There are several generalizations of the classical theory of Sobolev spaces as they are necessary for the applications to Carnot-Carathéodory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms. The aim of this paper is to present a unified approach to the theory of Sobolev spaces that covers applications to many of those areas. The variety of different areas of applications forces a very general setting. We are given a metric space X equipped with a doubling measure ¯. A generalization of a Sobolev function and its gradient is a pair u 2 L 1 loc (X), 0 g 2 L p (X) such that for every ball B ae X the Poincar'e-type inequality Z B ju \Gamma uB j d¯ Cr `Z oeB g p d¯ ' 1=p holds, where r is the radius of B and oe 1, C ? 0 are fixed constants. Working in the above setting we show that basically...

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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

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We prove essential self-adjointness for semi-bounded below magnetic Schrödinger operators on complete Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. Some singularities of the scalar potential are allowed. This is an extension of the Povzner–Wienholtz–Simader theorem. The proof uses the scheme of Wienholtz but requires a refined invariant integration by parts technique, as well as a use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.

In image sequence analysis, variational optical ow computations require the solution of a parameter dependent optimization problem with a data term and a regularizer. In this paper we study existence and uniqueness of the optimizers. Our studies rely on quasiconvex functionals on the spaces W

Introduction Let (M; g) be a Riemannian manifold (i.e. M is a C 1 -manifold, g = (g ij ) is a Riemannian metric on M ), dim M = n. We will always assume that M is connected. Let denote the Laplace-Beltrami operator on scalar functions on M i.e. u = 1 p g @ @x i ( p gg ij @u @x j ) where x 1 ; : : : ; x n are local coordinates, (g ij ) is the inverse matrix to g ij , g = det(g ij ) and we use the usual summation convention. The main object of our study will be the Schrodinger operator H = + V (x) (1.

We consider a magnetic Schrödinger operator H in R n or on a Riemannian manifold M of bounded geometry. Sufficient conditions for the spectrum of H to be discrete are given in terms of behavior at infinity for some effective potentials Veff which are expressed through electric and magnetic fields. These conditions can be formulated in the form Veff (x) → + ∞ as x → ∞. They generalize the classical result by K.Friedrichs (1934), and include earlier results of J. Avron, I. Herbst and B. Simon (1978), A. Dufresnoy (1983) and A. Iwatsuka (1990) which were obtained in the absence of an electric field. More precise sufficient conditions can be formulated in terms of the Wiener capacity and extend earlier work by A.M. Molchanov (1953) and V. Kondrat’ev and M. Shubin (1999) who considered the case of the operator without a magnetic field. These conditions become necessary and sufficient in case there is no magnetic field and the electric potential is semi-bounded below.

Our aim in this paper is to give geometrical characterizations of domains which support Sobolev-Poincaré type imbeddings. The classical Sobolev-Poincaré imbeddings for a “nice” bounded domain Ω ⊂ Rn depend on whether the exponent p is less than, equal to, or greater than n (throughout this paper, n ≥ 2). In the case 1 ≤ p < n, we get the Sobolev-Poincaré