14,841 research outputs found
Solvability of the Dirichlet, Neumann and the regularity problems for parabolic equations with H\"older continuous coefficients
We establish the -solvability of Dirichlet, Neumann and regularity
problems for divergence-form heat (or diffusion) equations with
H\"older-continuous diffusion coefficients, on bounded Lipschitz domains in
. This is achieved through the demonstration of invertibility of
the relevant layer-potentials which is in turn based on Fredholm theory and a
new systematic approach which yields suitable parabolic Rellich-type estimates
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Many PDEs involving fractional Laplacian are naturally set in unbounded
domains with underlying solutions decay very slowly, subject to certain power
laws. Their numerical solutions are under-explored. This paper aims at
developing accurate spectral methods using rational basis (or modified mapped
Gegenbauer functions) for such models in unbounded domains. The main building
block of the spectral algorithms is the explicit representations for the
Fourier transform and fractional Laplacian of the rational basis, derived from
some useful integral identites related to modified Bessel functions. With these
at our disposal, we can construct rational spectral-Galerkin and direct
collocation schemes by pre-computing the associated fractional differentiation
matrices. We obtain optimal error estimates of rational spectral approximation
in the fractional Sobolev spaces, and analyze the optimal convergence of the
proposed Galerkin scheme. We also provide ample numerical results to show that
the rational method outperforms the Hermite function approach
Hitchhiker's guide to the fractional Sobolev spaces
This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the
relations among some of their possible definitions and their role in the trace
theory. We prove continuous and compact embeddings, investigating the problem
of the extension domains and other regularity results. Most of the results we
present here are probably well known to the experts, but we believe that our
proofs are original and we do not make use of any interpolation techniques nor
pass through the theory of Besov spaces. We also present some counterexamples
in non-Lipschitz domains
Principal eigenvalue of the fractional Laplacian with a large incompressible drift
We add a divergence-free drift with increasing magnitude to the fractional
Laplacian on a bounded smooth domain, and discuss the behavior of the principal
eigenvalue for the Dirichlet problem. The eigenvalue remains bounded if and
only if the drift has non-trivial first integrals in the domain of the
quadratic form of the fractional Laplacian.Comment: 19 page
A brief survey of Nigel Kalton's work on interpolation and related topics
This is the third of a series of papers surveying some small part of the
remarkable work of our friend and colleague Nigel Kalton. We have written it as
part of a tribute to his memory. It does not contain new results. This time,
rather than concentrating on one particular paper, we attempt to give a general
overview of Nigel's many contributions to the theory of interpolation of Banach
spaces, and also, significantly, quasi-Banach spaces.Comment: 11 page
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