35 research outputs found
Refined asymptotics for the infinite heat equation with homogeneous Dirichlet boundary conditions
The nonnegative viscosity solutions to the infinite heat equation with
homogeneous Dirichlet boundary conditions are shown to converge as time
increases to infinity to a uniquely determined limit after a suitable time
rescaling. The proof relies on the half-relaxed limits technique as well as
interior positivity estimates and boundary estimates. The expansion of the
support is also studied
The mixed problem in L^p for some two-dimensional Lipschitz domains
We consider the mixed problem for the Laplace operator in a class of
Lipschitz graph domains in two dimensions with Lipschitz constant at most 1.
The boundary of the domain is decomposed into two disjoint sets D and N. We
suppose the Dirichlet data, f_D has one derivative in L^p(D) of the boundary
and the Neumann data is in L^p(N). We find conditions on the domain and the
sets D and N so that there is a p_0>1 so that for p in the interval (1,p_0), we
may find a unique solution to the mixed problem and the gradient of the
solution lies in L^p
On thin plate spline interpolation
We present a simple, PDE-based proof of the result [M. Johnson, 2001] that
the error estimates of [J. Duchon, 1978] for thin plate spline interpolation
can be improved by . We illustrate that -matrix
techniques can successfully be employed to solve very large thin plate spline
interpolation problem
Positive solutions of Schr\"odinger equations and fine regularity of boundary points
Given a Lipschitz domain in and a nonnegative
potential in such that is bounded
in we study the fine regularity of boundary points with respect to
the Schr\"odinger operator in . Using potential
theoretic methods, several conditions equivalent to the fine regularity of are established. The main result is a simple (explicit if
is smooth) necessary and sufficient condition involving the size of
for to be finely regular. An essential intermediate result consists in
a majorization of for
positive harmonic in and . Conditions for
almost everywhere regularity in a subset of are also
given as well as an extension of the main results to a notion of fine
-regularity, if , being two potentials, with and a second order elliptic operator.Comment: version 1. 23 pages version 3. 28 pages. Mainly a typo in Theorem 1.1
is correcte
Maximal L p -regularity for the Laplacian on Lipschitz domains
We consider the Laplacian with Dirichlet or Neumann boundary
conditions on bounded Lipschitz domains ?, both with the following two domains of
definition:D1(?) = {u ? W1,p(?) : ?u ? Lp(?), Bu = 0}, orD2(?) = {u ? W2,p(?) :
Bu = 0}, where B is the boundary operator.We prove that, under certain restrictions
on the range of p, these operators generate positive analytic contraction semigroups
on Lp(?) which implies maximal regularity for the corresponding Cauchy problems.
In particular, if ? is bounded and convex and 1 < p ? 2, the Laplacian with domain
D2(?) has the maximal regularity property, as in the case of smooth domains. In the
last part,we construct an example that proves that, in general, the DirichletâLaplacian
with domain D1(?) is not even a closed operator
Bethe-Sommerfeld conjecture for periodic operators with strong perturbations
We consider a periodic self-adjoint pseudo-differential operator
, , in which satisfies the following conditions:
(i) the symbol of is smooth in \bx, and (ii) the perturbation has
order less than . Under these assumptions, we prove that the spectrum of
contains a half-line. This, in particular implies the Bethe-Sommerfeld
Conjecture for the Schr\"odinger operator with a periodic magnetic potential in
all dimensions.Comment: 61 page
A limit model for thermoelectric equations
We analyze the asymptotic behavior corresponding to the arbitrary high
conductivity of the heat in the thermoelectric devices. This work deals with a
steady-state multidimensional thermistor problem, considering the Joule effect
and both spatial and temperature dependent transport coefficients under some
real boundary conditions in accordance with the Seebeck-Peltier-Thomson
cross-effects. Our first purpose is that the existence of a weak solution holds
true under minimal assumptions on the data, as in particular nonsmooth domains.
Two existence results are studied under different assumptions on the electrical
conductivity. Their proofs are based on a fixed point argument, compactness
methods, and existence and regularity theory for elliptic scalar equations. The
second purpose is to show the existence of a limit model illustrating the
asymptotic situation.Comment: 20 page
The transmission problem on a three-dimensional wedge
We consider the transmission problem for the Laplace equation on an infinite three-dimensional wedge, determining the complex parameters for which the problem is well-posed, and characterizing the infinite multiplicity nature of the spectrum. This is carried out in two formulations leading to rather different spectral pictures. One formulation is in terms of square integrable boundary data, the other is in terms of finite energy solutions. We use the layer potential method, which requires the harmonic analysis of a non-commutative non-unimodular group associated with the wedge