948 research outputs found
From spectral cluster to uniform resolvent estimates on compact manifolds
It is well known that uniform resolvent estimates imply spectral cluster
estimates. We show that the converse is also true in some cases. In particular,
the universal spectral cluster estimates of Sogge \cite{MR930395} for the
Laplace--Beltrami operator on compact Riemannian manifolds without boundary
directly imply the uniform Sobolev inequality of Dos Santos Ferreira, Kenig and
Salo \cite{MR3200351}, without any reference to parametrices. This observation
also yields new resolvent estimates for manifolds with boundary or with
nonsmooth metrics, based on spectral cluster bounds of Smith--Sogge
\cite{MR2316270} and Smith, Koch and Tataru~\cite{MR2443996}, respectively. We
also convert the recent spectral cluster bounds of Canzani and Galkowski
\cite{Canzani--Galkowski} to improved resolvent bounds. Moreover, we show that
the resolvent estimates are stable under perturbations and use this to
establish uniform Sobolev and spectral cluster inequalities for Schr\"odinger
operators with singular potentials.Comment: Theorem 4.1 now also converts improved spectral cluster bounds to
improved resolvent bounds. A new application of the recent spectral cluster
bounds of Canzani and Galkowski is included. Some minor typos correcte
Geometry of PDE's. IV: Navier–Stokes equation and integral bordism groups
AbstractFollowing our previous results on this subject [R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(I): Webs on PDE's and integral bordism groups. The general theory, Adv. Math. Sci. Appl. 17 (2007) 239–266; R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(II): Webs on PDE's and integral bordism groups. Applications to Riemannian geometry PDE's, Adv. Math. Sci. Appl. 17 (2007) 267–285; A. Prástaro, Geometry of PDE's and Mechanics, World Scientific, Singapore, 1996; A. Prástaro, Quantum and integral (co)bordism in partial differential equations, Acta Appl. Math. (5) (3) (1998) 243–302; A. Prástaro, (Co)bordism groups in PDE's, Acta Appl. Math. 59 (2) (1999) 111–201; A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing Co, Singapore, 2004, 500 pp.; A. Prástaro, Geometry of PDE's. I: Integral bordism groups in PDE's, J. Math. Anal. Appl. 319 (2006) 547–566; A. Prástaro, Geometry of PDE's. II: Variational PDE's and integral bordism groups, J. Math. Anal. Appl. 321 (2006) 930–948; A. Prástaro, Th.M. Rassias, Ulam stability in geometry of PDE's, Nonlinear Funct. Anal. Appl. 8 (2) (2003) 259–278; I. Stakgold, Boundary Value Problems of Mathematical Physics, I, The MacMillan Company, New York, 1967; I. Stakgold, Boundary Value Problems of Mathematical Physics, II, Collier–MacMillan, Canada, Ltd, Toronto, Ontario, 1968], integral bordism groups of the Navier–Stokes equation are calculated for smooth, singular and weak solutions, respectively. Then a characterization of global solutions is made on this ground. Enough conditions to assure existence of global smooth solutions are given and related to nullity of integral characteristic numbers of the boundaries. Stability of global solutions are related to some characteristic numbers of the space-like Cauchy data. Global solutions of variational problems constrained by (NS) are classified by means of suitable integral bordism groups too
Partitions of Minimal Length on Manifolds
We study partitions on three dimensional manifolds which minimize the total
geodesic perimeter. We propose a relaxed framework based on a
-convergence result and we show some numerical results. We compare our
results to those already present in the literature in the case of the sphere.
For general surfaces we provide an optimization algorithm on meshes which can
give a good approximation of the optimal cost, starting from the results
obtained using the relaxed formulation
Dynamical System Methods in Fluid Dynamics
The workshop was organized around the infusion of new techniques from dynamical systems, geometric methods, multiscale analysis, scientific computation, and control theory into traditional methods in fluid mechanics. It was well attended with about 45 participants with broad geographic representation from all continents. There was an excellent blend of senior researchers, students, postdocs and junior faculty
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