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  • 1.
    Avelin, Benny
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Boundary Behavior of p-Laplace Type Equations2013Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of six scientific papers, an introduction and a summary. All six papers concern the boundary behavior of non-negative solutions to partial differential equations.

    Paper I concerns solutions to certain p-Laplace type operators with variable coefficients. Suppose that u is a non-negative solution that vanishes on a part Γ of an Ahlfors regular NTA-domain. We prove among other things that the gradient Du of u has non-tangential limits almost everywhere on the boundary piece Γ, and that log|Du| is a BMO function on the boundary.  Furthermore, for Ahlfors regular NTA-domains that are uniformly (N,δ,r0)-approximable by Lipschitz graph domains we prove a boundary Harnack inequality provided that δ is small enough. 

    Paper II concerns solutions to a p-Laplace type operator with lower order terms in δ-Reifenberg flat domains. We prove that the ratio of two non-negative solutions vanishing on a part of the boundary is Hölder continuous provided that δ is small enough. Furthermore we solve the Martin boundary problem provided δ is small enough.

    In Paper III we prove that the boundary type Riesz measure associated to an A-capacitary function in a Reifenberg flat domain with vanishing constant is asymptotically optimal doubling.

    Paper IV concerns the boundary behavior of solutions to certain parabolic equations of p-Laplace type in Lipschitz cylinders. Among other things, we prove an intrinsic Carleson type estimate for the degenerate case and a weak intrinsic Carleson type estimate in the singular supercritical case.

    In Paper V we are concerned with equations of p-Laplace type structured on Hörmander vector fields. We prove that the boundary type Riesz measure associated to a non-negative solution that vanishes on a part Γ of an X-NTA-domain, is doubling on Γ.

    Paper VI concerns a one-phase free boundary problem for linear elliptic equations of non-divergence type. Assume that we know that the positivity set is an NTA-domain and that the free boundary is a graph. Furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction. We prove that the graph giving the free boundary is Lipschitz continuous.

    List of papers
    1. Estimates for Solutions to Equations of p-Laplace type in Ahlfors regular NTA-domains
    Open this publication in new window or tab >>Estimates for Solutions to Equations of p-Laplace type in Ahlfors regular NTA-domains
    2014 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 266, no 9, p. 5955-6005Article in journal (Refereed) Published
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:uu:diva-163517 (URN)10.1016/j.jfa.2014.02.027 (DOI)000334652000018 ()
    Available from: 2011-12-13 Created: 2011-12-12 Last updated: 2017-12-08Bibliographically approved
    2. Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms
    Open this publication in new window or tab >>Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms
    2011 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, no 1, p. 264-291Article in journal (Refereed) Published
    National Category
    Mathematics
    Identifiers
    urn:nbn:se:uu:diva-163370 (URN)10.1016/j.jde.2010.09.011 (DOI)
    Available from: 2011-12-12 Created: 2011-12-12 Last updated: 2017-12-08Bibliographically approved
    3. Optimal doubling, Reifenberg flatness and operators of p-Laplace type
    Open this publication in new window or tab >>Optimal doubling, Reifenberg flatness and operators of p-Laplace type
    2011 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, no 17, p. 5943-5955Article in journal (Refereed) Published
    Abstract [en]

    In this paper we consider operators of p-Laplace type of the form ∇·A(x,∇u) = 0. ConcerningA we assume, for p ∈ (1,∞) fixed, an appropriate ellipticity type condition, H¨older continuityin x and that A(x, ) = ||p−1A(x, /||) whenever x ∈ Rn and ∈ Rn \ {0}. Let  ⊂ Rn be abounded domain, let D be a compact subset of . We say that ˆu = ˆup,D, is the A-capacitaryfunction for D in  if ˆu ≡ 1 on D, ˆu ≡ 0 on @ in the sense of W1,p0 () and ∇·A(x,∇ˆu) = 0 in \D in the weak sense. We extend ˆu to Rn \  by putting ˆu ≡ 0 on Rn \ . Then there existsa unique finite positive Borel measure ˆμ on Rn, with support in @, such thatZ hA(x,∇ˆu),∇i dx = −Z dˆμ whenever ∈ C∞0 (Rn \ D).In this paper we prove that if  is Reifenberg flat with vanishing constant, thenlimr→0infw∈∂ˆμ(B(w, r))ˆμ(B(w, r))= limr→0supw∈∂ˆμ(B(w, r))ˆμ(B(w, r))= n−1,for every , 0 < ≤ 1. In particular, we prove that ˆμ is an asymptotically optimal doublingmeasure on @.

    National Category
    Mathematics
    Identifiers
    urn:nbn:se:uu:diva-163435 (URN)10.1016/j.na.2011.05.061 (DOI)
    Available from: 2012-11-30 Created: 2011-12-12 Last updated: 2017-12-08Bibliographically approved
    4. Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
    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
    5. Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures
    Open this publication in new window or tab >>Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures
    2013 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 85, p. 149-159Article in journal (Refereed) Published
    Abstract [en]

    Let be a system of C vector fields in Rn satisfying Hörmander’s finite rank condition and let Ω be a non-tangentially accessible domain with respect to the Carnot–Carathéodory distance d induced by X. We prove the doubling property of certain boundary measures associated to non-negative solutions, which vanish on a portion of Ω, to the equation

    Given p, 1<p<, fixed, we impose conditions on the function A=(A1,…,Am):Rn×RmRm, which imply that the equation is a quasi-linear partial differential equation of p-Laplace type structured on vector fields satisfying the classical Hörmander condition. In the case p=2 and for linear equations, our result coincides with the doubling property of associated elliptic measures. To prove our result we establish, and this is of independent interest, a Wolff potential estimate for subelliptic equations of p-Laplace type.

    National Category
    Mathematical Analysis
    Identifiers
    urn:nbn:se:uu:diva-186268 (URN)10.1016/j.na.2013.02.023 (DOI)000318378700013 ()
    Available from: 2013-03-26 Created: 2012-11-28 Last updated: 2017-12-07Bibliographically approved
    6. On a one-phase free boundary problem
    Open this publication in new window or tab >>On a one-phase free boundary problem
    2013 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, p. 181-191Article in journal (Other academic) Published
    Abstract [en]

    In this paper we extend a result regarding the free boundary regularity in a one-phaseproblem, by De Silva and Jerison [DJ], to non-divergence linear equations of second order.Roughly speaking we prove that the free boundary is given by a Lipschitz graph.

    Keywords
    One-phase, free boundary, NTA, non-divergence, linear
    National Category
    Mathematical Analysis
    Research subject
    Mathematics
    Identifiers
    urn:nbn:se:uu:diva-186265 (URN)10.5186/aasfm.2013.3815 (DOI)000316239200009 ()
    Available from: 2012-11-30 Created: 2012-11-28 Last updated: 2017-12-07Bibliographically approved
  • 2.
    Avelin, Benny
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    On a one-phase free boundary problem2013In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, p. 181-191Article in journal (Other academic)
    Abstract [en]

    In this paper we extend a result regarding the free boundary regularity in a one-phaseproblem, by De Silva and Jerison [DJ], to non-divergence linear equations of second order.Roughly speaking we prove that the free boundary is given by a Lipschitz graph.

  • 3.
    Avelin, Benny
    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..
    On time dependent domains for the degenerate p-parabolic equation: Carleson estimate and Holder continuity2016In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 364, no 1-2, p. 667-686Article in journal (Refereed)
    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.

  • 4.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Gianazza, Ugo
    Dipartimento di Matematica "F. Casorati", Università di Pavia.
    Salsa, Sandro
    Dipartimento di Matematica "F. Brioschi", Politecnico di Milano.
    Boundary Estimates for Certain Degenerate and Singular Parabolic Equations2016In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, no 2, p. 381-424Article in journal (Refereed)
    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.

  • 5.
    Avelin, Benny
    et al.
    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.
    A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity2015In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 3, p. 531-545Article in journal (Refereed)
    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. 

  • 6.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Hed, Lisa
    Department of mathematics and mathematical statistics, Umeå University.
    Persson, Håkan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Approximation and Bounded Plurisubharmonic Exhaustion Functions Beyond Lipschitz DomainsManuscript (preprint) (Other academic)
    Abstract [en]

    Using techniques from the analysis of PDEs to studythe boundary behaviour of functions on domains with low boundaryregularity, we extend results by Fornaæss-Wiegerinck (1989)on plurisubharmonic approximation and by Demailly (1987) onthe existence on bounded plurisubharmonic exhaustion functionsto domains beyond Lipschitz boundary regularity.

  • 7.
    Avelin, Benny
    et al.
    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.
    Hed, Lisa
    Umeå University.
    Persson, Håkan
    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)
    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.

  • 8.
    Avelin, Benny
    et al.
    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.
    Julin, Vesa
    Univ Jyvaskyla, Dept Math & Stat, POB 35, Jyvaskyla 40014, Finland..
    A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term2017In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 272, no 8, p. 3176-3215Article in journal (Refereed)
    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.

  • 9.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Kuusi, Tuomo
    Aalto Univ, Dept Math & Syst Anal, POB 11100, Aalto 00076, Finland..
    Mingione, Giuseppe
    Univ Parma, Dipartimento Matemat & Informat, Parco Area Sci 53-A, I-43124 Parma, Italy..
    Nonlinear Caldern-Zygmund Theory in the Limiting Case2018In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 227, no 2, p. 663-714Article in journal (Refereed)
    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.

  • 10.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Kuusi, Tuomo
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Boundary behavior of solutions to the parabolic p-Laplace equation2019In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 12, no 1, p. 1-42Article in journal (Refereed)
    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.

  • 11.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Lukkari, Teemu
    Aalto Univ, Finland.
    A comparison principle for the porous medium equation and its consequences2017In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 33, no 2, p. 573-594Article in journal (Refereed)
    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.

  • 12.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Lukkari, Teemu
    Lower semicontinuity of weak supersolutions to the porous medium equation2015In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, no 8, p. 3475-3486Article in journal (Refereed)
    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.

  • 13. Avelin, Benny
    et al.
    Lundström, Niklas L. P.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
    Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms2011In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, no 1, p. 264-291Article in journal (Refereed)
  • 14. Avelin, Benny
    et al.
    Lundström, Niklas L.P
    Nyström, Kaj
    Optimal doubling, Reifenberg flatness and operators of p-Laplace type2011In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, no 17, p. 5943-5955Article in journal (Refereed)
    Abstract [en]

    In this paper we consider operators of p-Laplace type of the form ∇·A(x,∇u) = 0. ConcerningA we assume, for p ∈ (1,∞) fixed, an appropriate ellipticity type condition, H¨older continuityin x and that A(x, ) = ||p−1A(x, /||) whenever x ∈ Rn and ∈ Rn \ {0}. Let  ⊂ Rn be abounded domain, let D be a compact subset of . We say that ˆu = ˆup,D, is the A-capacitaryfunction for D in  if ˆu ≡ 1 on D, ˆu ≡ 0 on @ in the sense of W1,p0 () and ∇·A(x,∇ˆu) = 0 in \D in the weak sense. We extend ˆu to Rn \  by putting ˆu ≡ 0 on Rn \ . Then there existsa unique finite positive Borel measure ˆμ on Rn, with support in @, such thatZ hA(x,∇ˆu),∇i dx = −Z dˆμ whenever ∈ C∞0 (Rn \ D).In this paper we prove that if  is Reifenberg flat with vanishing constant, thenlimr→0infw∈∂ˆμ(B(w, r))ˆμ(B(w, r))= limr→0supw∈∂ˆμ(B(w, r))ˆμ(B(w, r))= n−1,for every , 0 < ≤ 1. In particular, we prove that ˆμ is an asymptotically optimal doublingmeasure on @.

  • 15. Avelin, Benny
    et al.
    Nyström, Kaj
    Estimates for Solutions to Equations of p-Laplace type in Ahlfors regular NTA-domains2014In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 266, no 9, p. 5955-6005Article in journal (Refereed)
  • 16.
    Avelin, Benny
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures2013In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 85, p. 149-159Article in journal (Refereed)
    Abstract [en]

    Let be a system of C vector fields in Rn satisfying Hörmander’s finite rank condition and let Ω be a non-tangentially accessible domain with respect to the Carnot–Carathéodory distance d induced by X. We prove the doubling property of certain boundary measures associated to non-negative solutions, which vanish on a portion of Ω, to the equation

    Given p, 1<p<, fixed, we impose conditions on the function A=(A1,…,Am):Rn×RmRm, which imply that the equation is a quasi-linear partial differential equation of p-Laplace type structured on vector fields satisfying the classical Hörmander condition. In the case p=2 and for linear equations, our result coincides with the doubling property of associated elliptic measures. To prove our result we establish, and this is of independent interest, a Wolff potential estimate for subelliptic equations of p-Laplace type.

  • 17.
    Avelin, Benny
    et al.
    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..
    Saari, Olli
    Aalto Univ, Dept Math & Syst Anal, Sch Sci, Aalto 00076, Finland..
    Characterizations of interior polar sets for the degenerate p-parabolic equation2017In: Journal of evolution equations (Printed ed.), ISSN 1424-3199, E-ISSN 1424-3202, Vol. 17, no 2, p. 827-848Article in journal (Refereed)
    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.

1 - 17 of 17
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