353 research outputs found
On the negative spectrum of the Robin Laplacian in corner domains
For a bounded corner domain , we consider the Robin Laplacian in
with large Robin parameter. Exploiting multiscale analysis and a
recursive procedure, we have a precise description of the mechanism giving the
ground state of the spectrum. It allows also the study of the bottom of the
essential spectrum on the associated tangent structures given by cones. Then we
obtain the asymptotic behavior of the principal eigenvalue for this singular
limit in any dimension, with remainder estimates. The same method works for the
Schr\"odinger operator in with a strong attractive
delta-interaction supported on . Applications to some Erhling's
type estimates and the analysis of the critical temperature of some
superconductors are also provided
Gradient-like parabolic semiflows on BUC(âN)
We prove that a class of weighted semilinear reaction diffusion equations on RN generates gradient-like semiflows on the Banach space of bounded uniformly continuous functions on RN. If N = 1 we show convergence to a single equilibrium. The key for getting the result is to show the exponential decay of the stationary solutions, which is obtained by means of a decay estimate of the kernel of the underlying semigrou
On the lowest eigenvalue of Laplace operators with mixed boundary conditions
In this paper we consider a Robin-type Laplace operator on bounded domains.
We study the dependence of its lowest eigenvalue on the boundary conditions and
its asymptotic behavior in shrinking and expanding domains. For convex domains
we establish two-sided estimates on the lowest eigenvalues in terms of the
inradius and of the boundary conditions
Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians
We consider the problem of minimising the th eigenvalue, , of
the (-)Laplacian with Robin boundary conditions with respect to all domains
in of given volume . When , we prove that the second
eigenvalue of the -Laplacian is minimised by the domain consisting of the
disjoint union of two balls of equal volume, and that this is the unique domain
with this property. For and , we prove that in many cases a
minimiser cannot be independent of the value of the constant in the
boundary condition, or equivalently of the volume . We obtain similar
results for the Laplacian with generalised Wentzell boundary conditions .Comment: 16 page
On the isoperimetric problem for the Laplacian with Robin and Wentzell boundary conditions
We consider the problem of minimising the eigenvalues of the Laplacian with Robin boundary conditions and generalised Wentzell boundary conditions with respect to the domain on which the problem is defined. For the Robin problem, when we extend the Faber-Krahn inequality of Daners [Math. Ann. 335 (2006), 767--785], which states that the ball minimises the first eigenvalue, to prove that the minimiser is unique amongst domains of class . The method of proof uses a functional of the level sets to estimate the first eigenvalue from below, together with a rearrangement of the ball's eigenfunction onto the domain and the usual isoperimetric inequality. We then prove that the second eigenvalue attains its minimum only on the disjoint union of two equal balls, and set the proof up so it works for the Robin -Laplacian. For the higher eigenvalues, we show that it is in general impossible for a minimiser to exist independently of . When establish a type of equivalence property between the Wentzell and Robin minimisers for all eigenvalues. This yields a minimiser of the second Wentzell eigenvalue. We also prove a Cheeger-type inequality for the first eigenvalue in this case
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