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Strömqvist, Martin
Alternative names
Publications (10 of 12) Show all publications
Strömqvist, M. (2019). Harnack's Inequality for Parabolic Nonlocal Equations. Annales de l'Institut Henri Poincare. Analyse non linéar, 36(6), 1709-1745
Open this publication in new window or tab >>Harnack's Inequality for Parabolic Nonlocal Equations
2019 (English)In: Annales de l'Institut Henri Poincare. Analyse non linéar, ISSN 0294-1449, E-ISSN 1873-1430, Vol. 36, no 6, p. 1709-1745Article in journal (Refereed) Published
Abstract [en]

The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results require the solution to be globally positive.

National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-346641 (URN)10.1016/j.anihpc.2019.03.003 (DOI)000488142500007 ()
Available from: 2018-03-20 Created: 2018-03-20 Last updated: 2019-10-25Bibliographically approved
Strömqvist, M. (2019). Local Boundedness of Solutions to Non-Local Parabolic Equations Modeled on the Fractional P−Laplacian. Journal of Differential Equations, 266(12), 7948-7959
Open this publication in new window or tab >>Local Boundedness of Solutions to Non-Local Parabolic Equations Modeled on the Fractional P−Laplacian
2019 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 266, no 12, p. 7948-7959Article in journal (Refereed) Published
Abstract [en]

We state and prove estimates for the local boundedness of subsolutions of non-local, possibly degenerate, parabolic integro-differential equations of the form

 partial derivative(t)u(x, t) + P.V integral(Rn) K(x, y, t)vertical bar u(x, t) - u(y, t)vertical bar(p-2)(u(x, t) - u(y, t)) dy = 0,

(x, t) is an element of R-n x R, where P.V. means in the principle value sense, p is an element of (1, infinity) and the kernel obeys K (x,y,t) approximate to vertical bar x-y vertical bar(n+ps) for some s is an element of (0, 1), uniformly in (x, y, t) is an element of R-n x R-n x R. 

National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-346640 (URN)10.1016/j.jde.2018.12.021 (DOI)000461785900007 ()
Available from: 2018-03-20 Created: 2018-03-20 Last updated: 2019-04-04Bibliographically approved
Castro, A. J. & Strömqvist, M. (2018). Homogenization of a parabolic Dirichlet problem by a method of Dahlberg. Publicacions matemàtiques, 62(2), 439-473
Open this publication in new window or tab >>Homogenization of a parabolic Dirichlet problem by a method of Dahlberg
2018 (English)In: Publicacions matemàtiques, ISSN 0214-1493, E-ISSN 2014-4350, Vol. 62, no 2, p. 439-473Article in journal (Refereed) Published
Abstract [en]

Consider the linear parabolic operator in divergence form: Hu := partial derivative(t)u (X, t) - div(A (X) del u (X, t)). We employ a method of Dahlberg to show that the Dirichlet problem for H in the upper half plane is well-posed for boundary data in L-p, for any elliptic matrix of coef-ficients A which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation partial derivative(t)u(epsilon) (X, t) - div(A (X/epsilon) del u(epsilon) (X, t)) in Lipschitz domains with L-p-boundary data.

National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-345204 (URN)10.5565/PUBLMAT6221805 (DOI)000435637100005 ()
Available from: 2018-03-08 Created: 2018-03-08 Last updated: 2018-09-10Bibliographically approved
Nyström, K. & Strömqvist, M. (2017). On the parabolic Lipschitz approximation of parabolic uniform rectifiable sets. Revista matemática iberoamericana, 33(4), 1397-1422
Open this publication in new window or tab >>On the parabolic Lipschitz approximation of parabolic uniform rectifiable sets
2017 (English)In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 33, no 4, p. 1397-1422Article in journal (Refereed) Published
Abstract [en]

We prove the existence of big pieces of regular parabolic Lip-schitz graphs for a class of parabolic uniform rectifiable sets satisfying what we call a synchronized two cube condition. An application to the fine properties of parabolic measure is given.

Keywords
Parabolic Lipschitz graph, parabolic uniform rectifiability, big pieces, parabolic measure, caloric measure
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-348942 (URN)10.4171/RMI/976 (DOI)000425897100012 ()
Available from: 2018-04-19 Created: 2018-04-19 Last updated: 2018-04-19Bibliographically approved
Karakhanyan, A. L. & Strömqvist, M. (2016). Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization. Calculus of Variations and Partial Differential Equations, 55(6)
Open this publication in new window or tab >>Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization
2016 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 55, no 6Article in journal (Refereed) Published
Abstract [en]

We consider the intersection of a convex surface Gamma with a periodic perforation of R-d, which looks like a sieve, given by T epsilon = boolean OR(d)(k is an element of Z) {epsilon k + a epsilon T} where T is a given compact set and a epsilon << epsilon is the size of the perforation in the epsilon-cell (0, epsilon)(d) subset of R-d. When epsilon tends to zero we establish uniform estimates for p- capacity, 1 < p < d, of the set Gamma n T-epsilon. Additionally, we prove that the intersections Gamma boolean AND {epsilon k + a(epsilon)T}(k) are uniformly distributed over Gamma and give estimates for the discrepancy of the distribution. As an application we show that the thin obstacle problem with the obstacle defined on the intersection of Gamma and the perforations, in a given bounded domain, is homogenizable when p < 1+ d/4. This result is new even for the classical Laplace operator.

National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-313544 (URN)10.1007/s00526-016-1088-2 (DOI)000390043500009 ()
Available from: 2017-02-01 Created: 2017-01-20 Last updated: 2017-11-29Bibliographically approved
Nyström, K. & Strömqvist, M. (2015). On the parabolic Lipschitz approximation of parabolic uniform rectifiable sets. Revista matemática iberoamericana
Open this publication in new window or tab >>On the parabolic Lipschitz approximation of parabolic uniform rectifiable sets
2015 (English)In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616Article in journal (Refereed) Accepted
Abstract [en]

We prove  the existence of big pieces of regular parabolic Lipschitz graphs for a class of parabolic uniform rectifiable sets satisfying what we call a synchronized in time two cube condition. An application to the fine properties of parabolic measure is given.

National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-263916 (URN)
Available from: 2015-10-04 Created: 2015-10-04 Last updated: 2017-12-01Bibliographically approved
Karakhanyan, A. & Strömqvist, M. (2014). Application of Uniform Distribution to Homogenization of a Thin Obstacle Problem with p-Laplacian. Communications in Partial Differential Equations, 39(10), 1870-1897
Open this publication in new window or tab >>Application of Uniform Distribution to Homogenization of a Thin Obstacle Problem with p-Laplacian
2014 (English)In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 39, no 10, p. 1870-1897Article in journal (Refereed) Published
Abstract [en]

In this paper we study the homogenization of p-Laplacian with thin obstacle in a perforated domain. The obstacle is defined on the intersection between a hyperplane and a periodic perforation. We construct the family of correctors for this problem and show that the solutions for the epsilon-problem converge to a solution of a minimization problem of similar form but with an extra term involving the mean capacity of the obstacle. The novelty of our approach is based on the employment of quasi-uniform convergence. As an application we obtain Poincare's inequality for perforated domains.

Keywords
Capacity, Free boundary, Homogenization, p-Laplacian, Perforated domains, Quasiuniform convergence, Thin obstacle, Uniform distributions
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-268041 (URN)10.1080/03605302.2014.895013 (DOI)000341003700004 ()2-s2.0-84906490732 (Scopus ID)
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-12-01Bibliographically approved
Strömqvist, M. (2014). Homogenization in Perforated Domains. (Doctoral dissertation). Stockholm: KTH Royal Institute of Technology
Open this publication in new window or tab >>Homogenization in Perforated Domains
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Homogenization theory is the study of the asymptotic behaviour of solutionsto partial differential equations where high frequency oscillations occur.In the case of a perforated domain the oscillations are due to variations in thedomain of the equation. The four articles that constitute this thesis are devotedto obstacle problems in perforated domains. Paper A treats an optimalcontrol problem where the objective is to control the solution to the obstacleproblem by the choice of obstacle. The optimal obstacle in the perforated domain,as well as its homogenized limit, are characterized in terms of certainauxiliary problems they solve. In papers B,C and D the authors solve homogenizationproblems in a perforated domain where the perforation is definedas the intersection between a periodic perforation and a hyper plane. Thetheory of uniform distribution is an indespensible tool in the analysis of theseproblems. Paper B treats the obstacle problem for the Laplace operator andthe authors use correctors to derive a homogenized equation. Paper D is ageneralization of paper B to the p-Laplacian. The authors employ capacitytechniques which are well adapted to the problem. In Paper C the obstaclevaries on the same scale as the perforations. In this setting the authorsemploy the theory of Gamma-convergence to prove a homogenization result.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. p. vii, 22
Series
TRITA-MAT-A ; 2014:11
National Category
Natural Sciences
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-268044 (URN)978-91-7595-213-0 (ISBN)
Public defence
2014-09-05, F3, Lindstedtsvägen 25, KTH, 13:00 (English)
Opponent
Supervisors
Note

QC 20140703

Available from: 2015-12-04 Created: 2015-12-01 Last updated: 2015-12-04Bibliographically approved
Strömqvist, M. & Koroleva, Y. (2013). Gamma-convergence of Oscillating Thin Obstacles. Eurasian Mathematical Journal, 4, 88-100
Open this publication in new window or tab >>Gamma-convergence of Oscillating Thin Obstacles
2013 (English)In: Eurasian Mathematical Journal, ISSN 2077-9879, Vol. 4, p. 88-100Article in journal (Refereed) Published
Keywords
obstacle problem, homogenization theory, Γ-convergence
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-268046 (URN)
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-12-01Bibliographically approved
Lee, K.-a., Strömqvist, M. & Yoo, M. (2013). Highly oscillating thin obstacles. Advances in Mathematics, 237, 286-315
Open this publication in new window or tab >>Highly oscillating thin obstacles
2013 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 237, p. 286-315Article in journal (Refereed) Published
Abstract [en]

The focus of this paper is on a thin obstacle problem where the obstacle is defined on the intersection between a hyper-plane Gamma in R-n and a periodic perforation T-epsilon of R-n, depending on a small parameters epsilon > 0. As epsilon -> 0, it is crucial to estimate the frequency of intersections and to determine this number locally. This is done using strong tools from uniform distribution. By employing classical estimates for the discrepancy of sequences of type {k alpha}(k=1)(infinity), alpha is an element of R, we are able to extract rather precise information about the set Gamma boolean AND T-epsilon. As epsilon -> 0, we determine the limit u of the solution u(epsilon) to the obstacle problem in the perforated domain, in terms of a limit equation it solves. We obtain the typical "strange term" behavior for the limit problem, but with a different constant taking into account the contribution of all different intersections, that we call the averaged capacity. Our result depends on the normal direction of the plane, but holds for a.e. normal on the unit sphere in R-n.

Keywords
Homogenization, Thin obstacle, Ergodicity, Discrepancy, Corrector
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-268042 (URN)10.1016/j.aim.2013.01.007 (DOI)000316512500008 ()2-s2.0-84874437735 (Scopus ID)
Available from: 2013-04-22 Created: 2015-12-01 Last updated: 2017-12-01Bibliographically approved
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