| 1. |
- Acker, A., et al.
(author)
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The multi-layer free boundary problem for the p-Laplacian in convex domains
- 2004
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In: Interfaces and free boundaries (Print). - 1463-9963 .- 1463-9971. ; 6:1, s. 81-103
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Journal article (peer-reviewed)abstract
- The main result of this paper concerns existence of classical solutions to the multi-layer Bernoulli free boundary problem with nonlinear joining conditions and the p-Laplacian as governing operator. The present treatment of the two-layer case involves technical refinements of the one-layer case, studied earlier by two of the authors. The existence treatment of the multi-layer case is largely based on a reduction to the two-layer case, in which uniform separation of the free boundaries plays a key role.
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| 2. |
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| 3. |
- Aleksanyan, Gohar
(author)
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Analysis of blow-ups for the double obstacle problem in dimension two
- 2019
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In: Interfaces and free boundaries (Print). - : European Mathematical Society Publishing House. - 1463-9963 .- 1463-9971. ; 21:2, s. 131-167
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Journal article (peer-reviewed)abstract
- In this article we study a normalised double obstacle problem with polynomial obstacles p(1)(x) <= p(2)(x), where the equality holds iff x = 0. In dimension two we give a complete classification of blow-up solutions. In particular, we see that there exists a new type of blow-ups, which we call double-cone solutions, since the coincidence sets {u = p(1)} and {u = p(2)} are cones with a common vertex. Furthermore, we show that if the solution to the double obstacle problem has a double-cone blow-up limit at the origin, then the blow-up is unique, and locally the free boundary consists of four C-1,C-gamma-curves, meeting at the origin.
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| 4. |
- Andersson, John, et al.
(author)
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Optimal regularity for the parabolic no-sign obstacle type problem
- 2013
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In: Interfaces and free boundaries (Print). - 1463-9963 .- 1463-9971. ; 15:4, s. 477-499
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Journal article (peer-reviewed)abstract
- We study the parabolic free boundary problem of obstacle type Delta u - partial derivative u/partial derivative t = f chi({u not equal 0}). Under the condition that f = H nu for some function nu with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution u. Both the regularity and the assumptions are optimal. Using this result and assuming that f is Dini continuous, we prove that the free boundary is, near so called low energy points, a C-1 graph. Our result completes the theory for this type of problems for the heat operator.
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| 5. |
- Brasco, L., et al.
(author)
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The fractional Cheeger problem
- 2014
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In: Interfaces and free boundaries (Print). - 1463-9963 .- 1463-9971. ; 16:3, s. 419-458
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Journal article (peer-reviewed)abstract
- Given an open and bounded set Omega subset of R-N, we consider the problem of minimizing the ratio between the s-perimeter and the N-dimensional Lebesgue measure among subsets of Omega. This is the non-local version of the well-known Cheeger problem. We prove various properties of optimal sets for this problem, as well as some equivalent formulations. In addition, the limiting behaviour of some nonlinear and non-local eigenvalue problems is investigated, in relation with this optimization problem. The presentation is as self-contained as possible.
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| 6. |
- de Queiroz, Olivaine S., et al.
(author)
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A free boundary problem with log term singularity
- 2017
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In: Interfaces and free boundaries (Print). - 1463-9963 .- 1463-9971. ; 19:3, s. 351-369
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Journal article (peer-reviewed)abstract
- We study a minimum problem for a non-differentiable functional whose reaction term does not have scaling properties. Specifically we consider the functional (sic)(v) = integral(Omega) (vertical bar del v vertical bar(2)/2 - v(+)(log v - 1))dx -> min which should be minimized in some natural admissible class of non-negative functions. Here, v(+) = max{0, v}. The Euler-Lagrange equation associated with (sic) is -Delta u = chi({u>0}) log u, which becomes singular along the free boundary partial derivative{u > O}. Therefore, the regularity results do not follow from classical methods. Besides, the logarithmic forcing term does not have scaling properties, which are very important in the study of free boundary theory. Despite these difficulties, we obtain optimal regularity of a minimizer and show that, close to every free boundary point, they exhibit a super-characteristic growth like r(2)vertical bar log r vertical bar. This estimate is crucial in the study of analytic and geometric properties of the free boundary.
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| 7. |
- Heintz, Alexey, 1955, et al.
(author)
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A convolution thresholding scheme for the Willmore flow
- 2008
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In: Interfaces and Free Boundaries. - 1463-9971 .- 1463-9963. ; 10:2, s. 139-153
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Journal article (peer-reviewed)abstract
- A convolution thresholding scheme fro geometric Willmore flows is suggested and the consistency is proved.
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| 8. |
- Jeon, Seongmin, et al.
(author)
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Almost minimizers for the thin obstacle problem with variable coefficients
- 2024
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In: Interfaces and free boundaries (Print). - : European Mathematical Society - EMS. - 1463-9963 .- 1463-9971. ; 26:3, s. 321-380
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Journal article (peer-reviewed)abstract
- We study almost minimizers for the thin obstacle problem with variable Holder continuous coefficients and zero thin obstacle, and establish their C-1,C-beta. regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.
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| 9. |
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| 10. |
- Kumazaki, Kota, et al.
(author)
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Global weak solvability, continuous dependence on data, and large time growth of swelling moving interfaces
- 2020
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In: Interfaces and free boundaries (Print). - : European Mathematical Society Publishing House. - 1463-9963 .- 1463-9971. ; 22:1, s. 27-49
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Journal article (peer-reviewed)abstract
- We prove a global existence result for weak solutions to a free boundary problem with flux boundary conditions describing swelling along a halfline. Additionally, we show that solutions are not only unique but also depend continuously on data and parameters. The key observation is that the structure of our system of partial differential equations allows us to show that the moving a priori unknown interface never disappears. As main ingredients of the global existence proof, we rely on a local weak solvability result for our problem (as reported in [7]), uniform energy estimates of the solution, integral estimates on quantities defined at the free boundary, as well as a fine pointwise lower bound for the position of the moving boundary. Some of the estimates are time independent. They allow us to explore the large-time behavior of the position of the moving boundary. The approach is specific to one-dimensional settings.
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