| 1. |
- Avelin, Benny, 1984-, et al.
(författare)
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A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term
- 2017
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Ingår i: Journal of Functional Analysis. - : Elsevier BV. - 0022-1236 .- 1096-0783. ; 272:8, s. 3176-3215
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Avelin, Benny, 1984-, et al.
(författare)
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A comparison principle for the porous medium equation and its consequences
- 2017
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Ingår i: Revista matemática iberoamericana. - 0213-2230 .- 2235-0616. ; 33:2, s. 573-594
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- Avelin, Benny, et al.
(författare)
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A Galerkin type method for kinetic Fokker-Planck equations based on Hermite expansions
- 2023
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Ingår i: Kinetic and Related Models. - 1937-5093 .- 1937-5077.
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain (0,T)×D×Rd, where D is either Td or Rd. Our approach is based on a Hermite expansion in the velocity variable only, with a hyperbolic system that appears as the truncation of the Brinkman hierarchy, as well as ideas from $\href{arXiv:1902.04037v2}{Alb+21}$ and additional energy-type estimates that we have developed. We also establish the regularity of the solution based on the regularity of the initial data and the source term.
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| 4. |
- 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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| 5. |
- Avelin, Benny, 1984-, et al.
(författare)
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Approximation and Bounded Plurisubharmonic Exhaustion Functions Beyond Lipschitz Domains
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 6. |
- Avelin, Benny, et al.
(författare)
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Approximation of plurisubharmonic functions
- 2016
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Ingår i: Complex Variables and Elliptic Equations. - : Informa UK Limited. - 1747-6933 .- 1747-6941. ; 61:1, s. 23-28
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 7. |
- Avelin, Benny, et al.
(författare)
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Boundary behavior of solutions to the parabolic p-Laplace equation
- 2019
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Ingår i: Analysis & PDE. - : Mathematical Sciences Publishers. - 2157-5045 .- 1948-206X. ; 12:1, s. 1-42
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 8. |
- Avelin, Benny, 1984-
(författare)
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Boundary behavior of solutions to the parabolic p-Laplace equation II
- 2020
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Ingår i: Journal of Functional Analysis. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0022-1236 .- 1096-0783. ; 279:1
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Tidskriftsartikel (refereegranskat)abstract
- This paper is the second installment in a series of papers concerning the boundary behavior of solutions to the p-parabolic equations. In this paper we are interested in the short time behavior of the solutions, which is in contrast with much of the literature, where all results require a waiting time. We prove a dichotomy about the decay-rate of non-negative solutions vanishing on the lateral boundary in a cylindrical C-1,C-1 domain. Furthermore we connect this dichotomy to the support of the boundary type Riesz measure related to the p-parabolic equation in NTA-domains, which has consequences for the continuation of solutions.
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| 9. |
- Avelin, Benny, 1984-
(författare)
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Boundary Behavior of p-Laplace Type Equations
- 2013
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 10. |
- Avelin, Benny, 1984-, et al.
(författare)
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Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
- 2016
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Ingår i: Journal of the European Mathematical Society (Print). - 1435-9855 .- 1435-9863. ; 18:2, s. 381-424
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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