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Publications (10 of 17) Show all publications
Avelin, B., Kuusi, T. & Nyström, K. (2019). Boundary behavior of solutions to the parabolic p-Laplace equation. Analysis & PDE, 12(1), 1-42
Open this publication in new window or tab >>Boundary behavior of solutions to the parabolic p-Laplace equation
2019 (English)In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 12, no 1, p. 1-42Article in journal (Refereed) Published
Abstract [en]

We establish boundary estimates for non-negative solutions to the $p$-parabolic equation in the degenerate range $p>2$. Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA-domains together with sharp boundary decay estimates. If the underlying domain is $C^{1,1}$-regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and Hölder continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waiting time phenomena present which plays a fundamental role throughout the paper. In particular, conditions on these waiting times rule out well-known examples of explicit solutions violating the boundary Harnack principle.

National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-265500 (URN)10.2140/apde.2019.12.1 (DOI)000446595400001 ()
Funder
Swedish Research Council, 637-2014-6822
Available from: 2015-10-30 Created: 2015-10-30 Last updated: 2018-11-28Bibliographically approved
Avelin, B., Kuusi, T. & Mingione, G. (2018). Nonlinear Caldern-Zygmund Theory in the Limiting Case. Archive for Rational Mechanics and Analysis, 227(2), 663-714
Open this publication in new window or tab >>Nonlinear Caldern-Zygmund Theory in the Limiting Case
2018 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 227, no 2, p. 663-714Article in journal (Refereed) Published
Abstract [en]

We prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems. Specifically, we deal with the limiting case of the classical theory of Caldern and Zygmund in the setting of nonlinear, possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions. A prototype of the results obtained here tells for instance that ifwith being a Borel measure with locally finite mass on the open subset and , thenThe case is obviously forbidden already in the classical linear case of the Poisson equation.

National Category
Mathematics Mechanical Engineering
Identifiers
urn:nbn:se:uu:diva-341484 (URN)10.1007/s00205-017-1171-7 (DOI)000419122300006 ()
Available from: 2018-02-28 Created: 2018-02-28 Last updated: 2018-02-28Bibliographically approved
Avelin, B. & Julin, V. (2017). A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term. Journal of Functional Analysis, 272(8), 3176-3215
Open this publication in new window or tab >>A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term
2017 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 272, no 8, p. 3176-3215Article in journal (Refereed) Published
Abstract [en]

This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in [26], to prove a generalized Carleson estimate. We also prove boundary Holder continuity and a boundary Harnack type inequality.

Keywords
Generalized Carleson estimate, Non-Lipschitz drift, Elliptic PDE
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-321179 (URN)10.1016/j.jfa.2016.12.026 (DOI)000396380600002 ()
Funder
Swedish Research Council, 637-2014-6822
Available from: 2017-05-02 Created: 2017-05-02 Last updated: 2017-05-02
Avelin, B. & Lukkari, T. (2017). A comparison principle for the porous medium equation and its consequences. Revista matemática iberoamericana, 33(2), 573-594
Open this publication in new window or tab >>A comparison principle for the porous medium equation and its consequences
2017 (English)In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 33, no 2, p. 573-594Article in journal (Refereed) Published
Abstract [en]

We prove a comparison principle for the porous medium equation in more general open sets in Rn+1 than space-time cylinders. We apply this result in two related contexts: we establish a connection between a potential theoretic notion of the obstacle problem and a notion based on a variational inequality. We also prove the basic properties of the PME capacity, in particular that there exists a capacitary extremal which gives the capacity for compact sets.

Keywords
Porous medium equation, PME, elliptic comparison principle, obstacle problem, parabolic capacity
National Category
Mathematical Analysis Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-329153 (URN)10.4171/RMI/950 (DOI)000403582900009 ()
Funder
Swedish Research Council, 637-2014-6822
Available from: 2017-09-15 Created: 2017-09-15 Last updated: 2017-09-15Bibliographically approved
Avelin, B. & Saari, O. (2017). Characterizations of interior polar sets for the degenerate p-parabolic equation. Journal of evolution equations (Printed ed.), 17(2), 827-848
Open this publication in new window or tab >>Characterizations of interior polar sets for the degenerate p-parabolic equation
2017 (English)In: Journal of evolution equations (Printed ed.), ISSN 1424-3199, E-ISSN 1424-3202, Vol. 17, no 2, p. 827-848Article in journal (Refereed) Published
Abstract [en]

This paper deals with different characterizations of sets of nonlinear parabolic capacity zero, with respect to the parabolic p-Laplace equation. Specifically we prove that certain interior polar sets can be characterized by sets of zero nonlinear parabolic capacity. Furthermore we prove that zero capacity sets are removable for bounded supersolutions and that sets of zero capacity have a relation to a certain parabolic Hausdorff measure.

Keywords
Parabolic capacity, Degenerate parabolic equations, Nonlinear potential theory, P-parabolic equation, P-Laplace, Parabolic Hausdorff measure, Interior polar sets, Removability, Characterization
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-329699 (URN)10.1007/s00028-016-0339-1 (DOI)000404158100009 ()
Funder
Swedish Research Council, 637-2014-6822
Available from: 2017-10-09 Created: 2017-10-09 Last updated: 2017-10-09Bibliographically approved
Avelin, B., Hed, L. & Persson, H. (2016). Approximation of plurisubharmonic functions. Complex Variables and Elliptic Equations, 61(1), 23-28
Open this publication in new window or tab >>Approximation of plurisubharmonic functions
2016 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 61, no 1, p. 23-28Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
plurisubharmonic functions, approximation, continuous boundary, boundary regularity, Mergelyan type approximation
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-251324 (URN)10.1080/17476933.2015.1053473 (DOI)000365643500003 ()
Available from: 2015-04-15 Created: 2015-04-15 Last updated: 2017-12-04Bibliographically approved
Avelin, B., Gianazza, U. & Salsa, S. (2016). Boundary Estimates for Certain Degenerate and Singular Parabolic Equations. Journal of the European Mathematical Society (Print), 18(2), 381-424
Open this publication in new window or tab >>Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
2016 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, no 2, p. 381-424Article in journal (Refereed) Published
Abstract [en]

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic p-Laplace equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part S-T of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations of porous medium type.

Keywords
Degenerate and singular parabolic equations; Harnack estimates; boundary Harnack inequality; Carleson estimate
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-186267 (URN)10.4171/JEMS/593 (DOI)000370249100005 ()
Available from: 2013-02-12 Created: 2012-11-28 Last updated: 2017-12-07Bibliographically approved
Avelin, B. (2016). On time dependent domains for the degenerate p-parabolic equation: Carleson estimate and Holder continuity. Mathematische Annalen, 364(1-2), 667-686
Open this publication in new window or tab >>On time dependent domains for the degenerate p-parabolic equation: Carleson estimate and Holder continuity
2016 (English)In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 364, no 1-2, p. 667-686Article in journal (Refereed) Published
Abstract [en]

In this paper we propose a definition of "parabolic NTA" for solutions to the degenerate p-parabolic equation. Given this definition we prove the Carleson estimate, originally proved for this equation in Avelin et al. (J Eur Math Soc, 2015) for cylindrical domains. Moreover we study a non-optimal, stronger "outer corkscrew" condition, such that we obtain Holder continuity up to the boundary, for non-negative solutions vanishing on a part of the boundary.

National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-298255 (URN)10.1007/s00208-015-1226-8 (DOI)000376064900024 ()
Available from: 2016-07-01 Created: 2016-07-01 Last updated: 2017-11-28Bibliographically approved
Avelin, B., Hed, L. & Persson, H. (2015). A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity. Potential Analysis, 43(3), 531-545
Open this publication in new window or tab >>A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity
2015 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 3, p. 531-545Article in journal (Refereed) Published
Abstract [en]

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. 

Place, publisher, year, edition, pages
Håkan Persson, 2015
Keywords
plurisubharmonic functions, continuous boundary, hyperconvexity, bounded exhaustion function, Hölder for all exponents, log-lipschitz, boundary regularity, Reinhardt domains.
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-251330 (URN)10.1007/s11118-015-9486-1 (DOI)000365769100010 ()
Available from: 2015-04-15 Created: 2015-04-15 Last updated: 2017-12-04Bibliographically approved
Avelin, B. & Lukkari, T. (2015). Lower semicontinuity of weak supersolutions to the porous medium equation. Proceedings of the American Mathematical Society, 143(8), 3475-3486
Open this publication in new window or tab >>Lower semicontinuity of weak supersolutions to the porous medium equation
2015 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, no 8, p. 3475-3486Article in journal (Refereed) Published
Abstract [en]

Weak supersolutions to the porous medium equation are defined by means of smooth test functions under an integral sign. We show that non-negative weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. This shows that weak supersolutions belong to a class of supersolutions defined by a comparison principle.

Keywords
Porous medium equation, supersolutions, comparison principle, lower semicontinuity, degenerate diffusion
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-259086 (URN)10.1090/proc/12727 (DOI)000357042200028 ()
Available from: 2015-07-29 Created: 2015-07-27 Last updated: 2017-12-04Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-1429-4965

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