In the case ω = 1, this space is denoted W (Ω). Sobolev spaces without weights occur as spaces of solutions for elliptic and parabolic partial differential equations. In various applications, we can meet boundary value problems for elliptic equations whose ellipticity is “disturbed” in the sense that some degeneration or singularity appears. This “bad” behaviour can be caused by the coefficient...

Using the geometric properties of Sobolev spaces of integer order and a duality condition, the covariance operators of a generalized random field and its dual can be factorized. Via this covariance factorization, a representation of the generalized random field is obtained as a stochastic equation driven by generalized white noise. This stochastic equation becomes a differential equation under ...

Pucci and Serrin [21] conjecture that certain space dimensions behave “critically” in a semilinear polyharmonic eigenvalue problem. Up to now only a considerably weakened version of this conjecture could be shown. We prove that exactly in these dimensions an embedding inequality for higher order Sobolev spaces on bounded domains with an optimal embedding constant may be improved by adding a “li...

Abstract. We introduce a new family of refined Sobolev-Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stocha...

We study a Γ -convergence problem related to a new characterization of Sobolev spaces W1,p(RN) (p > 1) established in H.-M. Nguyen [H.-M. Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689–720] and J. Bourgain and H.-M. Nguyen [J. Bourgain, H.-M. Nguyen, A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 75–80]. We can also hand...

We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz’ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also gi...

We mainly study multivariate (uniform or Gaussian) integration defined for integrand spaces Fd such as weighted Sobolev spaces of functions of d variables with smooth mixed derivatives. The weight #j moderates the behavior of functions with respect to the jth variable. For #j #1, we obtain the classical Sobolev spaces whereas for decreasing #j 's the weighted Sobolev spaces consist of functions...

In a previous paper, the authors showed that the information complexity of the Fredholm problem of the second kind is essentially the same as that of the approximation problems over the spaces of kernels and right-hand sides. This allowed us to give necessary and sufficient conditions for the Fredholm problem to exhibit a particular level of tractability (for information complexity) over weight...

We prove dimension-invariant imbedding theorems for Sobolev spaces using extrapolation means.

Abstract. We propose a mathematical framework to effectively study lattice materials with periodic and non-periodic structures over entire spaces in one, two, and three dimensions. The existence and uniqueness of solutions for periodic lattice problems with absolute terms are proved in discrete Sobolev spaces. By Fourier transform discrete lattice problems are converted to semi-discrete problem...