A discrete norm on a Lipschitz surface and the Sobolev straightening of a boundary

Let a piece of the boundary of a Lipschitz domain be parameterized conventionally and let the traces of functions in the Sobolev space W _p ² be written out through this parameter. In this space, we propose a discrete (diadic) norm generalizing A. Kamont’s norm in the plane case. We study the conditions when the space of traces coincides with the corresponding space for the plane boundary.     

Periodization strategy may fail in high dimensions

We discuss periodization of smooth functions f of d variables for approximation of multivariate integrals. The benefit of periodization is that we may use lattice rules, which have recently seen significant progress. In particular, we know how to construct effectively a generator of the rank-1 lattice rule with n points whose worst case error enjoys a nearly optimal bound C _d,p n ^−p . Here C _d,p is independent of d or depends at most polynomially on d, and p can be arbitrarily close to the smoothness of functions belonging to a weighted Sobolev space with an appropriate condition on the weights. If F denotes the periodization for f then the error of the lattice rule for a periodized function F is bounded by C _d,p n ^−p ∣∣F∣∣ with the norm of F given in the same Sobolev space. For small or moderate d, the norm of F is not much larger than the norm of f. This means that for small or moderate d, periodization is successful and allows us to use optimal properties of lattice rules also for non-periodic functions. The situation is quite different if d is large since the norm of F can be exponentially larger than the norm of f. This can already be seen for f = 1. Hence, the upper bound of the worst case error of the lattice rule for periodized functions is quite bad for large d. We conjecture not only that this upper bound is bad, but also that all lattice rules fail for large d. That is, if we fix the number of points n and let d go to infinity then the worst case error of any lattice rule is bounded from below by a positive constant independent of n. We present a number of cases suggesting that this conjecture is indeed true, but the most interesting case, when the sum of the weights of the corresponding Sobolev space is bounded in d, remains open.      

Some Inequalities in Classical Spaces with Mixed Norms

We consider some inequalities in such classical Banach Function Spaces as Lorentz, Marcinkiewicz, and Orlicz spaces. Our aim is to explore connections between the norm of a function of two variables on the product space and the mixed norm of the same function, where mixed norm is calculated in function spaces on coordinate spaces, first in one variable, then in the other. This issue is motivated by various problems of functional analysis and theory of functions. We will currently mention just geometry of spaces of vector-valued functions and embedding theorems for Sobolev and Besov spaces generated by metrics which differ from L ^p. Our main results are actually counterexamples for Lorentz spaces versus the natural intuition that arises from the easier case of Orlicz spaces (Section 2). In the Appendix we give a proof for the Kolmogorov–Nagumo theorem on change of order of mixed norm calculation in its most general form. This result shows that L ^p is the only space where it is possible to change this order.     

Riemann-Hilbert and Poincaré problems with discontinuous boundary conditions for some model systems of partial differential equations

We study the solvability of the Riemann-Hilbert and Poincaré problems for systems of Cauchy-Riemann and Bitsadze equations in Sobolev spaces. For a generalized system of Cauchy-Riemann equations, we pose a boundary value problem and prove its unique solvability in the Sobolev space W ₂¹ (D). By supplementing the Riemann-Hilbert boundary conditions with some new conditions, we obtain a statement of the Poincaré problem with discontinuous boundary conditions for a system of second-order Bitsadze equations; we also prove the unique solvability of this problem in Sobolev spaces.     

On the stability of the collocation method for the double layer operator on polyhedral domains in R3

We develop a new method to give estimates for the double layer operator on cones in R³. Here we use weighted norms which are equivalent to the usual L^∞‐norm. This result includes the weighted norms which were constructed by Wendland and Kral for the case of rectangular cones. If all vertices in a polyhedral domain (resp. their corresponding cones) allow the construction of a weighted norm, such that the double layer operator has norm smaller than one half, we can prove the stability of the collocation method with piecewise constant trial functions.     

Characterization of Orlicz–Sobolev space

We give a new characterization of the Orlicz–Sobolev space W ^1,Ψ(R ⁿ ) in terms of a pointwise inequality connected to the Young function Ψ. We also study different Poincaré inequalities in the metric measure space.     

Compact Sobolev embedding theorems involving symmetry and its application

Given a bounded regular domain with cylindrical symmetry, functions having such symmetry and belonging to W ^1,p can be embedded compactly into some weighted L ^q spaces, with q superior to the critical Sobolev exponent. A similar result is also obtained for variable exponent Sobolev space W ^1,p(x). Furthermore, we give a simple application to the p(x)-Laplacian problem.     

Stochastic differential equations with fractal noise

Stochastic differential equations in ?ⁿ with random coefficients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space W^β₂ for some β > 1/2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution procedure is developed which combines the stochastic Itô calculus with fractional calculus via norm estimates of associated integral operators in W^α ₂ for 0 < α < 1. Linear equations are considered as a special case. This approach leads to fast computer algorithms basing on Picard's iteration method. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sobolev spaces on an arbitrary metric space

We define Sobolev space W ^1,p for 1<p on an arbitrary metric space with finite diameter and equipped with finite, positive Borel measure. In the Euclidean case it coincides with standard Sobolev space. Several classical imbedding theorems are special cases of general results which hold in the metric case. We apply our results to weighted Sobolev space with Muckenhoupt weight.This work is supported by KBN grant no. 2 1057 91 01     

One Initial Boundary-Value Problem for Integro-Differential Equation of the Second Order With Power Nonlinearity

In the Sobolev space W _∞ ² (?⁺) we investigate one initial boundary-value problem for integro-differential equation of the second order with power nonlinearity on a semi-axis. Assuming that summary-difference even kernel serves for the considered kernel as minorant in the sense of M. A. Krasnosel’skii, we prove the existence of a nonnegative (nontrivial) solution in the Sobolev spaceW _∞ ² (?⁺). We also calculate the limits of constructed solution at infinity.     

Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem

A singularly perturbed semilinear reaction-diffusion problem in the unit cube, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio condition is imposed. This result is obtained by combining (i) sharp bounds on the Green’s function of the continuous differential operator in the Sobolev W ^1,1 and W ^2,1 norms and (ii) a special representation of the residual in terms of an arbitrary current mesh and the current computed solution. Numerical results on a priori chosen meshes are presented that support our theoretical estimate.     

Variational Problems with a p-Homogeneous Energy

We investigate existence, uniqueness and positivity of minimizers or critical points for an energy functional which contains only p-homogeneous and linear terms, 1p-homogeneous part of the energy functional is that it be given by the p-th power of an equivalent, uniformly convex norm on the underlying Sobolev space. Finally, continuous dependence of minimizers on the energy functional is established.     

The Embedding W n,1 into C 0 on Compact Riemannian Manifolds

We study the best constant in the inequality corresponding to the Sobolev embedding W ^n,1(R ⁿ ) into the space of bounded continuous functions C ₀(R ⁿ ). Then, we adapt this inequality on compact Riemannian manifolds and discuss on its optimality.     

Fixed points of asymptotically regular semigroups in Banach spaces

In this paper we study in Banach spaces the existence of fixed points of (nonlinear) asymptotically regular semigroups. We establish for these semigroups some fixed point theorems in spaces with weak uniform normal structure, in a Hilbert space, inL ^p spaces, in Hardy spacesH ^p and in Sobolev spacesW ^r.p for 1<p<∞ andr≥0, in spaces with Lifshitz’s constant greater than one. These results are the generalizations of [8, 10, 16].     

Analysis of finite element methods for second order boundary value problems using mesh dependent norms

This paper presents a new approach to the analysis of finite element methods based onC ⁰-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL ₂ norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL ₂ norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.     

On a Strong Solution in the Method of Spherical Harmonics for a Nonstationary Transport Equation

We prove existence of a strong generalized solution in the Sobolev space W ₂ ¹ to the nonstationary problem for the system of the method of spherical harmonics (MSH) corresponding to the radiation transport problem.     

RECOVERING SINGULARITIES FROM BACKSCATTERING IN TWO DIMENSIONS

We have shown that in two dimensions the leading singularities of the quantum mechanical scattering potential are determined by the backscattering data. We assume that the short range potential belongs to a suitable weighted Sobolev space, and by estimating the iterative terms in the Born-expansion we are able to show, that for example for Heaviside-type singularities across a smooth hypersurface, both the location and the size of the jump are recovered from backscattering.

The main part of the proof consists in getting sharp enough estimates for the first non-linear Born-term. These estimates are proven using a recent characterization of W ^1,p -functions due to P. Hajlasz, and a modification of the classical Triebel's Maximal Inequality.     

The hardy-littlewood maximal function of a sobolev function

We prove that the Hardy-Littlewood maximal operator is bounded in the Sobolev spaceW ^1,p (R ⁿ ) for 1<p≤∞. As an application we study a weak type inequality for the Sobolev capacity. We also prove that the Hardy-Littlewood maximal function of a Sobolev function is quasi-continuous.     

Algebra properties for Sobolev spaces — applications to semilinear PDEs on manifolds

In this work, we aim to prove algebra properties for generalized Sobolev spaces W ^s,p ?? L ^?? on a Riemannian manifold (or more general homogeneous type space as graphs), where W ^s,p is of Bessel-type W ^s,p := (1+L)^?s/m (L ^p ) with an operator L generating a heat semigroup satisfying off-diagonal decays. We do not require any assumption on the gradient of the semigroup. Instead, we propose two different approaches (one by paraproducts associated to the heat semigroup and another one using functionals). We also study the action of nonlinearities on these spaces and give applications to semi-linear PDEs. These results are new on Riemannian manifolds (with a non-bounded geometry) and even in euclidean space for Sobolev spaces associated to second order uniformly elliptic operators in divergence form.     

On a nonlocal generalization of the biharmonic Dirichlet problem

We consider a mixed problem with the Dirichlet boundary conditions and integral conditions for the biharmonic equation. We prove the existence and uniqueness of a generalized solution in the weighted Sobolev space W ₂². We show that the problem can be viewed as a generalization of the Dirichlet problem.