Boundary element methods for variational inequalities

In this paper we present a priori error estimates for the Galerkin solution of variational inequalities which are formulated in fractional Sobolev trace spaces, i.e. in $\widetilde{H}^{1/2}(\Gamma )$ . In addition to error estimates in the energy norm we also provide, by applying the Aubin–Nitsche trick for variational inequalities, error estimates in lower order Sobolev spaces including $L_2(\Gamma )$ . The resulting discrete variational inequality is solved by using a semi-smooth Newton method, which is equivalent to an active set strategy. A numerical example is given which confirms the theoretical results.     

Solutions for nonlinear elliptic equations with variable growth and degenerate coercivity

In this paper, based on the theory of variable exponent Sobolev space, we study a class of nonlinear elliptic equations with principal part having degenerate coercivity and obtain some existence and regularity results for the solutions.     

Regularity of the inverse of a homeomorphism of a Sobolev–Orlicz space

Given a homeomorphism ? ∈ W _M ¹ , we determine the conditions that guarantee the belonging of the inverse of ? in some Sobolev–Orlicz space W _F ¹ . We also obtain necessary and sufficient conditions under which a homeomorphism of domains in a Euclidean space induces the bounded composition operator of Sobolev–Orlicz spaces defined by a special class of N-functions. Using these results, we establish requirements on a mapping under which the inverse homeomorphism also induces the bounded composition operator of another pair of Sobolev–Orlicz spaces which is defined by the first pair.     

Estimates of anisotropic Sobolev spaces with mixed norms for the Stokes system in a half-space

We are concerned with the non-stationary Stokes system with non-homogeneous external force and non-zero initial data in \({\mathbb {R}}^n_+ \times (0,T)\). We obtain new estimates of solutions including pressure in terms of mixed anisotropic Sobolev spaces. As an application, some anisotropic Sobolev estimates are presented for weak solutions of the Navier–Stokes equations in a half-space in dimension three.     

Bessel Potentials in Ahlfors Regular Metric Spaces

In this paper we introduce Bessel potentials and the Sobolev potential spaces resulting from them in the context of Ahlfors regular metric spaces. The Bessel kernel is defined using a Coifman type approximation of the identity, and we show integration against it improves the regularity of Lipschitz, Besov and Sobolev-type functions. For potential spaces, we prove density of Lipschitz functions, and several embedding results, including Sobolev-type embedding theorems. Finally, using singular integrals techniques such as the T1 theorem, we find that for small orders of regularity Bessel potentials are inversible, its inverse in terms of the fractional derivative, and show a way to characterize potential spaces, concluding that a function belongs to the Sobolev potential space if and only if itself and its fractional derivative are in L^p. Moreover, this characterization allows us to prove these spaces in fact coincide with the classical potential Sobolev spaces in the Euclidean case.     

Boundary values of differentiable functions defined on an arbitrary domain of a Carnot group

We study the boundary values of the functions of the Sobolev function spaces W _∞ ^l and the Nikol’ski? function spaces H _∞ ^l which are defined on an arbitrary domain of a Carnot group. We obtain some reversible characteristics of the traces of the spaces under consideration on the boundary of the domain of definition and sufficient conditions for extension of the functions of these spaces outside the domain of definition. In some cases these sufficient conditions are necessary.     

Approximation by Linear Fractional Transformations of Simple Partial Fractions and Their Differences

Let f be a function and ρ be a simple partial fraction of degree at most n. Under linear-fractional transformations, the difference f ? ρ becomes the difference of another function and a certain simple partial fraction of degree at most n with a quadratic weight. We study applications of this important property. We prove a theorem on uniqueness of interpolating simple partial fraction, generalizing known results, and obtain estimates for the best uniform approximation of certain functions on the real semi-axis ?⁺. For continuous functions of rather common type we first obtain estimates of the best approximation by differences of simple partial fractions on ?⁺. For odd functions we obtain such estimates on the whole axis ?.     

Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems

We consider a quadratic optimal control problem governed by a nonautonomous affine ordinary differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we provide a first order expansion for the penalized states and adjoint states around the state and adjoint state of the original problem. Our main argument relies on the following fact: if the optimal control satisfies strict complementarity conditions for its Hamiltonian except for a set of times with null Lebesgue measure, the functional estimates for the penalized optimal control problem can be derived from the estimates of a related finite dimensional problem. Our results provide several types of efficiency measures of the penalization technique: error estimates of the control for L ^s norms (s in [1, +∞]), error estimates of the state and the adjoint state in Sobolev spaces W ^1,s (s in [1, +∞)) and also error estimates for the value function. For the L ¹ norm and the logarithmic penalty, the sharpest results are given, by establishing an error estimate for the penalized control of order ${O(\varepsilon|\log\epsilon|)}$ where ${\varepsilon >0 }$ is the (small) penalty parameter.     

Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations

In this paper, we derive recovery type superconvergence analysis and a posteriori error estimates for the finite element approximation of the distributed optimal control governed by Stokes equations. We obtain superconvergence results and asymptotically exact a posteriori error estimates by applying two recovery methods, which are the patch recovery technique and the least-squares surface fitting method. Our results are based on some regularity assumption for the Stokes control problems and are applicable to the first order conforming finite element method with regular but nonuniform partitions.     

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.     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.     

Analytic semigroups generated in L 1(Ω) by second order elliptic operators via duality methods

Given an open domain (possibly unbounded) Ω?R ⁿ , we prove that uniformly elliptic second order differential operators, under nontangential boundary conditions, generate analytic semigroups in L ¹(Ω). We use a duality method, and, further, give estimates of first order derivatives for the resolvent and the semigroup, through properties of the generator in Sobolev spaces of negative order.     

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">P</Emphasis></Superscript>

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L² spaces and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of L ^p spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator to prove that they have a bounded holomorphic functional calculus in those L ^p spaces. We also obtain functional calculus results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining L ^p results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and L ^p bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L² extends to L ^p for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L² proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces and about the relationship between conical and vertical square functions.     

First Order Differential Operators Associated to the Space of Monogenic Functions in Parameter-Depending Clifford Algebras

This paper is aimed at finding all linear first order partial differential operators \({\mathcal{F}}\) with parameter-depending Clifford-algebra-valued coefficients, that are associated to the generalized Cauchy-Riemann operator. In order to obtain the conditions on the coefficients of \({\mathcal{F}}\), a Leibniz rule for functions with values in a more general Clifford-type algebra is proved. Using the theory of associated spaces, we show the construction of solutions of initial value problems involving these operators.     

The Properties of Functions and Approximation by Summation Rational Operators on the Real Axis

In the space $C(\mathbb{R})$ , we construct special summation operators of Jackson type with rational kernels and under a suitable choice of poles we obtain order estimates for deviations of these operators from functions belonging to certain classes.     

Sobolev embedding in ℂn and the\overline \partial - equation

We prove a Sobolev embedding theorem for functions that are in a Sobolev space while their is in L^t, fort large enough. This allows us to deduceL ^p or Lipschitz estimates with loss from classical Sobolev estimates for the solution of in weakly pseudo-convex domains.     

Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

In this paper we discuss Sobolev bounds on functions that vanish at scattered points in a bounded, Lipschitz domain that satisfies a uniform interior cone condition. The Sobolev spaces involved may have fractional as well as integer order. We then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions (RBFs). These estimates include situations in which the target function does not belong to the native space of the RBF.

     

Small Ball Estimates for Quasi-Norms

This note contains two types of small ball estimates for random vectors in finite-dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability of random vectors under some smoothness assumptions on their density function. In the second part, we obtain Littlewood–Offord type estimates for quasi-norms. This generalizes results which were previously obtained in Friedland and Sodin (C R Math Acad Sci Paris 345(9):513–518, 2007), and Rudelson and Vershynin (Commun Pure Appl Math 62(12):1707–1739, 2009).