectral problem and Robin boundary conditions
The hybrid spectral problem where the field satisfies Dirichlet conditions
(D) on part of the boundary of the relevant domain and Neumann (N) on the
remainder is discussed in simple terms. A conjecture for the C_1 coefficient is
presented and the conformal determinant on a 2-disc, where the D and N regions
are semi-circles, is derived. Comments on higher coefficients are made.
A hemisphere hybrid problem is introduced that involves Robin boundary
conditions and leads to logarithmic terms in the heat--kernel expansion which
are evaluated explicitly.Comment: 24 pages. Typos and a few factors corrected. Minor comments added.
Substantial Robin additions. Substantial revisio
On the essential spectrum of Nadirashvili-Martin-Morales minimal surfaces
We show that the spectrum of a complete submanifold properly immersed into a
ball of a Riemannian manifold is discrete, provided the norm of the mean
curvature vector is sufficiently small. In particular, the spectrum of a
complete minimal surface properly immersed into a ball of is
discrete. This gives a positive answer to a question of Yau.Comment: This article is an improvement of an earlier version titled On the
spectrum of Martin-Morales minimal surfaces. 7 page
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