Search

We study reflected diffusion on uniform domains where the underlying space admits a symmetric diffusion that satisfies sub-Gaussian heat kernel estimates. A celebrated theorem of Jones (Acta Math 147(1-2):71–88, 1981) states that uniform domains in Euclidean space are extension domains for Sobolev spaces. In this work, we obtain a similar extension property for metric spaces equipped with a Dirichlet form whose heat kernel satisfies a sub-Gaussian estimate. We introduce a scale-invariant version of this extension property and apply it to show that the reflected diffusion process on such a uniform domain inherits various properties from the ambient space, such as Harnack inequalities, cutoff energy inequality, and sub-Gaussian heat kernel bounds. In particular, our work extends Neumann heat kernel estimates of Gyrya and Saloff-Coste (Astérisque 336:145, 2011) beyond the Gaussian space-time scaling. Furthermore, our estimates on the extension operator imply that the energy measure of the boundary of a uniform domain is always zero. This property of the energy measure is a broad generalization of Hino's result (Probab Theory Relat Fields 156:739–793, 2013) that proves the vanishing of the energy measure on the outer square boundary of the standard Sierpiński carpet equipped with the self-similar Dirichlet form. [ABSTRACT FROM AUTHOR]

Let Ω and Ω ˜ be domains in the Euclidean space ℝn. We study the boundary behavior of weak (p, q)-quasiconformal mappings φ : Ω → Ω ˜ , n – 1 < q ≤ p < n. The suggested method is based on the capacitary distortion properties of the weak (p, q)-quasiconformal mappings. [ABSTRACT FROM AUTHOR]

We establish some equivalent conditions for a homeomorphism φ : D → D ′ of Euclidean domains in R n , n ≥ 2 , to induce a bounded composition operator φ ∗ : L p 1 (D ′ ; ω) ∩ Lip l (D ′) → L q 1 (D ; θ) , where 1 < q ≤ p < ∞ , by the composition rule: φ ∗ (f) = f ∘ φ . Here ω : D ′ → (0 , ∞) is an arbitrary weight function on the domain D ′ , and θ : D → (0 , ∞) is some weight function in Muckenhoupt's A q -class on the domain D. Moreover, we prove that the class of homeomorphisms under consideration is completely determined by the controlled variation of the weighted capacity of cubical condensers whose shells are concentric cubes. [ABSTRACT FROM AUTHOR]

The study of certain differential operators between Sobolev spaces of sections of vector bundles on compact manifolds equipped with rough metric is closely related to the study of locally Sobolev functions on domains in the Euclidean space. In this paper, we present a coherent rigorous study of some of the properties of locally Sobolev-Slobodeckij functions that are especially useful in the study of differential operators between sections of vector bundles on compact manifolds with rough metric. The results of this type in published literature generally can be found only for integer order Sobolev spaces W m , p or Bessel potential spaces H s . Here, we have presented the relevant results and their detailed proofs for Sobolev-Slobodeckij spaces W s , p where s does not need to be an integer. We also develop a number of results needed in the study of differential operators on manifolds that do not appear to be in the literature. [ABSTRACT FROM AUTHOR]

We study covariant Sobolev spaces and ∇ -differential operators with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate-free approach that uses connections (which are typically denoted ∇ ). These concepts arise naturally from geometric partial differential equations, including some that are formulated on plain Euclidean domains, for instance, from problems formulated on the boundary of smooth domains or in relation to the weighted Sobolev spaces used to study PDEs on polyhedral domains. We prove several basic properties of the covariant Sobolev spaces and of the ∇ -differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and the independence of the covariant Sobolev spaces on the choices of the connection ∇ , as long as the new connection is obtained using a totally bounded perturbation. We also introduce the Fréchet finiteness condition (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our covariant Sobolev spaces and of our ∇ -differential operators. We also introduce and study the notion of a ∇ -bidifferential operator (a bilinear version of differential operators), obtaining results similar to those obtained for ∇ -differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience. [ABSTRACT FROM AUTHOR]

We analyze several types of Galerkin approximations of a Gaussian random field Z : D × Ω → R indexed by a Euclidean domain D ⊂ R d whose covariance structure is determined by a negative fractional power L - 2 β of a second-order elliptic differential operator L : = - ∇ · (A ∇) + κ 2 . Under minimal assumptions on the domain D , the coefficients A : D → R d × d , κ : D → R , and the fractional exponent β > 0 , we prove convergence in L q (Ω ; H σ (D)) and in L q (Ω ; C δ (D ¯)) at (essentially) optimal rates for (1) spectral Galerkin methods and (2) finite element approximations. Specifically, our analysis is solely based on H 1 + α (D) -regularity of the differential operator L, where 0 < α ≤ 1 . For this setting, we furthermore provide rigorous estimates for the error in the covariance function of these approximations in L ∞ (D × D) and in the mixed Sobolev space H σ , σ (D × D) , showing convergence which is more than twice as fast compared to the corresponding L q (Ω ; H σ (D)) -rate. We perform several numerical experiments which validate our theoretical results for (a) the original Whittle–Matérn class, where A ≡ Id R d and κ ≡ const. , and (b) an example of anisotropic, non-stationary Gaussian random fields in d = 2 dimensions, where A : D → R 2 × 2 and κ : D → R are spatially varying. [ABSTRACT FROM AUTHOR]

We prove that each homeomorphism of Euclidean domains in , , belonging to the Sobolev class , where , and having finite distortion induces a bounded composition operator from the weighted Sobolev space into for some weight function . This implies that in the cases and as well as and the inverse belongs to the Sobolev class , has finite distortion, and is differentiable -almost everywhere in . We apply this result to -homeomorphisms; the method of proof also works for homeomorphisms of Carnot groups. Moreover, we prove that the class of -homeomorphisms is completely determined by the controlled variation of the capacity of cubical condensers whose shells are concentric cubes. [ABSTRACT FROM AUTHOR]

We survey a few trace theorems for Sobolev spaces on N-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries, including Besov's Theorem and other characterizations of traces on planar domains, polygons in particular, in the spirit of the work of P. Grisvard. Finally, we present a recent approach, originally developed by G. Auchmuty in the case of the Sobolev space H1(Ω) on a Lipschitz domain Ω, and which we have further developed for the trace spaces of Hk(Ω), k ⩾ 2, by using Fourier expansions associated with the eigenfunctions of new multi-parameter polyharmonic Steklov problems. [ABSTRACT FROM AUTHOR]

We establish an embedding theorem for spaces of functions of positive smoothness defined on irregular domains of n-dimensional Euclidean space in spaces of the same type and obtain some closely related results. [ABSTRACT FROM AUTHOR]

In the paper, we consider different generalizations of the Dirichlet problem in an arbitrary domain of n-dimensional Euclidean space Rn for a magnetic Schrödinger operator. Their equivalence under different conditions on magnetic and electric potentials are proved. Magnetic Sobolev space of first order is introduced and it is proved that this space is topologically equivalent to the ordinary Sobolev space of first order. The interval of a real axis for a spectral parameter, where the Dirichlet first generalized problem has a unique solution, is shown. The Green operator for the Dirichlet first and second generalized problems is introduced and its boundedness from the first order conjugated Sobolev space to the first order Sobolev space is proved. [ABSTRACT FROM AUTHOR]

We prove that Burenkov's extension operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n-dimensional Euclidean space. [ABSTRACT FROM AUTHOR]

An embedding of the Sobolev spaces W (ℝ) in Lizorkin-type spaces of locally integrable functions of smoothness zero is obtained; a similar assertion for Riesz and Bessel potentials is presented. The embedding theorem is extended to Sobolev spaces on irregular domains in n-dimensional Euclidean space. The statement of the theorem depends on geometric parameters of the domain of functions. [ABSTRACT FROM AUTHOR]

We obtain necessary and sufficient conditions for a homeomorphism of domains in a Euclidean space to generate a bounded embedding operator of the Orlicz-Sobolev spaces defined by a special class of N-functions. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with the existence of extremals for the Adams inequality. We first establish an upper bound for the classical Adams functional along of all concentrated sequences in the higher order Sobolev space with homogeneous Navier boundary conditions W N m , n m (Ω) , which in particular includes the classical Sobolev space W 0 m , n m (Ω) , where Ω is a smooth bounded domain in Euclidean n -space. Secondly, based on the Concentration-compactness alternative due to Do Ó and Macedo, we prove the existence of extremals for the Adams inequality under Navier boundary conditions for second order derivatives at least for higher dimensions when Ω is an Euclidean ball. [ABSTRACT FROM AUTHOR]

In this paper, the authors prove several equivalent characterizations of Sobolev spaces of even integer orders on R n , using the average B t f ( x ) ≔ 1 | B ( x , t ) | ∫ B ( x , t ) f ( y ) d y of a function f over the ball B ( x , t ) ≔ { y ∈ R n : | y − x | < t } with x ∈ R n and t ∈ ( 0 , ∞ ) . These characterizations rely only on the metric and the Lebesgue measure on R n and are simpler than those obtained recently by Alabern et al. (2012). Moreover, these results may shed new light on the theory of high order Sobolev spaces on spaces of homogeneous type. [ABSTRACT FROM AUTHOR]