| 1. |
- Fotouhi, Morteza, et al.
(författare)
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Remarks on the decay/growth rate of solutions to elliptic free boundary problems of obstacle type
- 2020
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Ingår i: MATHEMATICS IN ENGINEERING. - : American Institute of Mathematical Sciences (AIMS). - 2640-3501. ; 2:4, s. 698-708
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Tidskriftsartikel (refereegranskat)abstract
- The purpose of this note is to present a "new" approach to the decay rate of the solutions to the no-sign obstacle problem from the free boundary, based on Weiss-monotonicity formula. In presenting the approach we have chosen to treat a problem which is not touched earlier in the existing literature. Although earlier techniques may still work for this problem, we believe this approach gives a shorter proof, and may have wider applications.
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| 2. |
- Indrei, Emanuel, et al.
(författare)
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NONTRANSVERSAL INTERSECTION OF FREE AND FIXED BOUNDARIES FOR FULLY NONLINEAR ELLIPTIC OPERATORS IN TWO DIMENSIONS
- 2016
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Ingår i: Analysis & PDE. - : Mathematical Sciences Publishers. - 2157-5045 .- 1948-206X. ; 9:2, s. 487-502
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Tidskriftsartikel (refereegranskat)abstract
- In the study of classical obstacle problems, it is well known that in many configurations, the free boundary intersects the fixed boundary tangentially. The arguments involved in producing results of this type rely on the linear structure of the operator. In this paper, we employ a different approach and prove tangential touch of free and fixed boundaries in two dimensions for fully nonlinear elliptic operators. Along the way, several n-dimensional results of independent interest are obtained, such as BMO-estimates, C-1,C-1-regularity up to the fixed boundary, and a description of the behavior of blow-up solutions.
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| 3. |
- Indrei, Emanuel, et al.
(författare)
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Regularity of solutions in semilinear elliptic theory
- 2017
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Ingår i: Bulletin of Mathematical Sciences. - : SPRINGER BASEL AG. - 1664-3607 .- 1664-3615. ; 7:1, s. 177-200
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Tidskriftsartikel (refereegranskat)abstract
- We study the semilinear Poisson equation Delta u = f (x, u) in B-1. (1) Our main results provide conditions on f which ensure that weak solutions of (1) belong to C-1,C-1(B-1/2). In some configurations, the conditions are sharp.
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| 4. |
- Indrei, E., et al.
(författare)
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Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems
- 2016
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Ingår i: Annales de l'Institut Henri Poincare. Analyse non linéar. - : Elsevier. - 0294-1449 .- 1873-1430. ; 33:5, s. 1259-1277
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Tidskriftsartikel (refereegranskat)abstract
- We consider fully nonlinear obstacle-type problems of the form. F(D2u,x)=f(x)a.e. in B1∩Ω,|D2u|≤Ka.e. in B1\Ω, where Ω is an open set and K>0. In particular, structural conditions on F are presented which ensure that W2,n(B1) solutions achieve the optimal C1,1(B1/2) regularity when f is Hölder continuous. Moreover, if f is positive on B-1, Lipschitz continuous, and u≠0⊂Ω, we obtain interior C1 regularity of the free boundary under a uniform thickness assumption on u=0. Lastly, we extend these results to the parabolic setting.
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| 5. |
- Magnani, Valentino, et al.
(författare)
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OPTIMAL REGULARITY OF SOLUTIONS TO NO-SIGN OBSTACLE-TYPE PROBLEMS FOR THE SUB-LAPLACIAN
- 2022
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Ingår i: Analysis & PDE. - : Mathematical Sciences Publishers. - 2157-5045 .- 1948-206X. ; 15:6, s. 1429-1456
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Tidskriftsartikel (refereegranskat)abstract
- We establish the optimal C-H(1,1) interior regularity of solutions to Delta(H)u = f chi{u not equal 0}, where Delta(H) denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on f, namely the group convolution f * Gamma is C-H(1,1), where Gamma is the fundamental solution of Delta(H). The C-H(1,1) regularity is understood in the sense of Folland and Stein. In the classical Euclidean setting, the first seeds of the above problem were already present in the 1991 paper of Sakai and are also related to quadrature domains. As a special instance of our results, when u is nonnegative and satisfies the above equation, we recover the C-H(1,1) regularity of solutions to the obstacle problem in stratified groups, which was previously established by Danielli, Garofalo and Salsa. Our regularity result is sharp: it can be seen as the subelliptic counterpart of the C-1,C-1 regularity result due to Andersson, Lindgren and Shahgholian.
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| 6. |
- Minne, Andreas, 1984-, et al.
(författare)
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Asymptotic mean value Laplacian in metric measure spaces
- 2020
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 491:2
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Tidskriftsartikel (refereegranskat)abstract
- We use the mean value property in an asymptotic way to provide a notion of a pointwise Laplacian, called AMV Laplacian, that we study in several contexts including the Heisenberg group and weighted Lebesgue measures. We focus especially on a class of metric measure spaces including intersecting submanifolds of R-n, a context in which our notion brings new insights; the Kirchhoff law appears as a special case. In the general case, we also prove a maximum and comparison principle, as well as a Green-type identity for a related operator.
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| 7. |
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| 8. |
- Minne, Andreas, 1984-
(författare)
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Non-linear Free Boundary Problems
- 2015
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of an introduction and four research papers related to free boundary problems and systems of fully non-linear elliptic equations.Paper A and Paper B prove optimal regularity of solutions to general elliptic and parabolic free boundary problems, where the operators are fully non-linear and convex. Furthermore, it is proven that the free boundary is continuously differentiable around so called "thick" points, and that the free boundary touches the fixed boundary tangentially in two dimensions.Paper C analyzes singular points of solutions to perturbations of the unstable obstacle problem, in three dimensions. Blow-up limits are characterized and shown to be unique. The free boundary is proven to lie close to the zero-level set of the corresponding blow-up limit. Finally, the structure of the singular set is analyzed.Paper D discusses an idea on how existence and uniqueness theorems concerning quasi-monotone fully non-linear elliptic systems can be extended to systems that are not quasi-monotone.
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| 9. |
- Minne, Andreas, et al.
(författare)
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Non-transversal intersection of free and fixed boundary for fully nonlinear elliptic operators in two dimensions
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In the study of classical obstacle problems, it is well known that in many configurations the free boundary intersects the fixed boundary tangentially. The arguments involved in producing results of this type rely on the linear structure of the operator. In this paper we employ a different approach and prove tangential touch of free and fixed boundary in two dimensions for fully nonlinear elliptic operators. Along the way, several n-dimensional results of independent interest are obtained such as BMO-estimates, C1,1 regularity up to the fixed boundary, and a description of the behavior of blow-up solutions.
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| 10. |
- Minne, Andreas, 1984-, et al.
(författare)
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Symmetrized and non-symmetrizedasymptotic mean value Laplacian in metric measure spaces
- 2023
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Ingår i: Proceedings of the Royal Society of Edinburgh. Section A Mathematics. - : Cambridge University Press (CUP). - 0308-2105 .- 1473-7124. ; , s. 1-38
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Tidskriftsartikel (refereegranskat)abstract
- The asymptotic mean value Laplacian - AMV Laplacian - extends the Laplace operator from to metric measure spaces through limits of averaging integrals. The AMV Laplacian is however not a symmetric operator in general. Therefore, we consider a symmetric version of the AMV Laplacian, and focus lies on when the symmetric and non-symmetric AMV Laplacians coincide. Besides Riemannian and 3D contact sub-Riemannian manifolds, we show that they are identical on a large class of metric measure spaces, including locally Ahlfors regular spaces with suitably vanishing distortion. In addition, we study the context of weighted domains of - where the two operators typically differ - and provide explicit formulae for these operators, including points where the weight vanishes.
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