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• 1.
Stockholm University, Faculty of Science, Department of Mathematics.
Mergelyan's theorem with polynomials non-vanishing on unions of sets2014In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 59, no 1, p. 99-109Article in journal (Refereed)

We consider the problem of approximating a function having no zeros on the interior of a set by polynomials having no zeros on the entire set.

• 2.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Univ Jyvaskyla, Dept Math & Stat, Jyvaskyla 40014, Finland.
Umeå University. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Approximation of plurisubharmonic functions2016In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 61, no 1, p. 23-28Article in journal (Refereed)

We extend a result by Fornaaess and Wiegerinck [Ark. Mat. 1989;27:257-272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.

• 3. Avelin, Benny
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Approximation of plurisubharmonic functions2016In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 61, no 1, p. 23-28Article in journal (Refereed)

We extend a result by Fornaaess and Wiegerinck [Ark. Mat. 1989;27:257-272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.

• 4.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Correction of The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications2019In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 64, no 10, p. 1756-1757Article in journal (Refereed)

We fill in a gap in the proofs of Theorems 1.1-1.4 in The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications, to appear in Complex Var. Elliptic Equ., doi:10.1080/17476933.2017.1410799.

• 5.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Removable singularities for hardy spaces1998In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 35, no 1, p. 1-25Article in journal (Refereed)

In this paper we study removable singularities for Hardy spaces of analytic funtions on general domains. Two different definitions are given. For compact sets they turn out to be equal and moreover independent of the surrounding domain, as was proved by D. A Hejhal For non-compact sets the difference between the definitions is studied. A survey of the present knowledge is given, except for the special cases of singularities lying on curves and singularities being self-similar Cantor sets, which the author deals with in other papers. Among the results is the non-removability for Hp of sets with dimension greater than ρ. 0 < ρ < 1. Many counterexamples are provided and the Hp capacities are introduced and studied.

• 6.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications2019In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 64, no 1, p. 40-63Article in journal (Refereed)

In this paper, boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points has capacity zero) is obtained for a large class of compactifications, but also two examples when it fails are given. This study is done for complete metric spaces equipped with doubling measures supporting a p-Poincare inequality, but the results are new also in unweighted Euclidean spaces.

• 7.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
A note on B-envelope of holomorphy and B-extendable domains2008In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 53, no 4, p. 307-309Article in journal (Refereed)

Let   be a Banach Algebra on a Riemann domain X over  . We show that under certain conditions on   and X, all functions in   can be extended to functions in  where   is the  -envelope of holomorphy.

• 8.
Univ Basilicata, Italy.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Univ Liverpool, England.
The L-p-dissipativity of first order partial differential operators2018In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 63, no 7-8, p. 945-960Article in journal (Refereed)

We find necessary and sufficient conditions for the L-p-dissipatiyity of the Dirichlet problem for systems of partial differential operators of the first order with complex locally integrable coefficients. As a by product we obtain sufficient conditions for a certain class of systems of the second order.

• 9.
Linköping University, Department of Clinical and Experimental Medicine, Division of Surgery, Orthopedics and Oncology. Linköping University, Faculty of Medicine and Health Sciences.
A sufficient condition for locally open polyanalytic functions2019In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 64, no 10, p. 1733-1738Article in journal (Refereed)

We prove that a q-analytic function with only isolated critical points is locally open. The converse of the main result is false in the sense that there are q-analytic functions that are locally open but have non-discrete critical sets.

• 10.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
Washington Univ, Dept Math, St Louis, MO 63130 USA.
A note on a conjecture concerning boundary uniqueness2015In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 60, no 7, p. 945-950Article in journal (Refereed)
• 11.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
New construction techniques for minimal surfaces2019In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 64, no 9, p. 1546-1563Article in journal (Refereed)

It is pointed out that despite the nonlinearity of the underlying equations, there do exist rather general methods that allow to generate new minimal surfaces from known ones.

• 12.
Royal Inst Technol, Sweden; Inst Hautes Etud Sci, France.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
New construction techniques for minimal surfaces2019In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 64, no 9, p. 1546-1563Article in journal (Refereed)

It is pointed out that despite the nonlinearity of the underlying equations, there do exist rather general methods that allow to generate new minimal surfaces from known ones.

• 13.
Stockholm University, Faculty of Science, Department of Mathematics.
The argument cycle and the coamoeba2013In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 58, no 3, p. 373-384Article in journal (Refereed)

We investigate the coamoeba of a complex algebraic variety V???(C*) n through the study of initial forms of the defining ideal. By use of a universal Grobner basis, we prove that the closure of the coamoeba is included in the union of coamoebas corresponding to all initial ideals. We also study complete intersections V of dimension n/2 more closely to get a lower bound for the multiplicity in V of a given point ? on the n:th torus. For this purpose, we associate a certain algebraic cycle, the argument cycle, to V and ? , and study its homology. In particular, we give a method to approximate the coamoeba when n?=?2.

• 14.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Besov spaces on closed sets by means of atomic decomposition2009In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 54, no 6, p. 585-611Article in journal (Refereed)
• 15.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Markov's inequality on compact sets and Lagrange interpolation2009In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 54, no 6, p. 613-622Article in journal (Refereed)
• 16.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
St Petersburg State Univ, Russia; St Petersburg State Polytech Univ, Russia; RAS, Russia. Steklov Math Inst, Russia.
A fractal graph model of capillary type systems2018In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 63, no 7-8, p. 1044-1068Article in journal (Refereed)

We consider blood flow in a vessel with an attached capillary system. The latter is modelled with the help of a corresponding fractal graph whose edges are supplied with ordinary differential equations obtained by the dimension-reduction procedure from a three-dimensional model of blood flow in thin vessels. The Kirchhoff transmission conditions must be satisfied at each interior vertex. The geometry and physical parameters of this system are described by a finite number of scaling factors which allow the system to have self-reproducing solutions. Namely, these solutions are determined by the factors values on a certain fragment of the fractal graph and are extended to its rest part by virtue of these scaling factors. The main result is the existence and uniqueness of self-reproducing solutions, whose dependence on the scaling factors of the fractal graph is also studied. As a corollary we obtain a relation between the pressure and flux at the junction, where the capillary system is attached to the blood vessel. This relation leads to the Robin boundary condition at the junction and this condition allows us to solve the problem for the flow in the blood vessel without solving it for the attached capillary system.

• 17.
University of Liverpool, Liverpool, UK.
Department of Mathematical Sciences, University of Liverpool, UK.
Uniform asymptotics of Green's kernels in perforated domains and meso-scale approximations2012In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 57, no 2-4, p. 137-154Article in journal (Refereed)

The article is a review of the authors' results on asymptotic approximations of Green's kernels for elliptic boundary value problems in perforated domains. A new feature is the uniformity of the asymptotics with respect to the independent variables. Formal asymptotic approximations are supplied with estimates of the remainder terms. For the case when the number of perforations or inclusions becomes large, a novel method of meso-scale asymptotic approximations is discussed, and uniform asymptotic approximations of Green's kernels as well as solutions of boundary value problems in multiply perforated domains are presented. Such approximations do not require periodicity or other typical constraints attributed to homogenization approximations.

• 18.
University of Liverpool.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Brezis-Gallouet-Wainger type inequality for irregular domains2011In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 56, no 10, p. 991-1002Article in journal (Refereed)

A Brezis–Gallouet–Wainger logarithmic interpolation-embedding inequality is proved for various classes of irregular domains, in particular, for power cusps and λ-John domains.

• 19.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Department of Mathematical Analysis and Function Theory, Peoples Friendship University, Moscow. Department of Mathematical Analysis and Function Theory, Peoples Friendship University, Moscow.
Two-sided hardy-type inequalities for monotone functions2010In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 55, no 8, p. 973-989Article in journal (Refereed)

We consider Hardy-type operators on the cones of monotone functions with general positive σ-finite Borel measure. Some two-sided Hardy-type inequalities are proved for the parameter -∞ < p < ∞. It is pointed out that such equivalences, in particular, imply a new characterization of the discrete Hardy inequality for the (most difficult) case 0 < q < p ≤ 1

• 20.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Pluripolar sets and subextension in Cegrell's classes2008In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 53, no 7, p. 675-684Article in journal (Refereed)

In this article, we prove that if E is a complete pluripolar set in Ω, then E = { = −∞} for some  (Ω). Moreover, we study the subextension in Cegrell's class p .

• 21.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Analysis of free boundaries for convertible bonds, with a call feature2014In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 59, no 7, p. 912-928Article in journal (Refereed)

Convertible bonds give rise to the so-called free boundary; i.e. an unknown boundary between continuation and conversion regions of the bond. The characteristic feature of such a bond, with an extra call feature, is that the free boundary may reach all the way to the fixed boundary. Our intention in this paper is to study the behaviour of the free boundary in the vicinity of a touching point with the fixed boundary. Along the lines of our analysis, we will also produce some results on regularity of solutions (value of the bond) up to the fixed boundary. Our methods are robust and of general nature, and can be applied to fully nonlinear equations. In particular, we shall obtain uniform results for the regularity of both solutions and their free boundaries.

• 22.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Diversifications of Serrin's and related symmetry problems2012In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 57, no 6, p. 653-665Article in journal (Refereed)

If D is a bounded C-1 domain (in R-n) for which the solution to the Dirichlet problem Delta u = -1, in D, u = 0 on partial derivative D has the property that, for given constants r, l>40, and for all x is an element of partial derivative D dist(x, Gamma(l)) = r, (Gamma(l) = {u = l}), then D is necessarily a ball. We prove this, and several other related symmetry results, using various known symmetry methods. The novelty of this article lies in the problem(s) rather than in the method(s). We also present (and in some cases also prove) a variety of possible formulations, that diversifies and generalizes Serrin's and other symmetry problems.

• 23.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Harmonic balls and the two-phase Schwarz function2013In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 58, no 6, p. 837-852Article in journal (Refereed)

In this article, we introduce the concept of harmonic balls in sub-domains of (n), through a mean-value property for a subclass of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions cannot have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exist, and gives rise to the generalization of the well-known Schwarz function to the case of a two-phase Schwarz function. Our primary goal is to derive simple properties for these problems, and tease the appetites of experts working on Schwarz function and related topics. Hopefully these two concepts will provoke further study of the topic.

• 24.
Royal Institute of Technology, Sweden.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Harmonic balls and the two-phase Schwarz function2013In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 58, no 6, p. 837-852Article in journal (Refereed)

In this article, we introduce the concept of harmonic balls in sub-domains of n , through a mean-value property for a subclass of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions cannot have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exist, and gives rise to the generalization of the well-known Schwarz function to the case of a two-phase Schwarz function. Our primary goal is to derive simple properties for these problems, and tease the appetites of experts working on Schwarz function and related topics. Hopefully these two concepts will provoke further study of the topic.

• 25.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Mother bodies of algebraic domains in the Complex plane2006In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 51, no 4, p. 357-369Article in journal (Refereed)

We give a definition of a mother body of a domain in the complex plane, and prove some continuityproperties of its potential in terms of the Schwarz function (which is explicitly assumedto exist). We end the article by studying the case of the ellipse, and use the previous results toprove existence and uniqueness of a mother body in this case, as well as a related existence resultabout graviequivalent measures for the ellipse.

• 26.
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
Sign-changing and symmetry-breaking solutions to singular problems2012In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 57, no 11, p. 1191-1208Article in journal (Refereed)

We consider the degenerate elliptic equation -div(vertical bar x vertical bar(-ap) vertical bar del(u)vertical bar(p-2) del(u)) - lambda vertical bar x vertical bar(-p(a+1))vertical bar u vertical bar(p-2)u = vertical bar x vertical bar(-bq) vertical bar u vertical bar(q-2)u in R-N related to the Caffarelli-Kohn-Nirenberg inequality. We show that it possesses infinitely many solutions which are sign-changing and nonradial. The solutions are obtained by constrained minimization on subspaces consisting of functions which have certain prescribed symmetry properties. We also extend these results to higher order equations.

• 27.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Riesz Potential Equations in Local Lp-spaces.2009In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 54, no 2, p. 125-151Article in journal (Refereed)

We consider the following equation for the Riesz potential of order one:

$\int_{\mathbf{R}^N} \frac{u(y) \, dy}{|x-y|^{N-1}} = f(x) \text{,} \qquad x \in \mathbf{R}^N \text{.}$

Uniqueness of solutions is proved in the class of solutions for which the integral is absolutely convergent for almost every x. We also prove anexistence result and derive an asymptotic formula for solutions near the origin.Our analysis is carried out in local Lp-spaces and Sobolev spaces, which allows us to obtain optimal results concerning the class of right-hand sides and solutions. We also apply our results to weighted Lp-spaces and homogenous Sobolev spaces.

• 28.
Linnaeus University, Faculty of Technology, Department of Mathematics.
Paley-Wiener properties for spaces of power series expansions2019In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941Article in journal (Refereed)

We extend Paley-Wiener results in the Bargmann setting deduced in Nabizadeh et al. [Paley-Wiener properties for spaces of entire functions, (preprint), arXiv:1806.10752.] to larger classes of power series expansions. At the same time, we deduce characterizations of all Pilipovic spaces and their distributions (and not only of low orders as in Nabizadeh et al. [Paley-Wiener properties for spaces of entire functions, (preprint), arXiv:1806.10752.]).

• 29.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the Blocki-Zwonek conjectures2015In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 60, no 9, p. 1270-1276Article in journal (Refereed)

Let Omega be a bounded pseudoconvex domain in C-n, and let g(Omega) (z, a) be the pluricomplex Green function with pole at a in Omega. It was conjectured by Blocki and Zwonek that the function given by beta = beta(Omega),(a) : (-infinity, 0) (sic) t -> beta(t) = log (lambda(n)({z is an element of Omega g(Omega) (z, a) < t})) is convex. Here.n is the Lebesgue measure in Cn. In this note we give an affirmative answer to this conjecture when Omega is biholomorphic to the unit ball or to the polydisc in C-n, n >= 1.

• 30.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Extension and approximation of m-subharmonic functions2018In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 63, no 6, p. 783-801Article in journal (Refereed)

Let  be a bounded domain, and let f be a real-valued function defined on the whole topological boundary . The aim of this paper is to find a characterization of the functions f which can be extended to the inside to a m-subharmonic function under suitable assumptions on . We shall do so using a function algebraic approach with focus on m-subharmonic functions defined on compact sets. We end this note with some remarks on approximation of m-subharmonic functions.

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