Consider the Sobolev embedding operator from the space of functions in W 1,p(I) with average zero into Lp , where I is a finite interval and p> 1. This operator plays an important role in recent work. The operator norm and its approximation numbers in closed form are calculated. The closed form of the norm and approximation numbers of several similar Sobolev embedding operators on a finite inte...

In this paper we present a scheme for the numerical solution of one-dimensional stochastic differential equations (SDEs) whose drift belongs to fractional Sobolev space negative regularity (a subspace Schwartz distributions). We obtain rate convergence in suitable $L^1$-norm and implement numerically. To best our knowledge is first study (and implement) solutions SDEs lives distributions. As by...

max(u,v)∈E |f(u)− f(v)| if p =∞. If G is connected, then the only functions f satisfying ||f ||E,p = 0 are constant functions, so || · ||E,p is a norm on each linear space of functions on VG which does not contain constants. Usually we shall consider the subspace in the space of all functions on VG given by ∑ v∈V f(v)dv = 0. The obtained normed space will be called a Sobolev space on G and will...

We study a semi-discrete Galerkin method for solving the single-layer equation Vu = f with an approximating subspace of piecewise constant functions. Error bounds in Sobolev norms kkk s with ?1 s < 1 2 are proven and are of the same order as for the original Galerkin method. The distinctive features of the present work are that we handle irregular meshes and do not rely on Fourier methods. The ...

where Ω is a bounded domain in Rn (1 ≤ n ≤ 3), f is a nonlinear map.We use the homogeneous Sobolev space H1 0 (Ω)(≡ H1 0 ) for the solution of (1). Also some appropriate assumptions are imposed on the map f . In order to treat the problem as the finite procedure, we use a finite element subspace Sh of H1 0 with mesh size h. Denoting the inner product on L2(Ω) by (·, ·), we define the H1 0 -proj...

Sato’s hyperfunctions are known to be represented as the boundary values of harmonic functions as well as those of holomorphic functions. The author obtains a bijective Poisson mapping P : S∗′(Rn) −→ S∗′(S∗Rn) ∩H(S∗Rn) where H(S∗Rn) is a kind of Hardy subspace of B(S∗Rn). Moreover, the author has an isomorphism between Sobolev spaces P : W (R) −→ W s+(n−1)/4(S∗Rn) ∩H(S∗Rn). There are some simil...

where ∆pu= div(|∇u|p−2∇u) is the p-Laplacian, 1 < p <∞. ByW1,p(Ω) we denote the usual Sobolev space with dual space (W1,p(Ω))∗, andW 0 (Ω) denotes its subspace whose elements have generalized homogeneous boundary values and whose dual space is given by W−1,p(Ω). We assume the following growth and asymptotic behaviour of the nonlinear right-hand side f of (1.1): (H1) f :Ω×R → R is a Carathéodory...

Molecule spaces have been introduced by Furioli and Terraneo [Funkcial. Ekvac., 45 (2002), pp. 141–160] to study some local behavior of solutions to the Navier–Stokes equations. In this paper we give a new characterization of these spaces and simplify Furioli and Terraneo’s result. Our analysis also provides a persistence result for Navier–Stokes in a subspace of L2(R3, (1 + |x|2)αdx), α < 5/2,...

We consider the Schr\"odinger operator on halfline with potential $(m^2-\frac14)\frac1{x^2}$, often called Bessel operator. assume that $m$ is complex. study domains of various closed homogeneous realizations In particular, we prove domain its minimal realization for $|\Re(m)|<1$ and unique $\Re(m)>1$ coincide second order Sobolev space. On other hand, if $\Re(m)=1$ space a subspace infinite co...