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• 1.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
A new variational characterization of Sobolev spaces2015In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 25, no 4, p. 2185-2195Article in journal (Refereed)

We obtain a new variational characterization of the Sobolev space $W_p^1(\Omega)$ (where $\Omega\subseteq\R^n$ and $p>n$). This is a generalization of a classical result of F. Riesz. We also consider some related results.

• 2.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On functions of bounded p-variation2009In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 356, no 2, p. 582-604Article in journal (Refereed)

We obtain estimates of the total p-variation (1<p<∞) and other related functionals for a periodic function fLp[0,1] in terms of its Lp-modulus of continuity ωp(f;δ). These estimates are sharp for any rate of the decay of ωp(f;δ). Moreover, the constant coefficients in them depend on parameters in an optimal way.

• 3.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On moduli of p-continuity2012In: Acta Mathematica Hungarica, ISSN 0236-5294, E-ISSN 1588-2632, Vol. 137, no 3, p. 191-213Article in journal (Refereed)

Moduli of p-continuity provide a measure of fractional smoothness of functions via p-variation. We prove sharp estimates of the modulus of p-continuty in terms of the modulus of q-continuity (1<p<q<\infty).

• 4.
Capital City Kiel, Transportation Infrastructures Department, Germany.
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013). Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013). Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013). Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
Modelling, simulation and parameter identification of active pollution reduction with photocatalytic asphalt2019In: Acta Polytechnica, ISSN 1210-2709, E-ISSN 1805-2363, Vol. 59, no 1, p. 51-58Article in journal (Refereed)

We develop and implement a numerical model to simulate the effect of photocatalytic asphalt on reducing the concentration of nitrogen monoxide (NO) due to the presence of heavy traffic in an urban environment. The contributions in this paper are threefold: we model and simulate the spread and breakdown of pollution in an urban environment, we provide a parameter estimation process that can be used to find missing parameters, and finally, we train and compare this simulation with different data sets. We analyse the results and provide an outlook on further research.

• 5.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Estimates of the total p-variation of bivariate functions2013In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 401, p. 218-231Article in journal (Refereed)

We obtain sharp estimates of the Hardy-Vitali type p-variation of a function of two variables in terms of its mixed modulus of continuity in Lp([0,1]2).

• 6.
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
Fubini-type properties of bivariate functions of bounded ρ-variation2014In: Acta Scientarum Mathematicarum, ISSN 0001-6969, Vol. 80, no 4, p. 467-475Article in journal (Refereed)

We investigate Fubini-type properties of the space of functions of bounded Hardy-Vitali-type p-variation. This leads us to consider mixed norm spaces of bivariate functions whose linear sections have bounded p-variation in the sense of Wiener

• 7.
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
Functions of Generalized Bounded Variation2013Doctoral thesis, monograph (Other academic)

This thesis is devoted to the study of different generalizations of the classical conception of a function of bounded variation.

First, we study the functions of bounded p-variation introduced by Wiener in 1924. We obtain estimates of the total p-variation (1<p<∞) and other related functionals for a periodic function f in Lp([0,1]) in terms of its Lp-modulus of continuity ω(f;δ)p. These estimates are sharp for any rate of decay of ω(f;δ)p. Moreover, the constant coefficients in them depend on parameters in an optimal way.

Inspired by these results, we consider the relationship between the Riesz type generalized variation vp,α(f) (1<p<∞, 0≤α≤1-1/p) and the modulus of p-continuity  ω1-1/p(f;δ). These functionals generate scales of spaces that connect the space of functions of bounded p-variation and the Sobolev space Wp1. We prove sharp estimates of vp,α(f) in terms of ω1-1/p(f;δ).

In the same direction, we study relations between moduli of p-continuity and q-continuity for 1<p<q<. We prove an inequality that estimates ω1-1/p(f;δ) in terms of ω1-1/q(f;δ). The inequality is sharp for any order of decay of ω1-1/q(f;δ).

Next, we study another generalization of bounded variation: the so-called bounded Λ-variation, introduced by Waterman in 1972. We investigate relations between the space ΛBV of functions of bounded Λ-variation, and classes of functions defined via integral smoothness properties. In particular, we obtain the necessary and sufficient condition for the embedding of the class Lip(α;p) into ΛBV. This solves a problem of Wang (2009).

We consider also functions of two variables. Applying our one-dimensional result, we obtain sharp estimates of the Hardy-Vitali type p-variation of a bivariate function in terms of its mixed modulus of continuity in Lp([0,1]2). Further, we investigate Fubini-type properties of the space Hp(2) of functions of bounded Hardy-Vitali p-variation. This leads us to consider the symmetric mixed norm space Vp[Vp]sym of functions of bounded iterated p-variation. For p>1, we prove that Hp(2) is not embedded into Vp[Vp]sym, and that Vp[Vp]sym is not embedded into Hp(2). In other words, Fubini-type properties completely fail in the class of functions of bounded Hardy-Vitali type p-variation for p>1.

• 8.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
On fractional smoothness of functions related to p-variation2013In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 16, no 1, p. 21-39Article in journal (Refereed)

This paper is concerned with the study of two functionals of variational type - the Riesz type generalized variation v_{p,\alpha}(f) (1<p<\infty, 0\le\alpha\le1-1/p) and the moduli of p-continuity \omega_{1-1/p}(f;d). These functionals generate scales of spaces connecting the class of functions of bounded p-variation and the Sobolev space W_p^1. Some limiting relations in these scales are proved. Sharp estimates of v_{p,\alpha}(f) in terms of \omega_{1-1/p}(f;d) are obtained.

• 9.
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
On functions of bounded Lambda-variation and integral smoothness2015In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 27, no 3, p. 1523-1538Article in journal (Refereed)

We obtain a necessary and sufficient condition for embeddings of integral Lipschitz classes Lip(alpha; p) into classes Lambda BV of functions of bounded Lambda- variation.

• 10.
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
A Priori Feedback Estimates for Multiscale Reaction-Diffusion Systems2018In: Numerical Functional Analysis and Optimization, ISSN 0163-0563, E-ISSN 1532-2467, Vol. 39, no 4, p. 413-437Article in journal (Refereed)

We study the approximation of a multiscale reaction–diusion system posed on both macroscopic and microscopic space scales. The coupling between the scales is done through micro– macro ux conditions. Our target system has a typical structure for reaction–diusion ow problems in media with distributed microstructures (also called, double porosity materials). Besides ensuring basic estimates for the convergence of two-scale semidiscrete Galerkin approximations, we provide a set of a priori feedback estimates and a local feedback error estimator that help in designing a distributed-high-errors strategy to allow for a computationally ecient zooming in and out from microscopic structures. The error control on the feedback estimates relies on two-scale-energy, regularity, and interpolation estimates as well as on a ne bookeeping of the sources responsible with the propagation of the (multiscale) approximation errors. The working technique based on a priori feedback estimates is in principle applicable to a large class of systems of PDEs with dual structure admitting strong solutions. A

• 11.
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013). Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
Well-posedness and inverse Robin estimate for a multiscale elliptic/parabolic system2018In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 97, no 1, p. 89-106Article in journal (Refereed)

We establish the well-posedness of a coupled micro–macro parabolic– elliptic system modeling the interplay between two pressures in a gas–liquid mixture close to equilibrium that is filling a porous media with distributed microstructures. Additionally, we prove a local stability estimate for the inverse micro–macro Robin problem, potentially useful in identifying quantitatively a micro–macro interfacial Robin transfer coefficient given microscopic measurements on accessible fixed interfaces. To tackle the solvability issue we use two-scale energy estimates and twoscale regularity/compactness arguments cast in the Schauder’s fixed point theorem. A number of auxiliary problems, regularity, and scaling arguments are used in ensuring the suitable Fréchet differentiability of the solution and the structure of the inverse stability estimate.

• 12.
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
University of South Carolina.
Nonlinear Nonnested Spline Approximation2017In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 45, no 2, p. 143-191Article in journal (Refereed)

Nonlinear approximation from regular piecewise polynomials (splines) supported on rings in R2 is studied. By definition, a ring is a set in R2 obtained by subtracting a compact convex set with polygonal boundary from another such a set, but without creating uncontrollably narrow elongated subregions. Nested structure of the rings is not assumed; however, uniform boundedness of the eccentricities of the underlying convex sets is required. It is also assumed that the splines have maximum smoothness. Bernstein type inequalities for this sort of splines are proved that allow us to establish sharp inverse estimates in terms of Besov spaces.

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