| 1. |
- Aikawa, Hiroaki, 1956, et al.
(författare)
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Martin boundary of a fractal domain
- 2003
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 18:4, s. 311-357
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Tidskriftsartikel (refereegranskat)abstract
- A uniformly John domain is a domain intermediate between a John domain and a uniform domain. We determine the Martin boundary of a uniformly John domain D as an application of a boundary Harnack principle. We show that a certain self-similar fractal has its complement as a uniformly John domain. In particular, the complement of the 3-dimensional Sierpinacuteski gasket is a uniform domain and its Martin boundary is homeomorphic to the Sierpinacuteski gasket itself.
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| 2. |
- Aleksanyan, Hayk, et al.
(författare)
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Perturbed Divisible Sandpiles and Quadrature Surfaces
- 2019
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Ingår i: Potential Analysis. - : SPRINGER. - 0926-2601 .- 1572-929X. ; 51:4, s. 511-540
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Tidskriftsartikel (refereegranskat)abstract
- The main purpose of the present paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). For this aim, we introduce a new model of Laplacian growth on the lattice DOUBLE-STRUCK CAPITAL Zd (d >= 2) which continuously deforms occupied regions of the divisible sandpile model of Levine and Peres (J. Anal. Math. 111(1), 151-219 2010), by redistributing the total mass of the system onto 1/m-sub-level sets of the odometer which is a function counting total emissions of mass from lattice vertices. In free boundary terminology this goes in parallel with singular perturbation, which is known to converge to a Bernoulli type free boundary. We prove that models, generated from a single source, have a scaling limit, if the threshold m is fixed. Moreover, this limit is a ball, and the entire mass of the system is being redistributed onto an annular ring of thickness 1/m. By compactness argument we show that when m tends to infinity sufficiently slowly with respect to the scale of the model, then in this case also there is scaling limit which is a ball, with the mass of the system being uniformly distributed onto the boundary of that ball, and hence we recover a quadrature surface in this case. Depending on the speed of decay of 1/m, the visited set of the sandpile interpolates between spherical and polygonal shapes. Finding a precise characterisation of this shape-transition phenomenon seems to be a considerable challenge, which we cannot address at this moment.
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| 3. |
- Ameur, Yacin, et al.
(författare)
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The Random Normal Matrix Model : Insertion of a Point Charge
- 2021
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 1572-929X .- 0926-2601. ; 2021
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Tidskriftsartikel (refereegranskat)abstract
- In this article, we study microscopic properties of a two-dimensional Coulomb gas ensemble near a conical singularity arising from insertion of a point charge in the bulk of the droplet. In the determinantal case, we characterize all rotationally symmetric scaling limits (“Mittag-Leffler fields”) and obtain universality of them when the underlying potential is algebraic. Applications include a central limit theorem for log|pn(ζ)| where pn is the characteristic polynomial of an n:th order random normal matrix.
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| 4. |
- Andreev, R., et al.
(författare)
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Kolmogorov-Chentsov Theorem and Differentiability of Random Fields on Manifolds
- 2014
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 41:3, s. 761-769
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Tidskriftsartikel (refereegranskat)abstract
- A version of the Kolmogorov-Chentsov theorem on sample differentiability and Hölder continuity of random fields on domains of cone type is proved, and the result is generalized to manifolds.
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| 5. |
- Arakelyan, A., et al.
(författare)
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Multi-Phase Quadrature Domains and a Related Minimization Problem
- 2016
-
Ingår i: Potential Analysis. - : Springer Netherlands. - 0926-2601 .- 1572-929X. ; , s. 1-21
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we introduce the multi-phase version of the so-called Quadrature Domains (QD), which refers to a generalized type of mean value property for harmonic functions. The well-established and developed theory of one-phase QD was recently generalized to a two-phase version, by one of the current authors (in collaboration). Here we introduce the concept of the multi-phase version of the problem, and prove existence as well as several properties of such solutions. In particular, we discuss possibilities of multi-junction points.
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| 6. |
- Avelin, Benny, et al.
(författare)
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A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity
- 2015
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Ingår i: Potential Analysis. - : Håkan Persson. - 0926-2601 .- 1572-929X. ; 43:3, s. 531-545
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Tidskriftsartikel (refereegranskat)abstract
- We show that bounded pseudoconvex domains that are Hölder continuous for all α < 1 are hyperconvex, extending the well-known result by Demailly (Math. Z. 184 1987) beyond Lipschitz regularity.
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| 7. |
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| 8. |
- Björn, Anders, et al.
(författare)
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Quasiopen and p-Path Open Sets, and Characterizations of Quasicontinuity
- 2017
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Ingår i: Potential Analysis. - : SPRINGER. - 0926-2601 .- 1572-929X. ; 46:1, s. 181-199
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we give various characterizations of quasiopen sets and quasicontinuous functions on metric spaces. For complete metric spaces equipped with a doubling measure supporting a p-Poincar, inequality we show that quasiopen and p-path open sets coincide. Under the same assumptions we show that all Newton-Sobolev functions on quasiopen sets are quasicontinuous.
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| 9. |
- Björn, Anders, 1966-, et al.
(författare)
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The Dirichlet Problem for p-minimizers on Finely Open Sets in Metric Spaces
- 2023
-
Ingår i: Potential Analysis. - : Springer. - 0926-2601 .- 1572-929X. ; 59, s. 1117-1140
-
Tidskriftsartikel (refereegranskat)abstract
- We initiate the study of fine p-(super)minimizers, associated with p-harmonic functions, on finely open sets in metric spaces, where infinity. After having developed their basic theory, we obtain the p-fine continuity of the solution of the Dirichlet problem on a finely open set with continuous Sobolev boundary values, as a by-product of similar pointwise results. These results are new also on unweighted . We build this theory in a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality.
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| 10. |
- Björn, Anders, et al.
(författare)
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The Variational Capacity with Respect to Nonopen Sets in Metric Spaces
- 2014
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Ingår i: Potential Analysis. - : Springer Verlag (Germany). - 0926-2601 .- 1572-929X. ; 40:1, s. 57-80
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Tidskriftsartikel (refereegranskat)abstract
- We pursue a systematic treatment of the variational capacity on metric spaces and give full proofs of its basic properties. A novelty is that we study it with respect to nonopen sets, which is important for Dirichlet and obstacle problems on nonopen sets, with applications in fine potential theory. Under standard assumptions on the underlying metric space, we show that the variational capacity is a Choquet capacity and we provide several equivalent definitions for it. On open sets in weighted R (n) it is shown to coincide with the usual variational capacity considered in the literature. Since some desirable properties fail on general nonopen sets, we introduce a related capacity which turns out to be a Choquet capacity in general metric spaces and for many sets coincides with the variational capacity. We provide examples demonstrating various properties of both capacities and counterexamples for when they fail. Finally, we discuss how a change of the underlying metric space influences the variational capacity and its minimizing functions.
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| 11. |
- Björn, Anders
(författare)
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Weak Barriers in Nonlinear Potential Theory
- 2007
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 27:4, s. 381-387
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Tidskriftsartikel (refereegranskat)abstract
- We characterize regular boundary points for p-harmonic functions using weak barriers. We use this to obtain some consequences on boundary regularity. The results also hold for A-harmonic functions under the usual assumptions on A, and for Cheeger p-harmonic functions in metric spaces.
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| 12. |
- Björn, Jana, et al.
(författare)
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Capacitary estimates for solutions of the Dirichlet problem for second order elliptic equations in divergence form
- 2000
-
Ingår i: Potential Analysis. - 0926-2601 .- 1572-929X. ; 12:1, s. 81-113
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Tidskriftsartikel (refereegranskat)abstract
- We consider the Dirichlet problem for A-harmonic functions, i.e. the solutions of the uniformly elliptic equation div(A(x)del u(x)) = 0 in an n-dimensional domain Omega, n greater than or equal to 3. The matrix A is assumed to have bounded measurable entries. We obtain pointwise estimates for the A-harmonic functions near a boundary point. The estimates are in terms of the Wiener capacity and the so called capacitary interior diameter. They imply pointwise estimates for the A-harmonic measure of the domain Omega, which in turn lead to a sufficient condition for the Holder continuity of A-harmonic functions at a boundary point. The behaviour of A-harmonic functions at infinity and near a singular point is also studied and theorems of Phragmen-Lindelof type, in which the geometry of the boundary is taken into account, are proved. We also obtain pointwise estimates for the Green function for the operator -div(A(.)del u(.)) in a domain Omega and for the solutions of the nonhomogeneous equation -div(A(x)del u(x)) = mu with measure on the right-hand side.
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| 13. |
- Bui, T. A., et al.
(författare)
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Weighted Bounds for Multilinear Square Functions
- 2017
-
Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 46:1, s. 135-148
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Tidskriftsartikel (refereegranskat)abstract
- Let with 1 < p (1), aEuro broken vertical bar, p (m) < a, 1/p (1)+a+1/p (m) =1/p and . In this paper, we investigate the weighted bounds with dependence on aperture alpha for multilinear square functions . We show that. This result extends the result in the linear case which was obtained by Lerner in 2014. Our proof is based on the local mean oscillation technique presented firstly to find the weighted bounds for Caldern-Zygmund operators. This method helps us avoiding intrinsic square functions in the proof of our main result.
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| 14. |
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| 15. |
- Christiansen, Jacob Stordal, et al.
(författare)
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Julia sets of orthogonal polynomials
- 2019
-
Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 1572-929X .- 0926-2601. ; 50:3, s. 401-413
-
Tidskriftsartikel (refereegranskat)abstract
- For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials {P n } to properties of the support. More precisely we relate the Julia set of P n to the outer boundary of the support, the filled Julia set to the polynomial convex hull K of the support, and the Green’s function associated with P n to the Green’s function for the complement of K.
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| 16. |
- Cinti, Chiara, et al.
(författare)
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A Note on Harnack Inequalities and Propagation Sets for a Class of Hypoelliptic Operators
- 2010
-
Ingår i: Potential Analysis. - : Springer Netherlands. - 0926-2601 .- 1572-929X. ; 33:4, s. 341-354
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we are concerned with Harnack inequalities for non-negative solutions u:Ω→ℝ to a class of second order hypoelliptic ultraparabolic partial differential equations in the form Lu:=∑j=1mX2ju+X0u−∂tu=0 where Ω is any open subset of ℝN + 1, and the vector fields X1, ..., Xm and X0−∂t are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed (x0, t0) ∈ Ω we give a geometric sufficient condition on the compact sets K⊆Ω for which the Harnack inequality supK u≤CK u(x0, t0) holds for all non-negative solutions u to the equation Lu=0. We also compare our result with an abstract Harnack inequality from potential theory.
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| 17. |
- Dimitrov, Dimitar K., et al.
(författare)
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Electrostatic Problems with a Rational Constraint and Degenerate Lame Equations
- 2020
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 52:4, s. 645-659
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Tidskriftsartikel (refereegranskat)abstract
- In this note we extend the classical relation between the equilibrium configurations of unit movable point charges in a plane electrostatic field created by these charges together with some fixed point charges and the polynomial solutions of a corresponding Lame differential equation. Namely, we find similar relation between the equilibrium configurations of unit movable charges subject to a certain type of rational or polynomial constraint and polynomial solutions of a corresponding degenerate Lame equation, see details below. In particular, the standard linear differential equations satisfied by the classical Hermite and Laguerre polynomials belong to this class. Besides these two classical cases, we present a number of other examples including some relativistic orthogonal polynomials and linear differential equations satisfied by those.
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| 18. |
- Dunkels, Andrejs
(författare)
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A note on the equilibrium potential of certain Dirichlet spaces
- 1997
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Ingår i: Potential Analysis. - 0926-2601 .- 1572-929X. ; 6:1, s. 99-104
-
Tidskriftsartikel (refereegranskat)abstract
- There exists an equilibrium potential, an element of a Dirichlet space, for any compact subset, with non-emtpy interior, of Rm. This potential is constant on the interior of the set. There is a corresponding measure that is 0 outside the set. We prove that the restriction of this equilibrium measure to the interior of the set is absolutely continuous, and we derive an explicit formula for its density.
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| 19. |
- Emamizadeh, Behrouz, et al.
(författare)
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A Two Phase Free Boundary Problem Related to Quadrature Domains
- 2011
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 34:2, s. 119-138
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we introduce a two phase version of the well-known Quadrature Domain theory, which is a generalized (sub)mean value property for (sub)harmonic functions. In concrete terms, and after reformulation into its PDE version the problem boils down to finding solution to -Delta u=(mu(+)-lambda(+))(chi{u>0}) - (mu(-) - lambda(-))(chi{u<0}) in IRN. where lambda(+/-) >0 are given constants and mu(+/-), are non-negative bounded Radon measures, with compact support. Our primary concern is to discuss existence and geometric properties of solutions.
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| 20. |
- Enstedt, M, et al.
(författare)
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Weighted spectral gap for magnetic Schrödinger operators with a potential term
- 2009
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 31:3, s. 215-226
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Tidskriftsartikel (refereegranskat)abstract
- We prove that singular Schrodinger equations with external magnetic field admit a representation with a positive Lagrangian density whenever their "nonmagnetic" counterpart is nonnegative. In this case the operator has a weighted spectral gap as long as the strength of the magnetic field is not identically zero. We provide estimates of the weight in the spectral gap, including the versions with L (p) -norm and with a magnetic gradient term, and applications to an increase of the best Hardy constant due to the presence of a magnetic field. The paper also shows existence of the ground state for the nonlinear magnetic Schrodinger equation with the periodic magnetic field.
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| 21. |
- Hakobyan, A, et al.
(författare)
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A uniqueness result for an overdetermined problem in non-linear parabolic potential theory
- 2004
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Ingår i: Potential Analysis. - 0926-2601 .- 1572-929X. ; 21:4, s. 405-414
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we study the question of uniqueness for an inverse problem, arising in the (thermal) linear and/or non-linear potential theory. The overdetermined problem we shall study is represented by (div(\delu\(p-2)delu) - D(t)u - chi(Omega) + mu)u = 0, where supp(mu) subset of Omega subset of R-n x (0,infinity), 1 < p < infinity, mu is an element of L-infinity, and Omega boolean AND {t = tau} is bounded for tau > 0. The problem has applications in shape-recognition in underground water/oil recovery, subject to shape-change during time intervals. The particular case u greater than or equal to 0, D(t)u greater than or equal to 0, and p = 2, is an example of the well-known Stefan.
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| 22. |
- Jonsson, Alf
(författare)
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Dirichlet forms and Brownian motion penetrating fractals
- 2000
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Ingår i: Potential Analysis. - Dordrecht : Kluwer Academic Publishers. - 0926-2601 .- 1572-929X. ; 13:1, s. 69-80
-
Tidskriftsartikel (refereegranskat)abstract
- Can a Brownian motion penetrate the two-dimensional Sierpinski gasket? This question was studied in [8], and an affirmative answer was given. In this paper, the problem is studied with a different approach, using Dirichlet forms and function space theory. The results obtained are somewhat different from, and from certain aspects more general than, the results in [8].
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| 23. |
- Kow, P. Z., et al.
(författare)
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Quadrature Domains for the Helmholtz Equation with Applications to Non-scattering Phenomena
- 2023
-
Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 60, s. 387-424
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps.
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| 24. |
- Kozlov, Vladimir, et al.
(författare)
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A Comparison Theorem for Nonsmooth Nonlinear Operators
- 2021
-
Ingår i: Potential Analysis. - : SPRINGER. - 0926-2601 .- 1572-929X. ; 54:3, s. 471-481
-
Tidskriftsartikel (refereegranskat)abstract
- We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearityfisL(p)function withp> 1. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class. An application to estimation of periodic water waves profiles is given.
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| 25. |
- Kozlov, Vladimir, 1954-, et al.
(författare)
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Heat content asymptotics for Riemannian manifolds with Zaremba boundary conditions
- 2007
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 26:3, s. 225-254
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Tidskriftsartikel (refereegranskat)abstract
- The existence of a full asymptotic expansion for the heat content asymptotics of an operator of Laplace type with classical Zaremba boundary conditions on a smooth manifold is established. The first three coefficients in this asymptotic expansion are determined in terms of geometric invariants, partial information is obtained about the fourth coefficient.
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| 26. |
- Li, Hong Quan, et al.
(författare)
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Weak Type (1,1) Bounds for Some Operators Related to the Laplacian with Drift on Real Hyperbolic Spaces
- 2017
-
Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 46:3, s. 463-484
-
Tidskriftsartikel (refereegranskat)abstract
- © 2016, The Author(s).The setting of this work is the n-dimensional hyperbolic space R+× Rn-1, where the Laplacian is given a drift in the R+ direction. We consider the operators defined by the horizontal Littlewood-Paley-Stein functions for the heat semigroup and the Poisson semigroup, and also the Riesz transforms of order 1 and 2. These operators are known to be bounded on Lp,1
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| 27. |
- Lindgren, Erik, 1980-, et al.
(författare)
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Perron's Method and Wiener's Theorem for a Nonlocal Equation
- 2017
-
Ingår i: Potential Analysis. - : SPRINGER. - 0926-2601 .- 1572-929X. ; 46:4, s. 705-737
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Tidskriftsartikel (refereegranskat)abstract
- We study the Dirichlet problem for non-homogeneous equations involving the fractional p-Laplacian. We apply Perron's method and prove Wiener's resolutivity theorem.
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| 28. |
- Lundström, Niklas L P, 1980-
(författare)
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Phragmén-Lindelöf theorems and p-harmonic measures for sets near low-dimensional hyperplanes
- 2016
-
Ingår i: Potential Analysis. - : Springer Netherlands. - 0926-2601 .- 1572-929X. ; 44:2, s. 313-330
-
Tidskriftsartikel (refereegranskat)abstract
- We prove estimates of a p-harmonic measure, p∈(n−m,∞], for sets in R n which are close to an m-dimensional hyperplane Λ⊂R n , m∈[0,n−1]. Using these estimates, we derive results of Phragmén-Lindelöf type in unbounded domains Ω⊂R n ∖Λ for p-subharmonic functions. Moreover, we give local and global growth estimates for p-harmonic functions, vanishing on sets in R n , which are close to an m-dimensional hyperplane.
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| 29. |
- Lundström, Niklas L. P., 1980-, et al.
(författare)
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Strong maximum principle and boundary estimates for nonhomogeneous elliptic equations
- 2024
-
Ingår i: Potential Analysis. - : Springer. - 0926-2601 .- 1572-929X. ; 60, s. 425-443
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Tidskriftsartikel (refereegranskat)abstract
- We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear nonhomogeneous degenerate elliptic equations on the formF(x, u, Du, D2u) = 0 under suitable assumptions allowing for non-Lipschitz growth in the gradient term. In case of smooth boundaries, we also prove a Hopf lemma, a boundary Harnack inequality, and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply, e.g., to weak solutions of an eigenvalue problem for the variable exponent p-Laplacian.
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| 30. |
- Luo, Gou, et al.
(författare)
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Weighted positivity of second order elliptic systems
- 2007
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Ingår i: Potential Analysis. - : Springer Netherlands. - 0926-2601 .- 1572-929X. ; 27:3, s. 251-270
-
Tidskriftsartikel (refereegranskat)abstract
- Integral inequalities that concern the weighted positivity of a differential operator have important applications in qualitative theory of elliptic boundary value problems. Despite the power of these inequalities, however, it is far from clear which operators have this property. In this paper, we study weighted integral inequalities for general second order elliptic systems in ℝ n (n ≥ 3) and prove that, with a weight, smooth and positive homogeneous of order 2–n, the system is weighted positive only if the weight is the fundamental matrix of the system, possibly multiplied by a semi-positive definite constant matrix.
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| 31. |
- Luo, Guo, et al.
(författare)
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Wiener Type Regularity of a Boundary Point for the 3D Lam, System
- 2010
-
Ingår i: POTENTIAL ANALYSIS. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 32:2, s. 133-151
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we study the 3D Lam, system and establish its weighted positive definiteness for a certain range of elastic constants. By modifying the general theory developed in Mazya (J Duke Math 115(3): 479-512, 2002), we then show, under the assumption of weighted positive definiteness, that the divergence of the classical Wiener integral for a boundary point guarantees the continuity of solutions to the Lam, system at this point.
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| 32. |
- Löbus, Jörg-Uwe
(författare)
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Mosco Type Convergence of Bilinear Forms and Weak Convergence of n-Particle Systems
- 2015
-
Ingår i: Potential Analysis. - : Springer Verlag (Germany). - 0926-2601 .- 1572-929X. ; 43:2, s. 241-267
-
Tidskriftsartikel (refereegranskat)abstract
- It is well known that Mosco (type) convergence is a tool in order to verify weak convergence of finite dimensional distributions of sequences of stochastic processes. In the present paper we are concerned with the concept of Mosco type convergence for non-symmetric stochastic processes and, in particular, n-particle systems in order to establish relative compactness.
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| 33. |
- Nyström, Kaj, 1969-
(författare)
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An estimate of a polynomial capacity
- 1998
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Ingår i: Potential Analysis. - 0926-2601 .- 1572-929X. ; 9:3, s. 217-227
-
Tidskriftsartikel (refereegranskat)
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| 34. |
- Peletier, M., et al.
(författare)
-
Large Deviations and Gradient Flows for the Brownian One-Dimensional Hard-Rod System
- 2021
-
Ingår i: Potential Analysis. - : Springer Nature. - 0926-2601 .- 1572-929X.
-
Tidskriftsartikel (refereegranskat)abstract
- We study a system of hard rods of finite size in one space dimension, which move by Brownian noise while avoiding overlap. We consider a scaling in which the number of particles tends to infinity while the volume fraction of the rods remains constant; in this limit the empirical measure of the rod positions converges almost surely to a deterministic limit evolution. We prove a large-deviation principle on path space for the empirical measure, by exploiting a one-to-one mapping between the hard-rod system and a system of non-interacting particles on a contracted domain. The large-deviation principle naturally identifies a gradient-flow structure for the limit evolution, with clear interpretations for both the driving functional (an ‘entropy’) and the dissipation, which in this case is the Wasserstein dissipation. This study is inspired by recent developments in the continuum modelling of multiple-species interacting particle systems with finite-size effects; for such systems many different modelling choices appear in the literature, raising the question how one can understand such choices in terms of more microscopic models. The results of this paper give a clear answer to this question, albeit for the simpler one-dimensional hard-rod system. For this specific system this result provides a clear understanding of the value and interpretation of different modelling choices, while giving hints for more general systems.
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| 35. |
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| 36. |
- Pettersson, R, et al.
(författare)
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Numerical approximation for a white noise driven SPDE with locally bounded drift
- 2005
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 1572-929X .- 0926-2601. ; 22:4, s. 375-393
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Tidskriftsartikel (refereegranskat)abstract
- A numerical scheme for a stochastic partial differential equation of heat equation type is considered where the drift is locally bounded and the dispersion may be state dependent. Uniform convergence in probability is obtained.
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| 37. |
- Sjödin, Tord, 1944-
(författare)
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Continuous functions and Riesz type potentials in homogeneous spaces
- 2015
-
Ingår i: Potential Analysis. - New York : Springer-Verlag New York. - 0926-2601 .- 1572-929X. ; 43:3, s. 495-511
-
Tidskriftsartikel (refereegranskat)abstract
- We develop a potential theory for a Riesz type kernel in a homogeneous space and characterize the compact sets K with capacity zero as the sets K for which every continous function f on K is the restriction to K of a continuous potential Uσfk of an absolutely continuous measure σ f supported in an arbitrarily small neighbourhood of K. The measure σ f can be choosen as a suitable restriction of a single measure σ that only depends on the set K and the kernel k.
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| 38. |
- Skrzypczak, Leszek, et al.
(författare)
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On Caffarelli-Kohn-Nirenberg Inequalities for Block-Radial Functions
- 2016
-
Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 45:1, s. 65-81
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Tidskriftsartikel (refereegranskat)abstract
- The paper provides weighted Sobolev inequalities of the Caffarelli-Kohn-Nirenberg type for functions with multi-radial symmetry. An elementary example of such inequality is the following inequality of Hardy type for functions u = u(r(1)(x), r(2)(x)), where r(1)(x) = root x(1)(2) + x(2)(2) and r(2)(x) = root x(3)(2) + x(4)(2) from the subspace (H) over dot((2,2))(1,3) (R-4) of the Sobolev space (H) over dot(1,3) (R-4), radially symmetric in variables (x(1), x(2)) and in variables (x(3), x(4)): integral(R4) u(2)/r(1)(x)r(2)(x) dx <= C integral(R4) vertical bar del u vertical bar(2)dx, Similarly to the previously studied radial case, the range of parameters in CKN inequalities can be extended, sometimes to infinity, providing a pointwise estimate similar to the radial estimate in Strauss (Comm. Math. Phys. 55, 149-162 1977). Furthermore, the "multi-radial" weights are a stronger singularity than radial weights of the same homogeneity, e.g. 1/r(1)(x)r(2)(x) >= 1/2 vertical bar x vertical bar(2).
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| 39. |
- Thim, Johan
(författare)
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Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces with Small Lipschitz Constant
- 2015
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Ingår i: Potential Analysis. - : Springer Verlag (Germany). - 0926-2601 .- 1572-929X. ; 43:1, s. 79-95
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Tidskriftsartikel (refereegranskat)abstract
- This article considers two weight estimates for the single layer potential - corresponding to the Laplace operator in R (N+1) - on Lipschitz surfaces with small Lipschitz constant. We present conditions on the weights to obtain solvability and uniqueness results in weighted Lebesgue spaces and weighted homogeneous Sobolev spaces, where the weights are assumed to be radial and doubling. In the case when the weights are additionally assumed to be differentiable almost everywhere, simplified conditions in terms of the logarithmic derivative are presented, and as an application, we prove that the operator corresponding to the single layer potential in question is an isomorphism between certain weighted spaces of the type mentioned above. Furthermore, we consider several explicit weight functions. In particular, we present results for power exponential weights which generalize known results for the case when the single layer potential is reduced to a Riesz potential, which is the case when the Lipschitz surface is given by a hyperplane.
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| 40. |
- Gudmundsson, Sigmundur, et al.
(författare)
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A note on the classification of holomorphic harmonic morphisms
- 1993
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Ingår i: Potential Analysis. - 1572-929X. ; 2:3, s. 295-298
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Tidskriftsartikel (refereegranskat)abstract
- In this note we give a complete classification of those holomorphic maps phgr:UrarrCopf n defined on open and connected subsets of Copf m which are harmonic morphisms.
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