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Avelin, Bennyorcid.org/0000-0002-1429-4965

Open this publication in new window or tab >>Boundary behavior of solutions to the parabolic p-Laplace equation### Avelin, Benny

### Kuusi, Tuomo

### Nyström, Kaj

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##### Abstract [en]

##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-265500 (URN)10.2140/apde.2019.12.1 (DOI)000446595400001 ()
#####

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##### Funder

Swedish Research Council, 637-2014-6822
Available from: 2015-10-30 Created: 2015-10-30 Last updated: 2018-11-28Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

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.

Open this publication in new window or tab >>Nonlinear Caldern-Zygmund Theory in the Limiting Case### Avelin, Benny

### Kuusi, Tuomo

### Mingione, Giuseppe

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 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]

##### National Category

Mathematics Mechanical Engineering
##### Identifiers

urn:nbn:se:uu:diva-341484 (URN)10.1007/s00205-017-1171-7 (DOI)000419122300006 ()
#####

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Available from: 2018-02-28 Created: 2018-02-28 Last updated: 2018-02-28Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Aalto Univ, Dept Math & Syst Anal, POB 11100, Aalto 00076, Finland..

Univ Parma, Dipartimento Matemat & Informat, Parco Area Sci 53-A, I-43124 Parma, Italy..

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.

Open this publication in new window or tab >>A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term### Avelin, Benny

### Julin, Vesa

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 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]

##### 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 ()
#####

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##### Funder

Swedish Research Council, 637-2014-6822
Available from: 2017-05-02 Created: 2017-05-02 Last updated: 2017-05-02

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Aalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland.

Univ Jyvaskyla, Dept Math & Stat, POB 35, Jyvaskyla 40014, Finland..

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.

Open this publication in new window or tab >>A comparison principle for the porous medium equation and its consequences### Avelin, Benny

### Lukkari, Teemu

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 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]

##### 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 ()
#####

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##### Funder

Swedish Research Council, 637-2014-6822
Available from: 2017-09-15 Created: 2017-09-15 Last updated: 2017-09-15Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Aalto Univ, Finland.

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.

Open this publication in new window or tab >>Characterizations of interior polar sets for the degenerate *p*-parabolic equation### Avelin, Benny

### Saari, Olli

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 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]

##### 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 ()
#####

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##### Funder

Swedish Research Council, 637-2014-6822
Available from: 2017-10-09 Created: 2017-10-09 Last updated: 2017-10-09Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Aalto Univ, Dept Math & Syst Anal, Sch Sci, Aalto 00076, Finland..

Aalto Univ, Dept Math & Syst Anal, Sch Sci, Aalto 00076, Finland..

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.

Open this publication in new window or tab >>Approximation of plurisubharmonic functions### Avelin, Benny

### Hed, Lisa

### Persson, Håkan

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 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]

##### 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 ()
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Available from: 2015-04-15 Created: 2015-04-15 Last updated: 2017-12-04Bibliographically approved

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.

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.

Open this publication in new window or tab >>Boundary Estimates for Certain Degenerate and Singular Parabolic Equations### Avelin, Benny

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Gianazza, Ugo

### Salsa, Sandro

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 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]

##### 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 ()
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Available from: 2013-02-12 Created: 2012-11-28 Last updated: 2017-12-07Bibliographically approved

Dipartimento di Matematica "F. Casorati", Università di Pavia.

Dipartimento di Matematica "F. Brioschi", Politecnico di Milano.

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.

Open this publication in new window or tab >>On time dependent domains for the degenerate p-parabolic equation: Carleson estimate and Holder continuity### Avelin, Benny

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 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]

##### National Category

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-298255 (URN)10.1007/s00208-015-1226-8 (DOI)000376064900024 ()
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Available from: 2016-07-01 Created: 2016-07-01 Last updated: 2017-11-28Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Univ Jyvaskyla, Dept Math & Stat, POB 35, Jyvaskyla 40014, Finland.;Aalto Univ, Inst Math, POB 11100, Aalto 00076, Finland..

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.

Open this publication in new window or tab >>A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity### Avelin, Benny

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Hed, Lisa

### Persson, Håkan

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 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]

##### 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 ()
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Available from: 2015-04-15 Created: 2015-04-15 Last updated: 2017-12-04Bibliographically approved

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.

Open this publication in new window or tab >>Lower semicontinuity of weak supersolutions to the porous medium equation### Avelin, Benny

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Lukkari, Teemu

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##### Abstract [en]

##### 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 ()
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Available from: 2015-07-29 Created: 2015-07-27 Last updated: 2017-12-04Bibliographically approved

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.