| 1. |
- Acker, A., et al.
(författare)
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The multi-layer free boundary problem for the p-Laplacian in convex domains
- 2004
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Ingår i: Interfaces and free boundaries (Print). - 1463-9963 .- 1463-9971. ; 6:1, s. 81-103
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Tidskriftsartikel (refereegranskat)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. |
- Apushkinskaya, D. E., et al.
(författare)
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Lipschitz property of the free boundary in the parabolic obstacle problem
- 2004
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Ingår i: St. Petersburg Mathematical Journal. - 1061-0022 .- 1547-7371. ; 15:3, s. 375-391
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Tidskriftsartikel (refereegranskat)abstract
- A parabolic obstacle problem with zero constraint is considered. It is proved, without any additional assumptions on a free boundary, that near the fixed boundary where the homogeneous Dirichlet condition is fulfilled, the boundary of the noncoincidence set is the graph of a Lipschitz function.
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| 3. |
- Blank, Ivan, et al.
(författare)
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Boundary Regularity and Compactness for Overdetermined Problems
- 2003
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Ingår i: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V. - 0391-173X .- 2036-2145. ; 2:4, s. 787-802
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Tidskriftsartikel (refereegranskat)abstract
- Let D be either the unit ball B-1(0) or the half ball B-1(+)(0), let f be a strictly positive and continuous function, and let u and Omega subset of D solve the following overdetermined problem: Delta u (x) = chi(Omega) (x) f (x) in D, 0 is an element of partial derivative Omega, u = vertical bar del u vertical bar = 0 in Omega(c), where chi(Omega) denotes the characteristic function of Omega, Omega(c) denotes the set D\Omega, and the equation is satisfied in the sense of distributions. When D = B-1(+)(0), then we impose in addition that u(x) 0 {(x', x(n))vertical bar x(n) = 0} We show that a fairly mild thickness assumption on Omega(c) will ensure enough compactness on it to give us "blow-up" limits, and we show how this compactness leads to regularity of partial derivative Omega. In the case where f is positive and Lipschitz, the methods developed in Caffarelli, Karp, and Shahgholian (2000) lead to regularity of partial derivative Omega under a weaker thickness assumption.
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| 4. |
- Caffarelli, L. A., et al.
(författare)
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Regularity of a free boundary with application to the Pompeiu problem
- 2000
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Ingår i: Annals of Mathematics. - : JSTOR. - 0003-486X .- 1939-8980. ; 151:1, s. 269-292
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Tidskriftsartikel (refereegranskat)abstract
- In the unit ball B(0, 1), let u and Omega (a domain in R-N) salve the following overdetermined problem: Delta u = chi(Omega) in B(0, 1), 0 is an element of partial derivative Omega, u = \del u\ = 0 in B(0, 1) \ Omega, where chi(Omega) denotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement of Omega does not develop cusp singularities at the origin then we prove partial derivative Omega is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.
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| 5. |
- Caffarelli, L., et al.
(författare)
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Free-boundary regularity for a problem arising in superconductivity
- 2004
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer Science and Business Media LLC. - 0003-9527 .- 1432-0673. ; 171:1, s. 115-128
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Tidskriftsartikel (refereegranskat)abstract
- This paper concerns regularity properties of the mean-field theory of superconductivity. The problem is reminiscent of the one studied earlier by L.A. Caffarelli, L. Karp and H. Shahgholian in connection with potential theory. The difficulty introduced in this paper is the existence of several patches, where on each patch the solution to the problem may have different constant values. However, using a refined analysis, we reduce the problem to the one-patch case, at least locally near ''regular'' free boundary points. Using a monotonicity formula, due to Georg S. Weiss, we characterize global solutions of a related equation. Hence earlier regularity results apply and we conclude the C-1 regularity of the free boundary.
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| 6. |
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| 7. |
- Danielli, D., et al.
(författare)
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A singular perturbation problem for the p-Laplace operator
- 2003
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Ingår i: Indiana University Mathematics Journal. - 0022-2518 .- 1943-5258. ; 52:2, s. 457-476
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we initiate the study of the nonlinear one phase singular perturbation problem div(\delu(epsilon)\(p-2)delu(epsilon)) = beta(epsilon)(u(epsilon)), (1 < p < infinity) in a domain Omega of R-N. We prove uniform Lipschitz regularity of uniformly bounded solutions. Once this is done we can pass to the limit to obtain a solution to the stationary case of a combustion problem with a nonlinearity of power type. (The case p = 2 has been considered earlier by several authors.).
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| 8. |
- Hakobyan, A, et al.
(författare)
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A uniqueness result for an overdetermined problem in non-linear parabolic potential theory
- 2004
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Ingår i: Potential Analysis. - 0926-2601 .- 1572-929X. ; 21:4, s. 405-414
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study the question of uniqueness for an inverse problem, arising in the (thermal) linear and/or non-linear potential theory. The overdetermined problem we shall study is represented by (div(\delu\(p-2)delu) - D(t)u - chi(Omega) + mu)u = 0, where supp(mu) subset of Omega subset of R-n x (0,infinity), 1 < p < infinity, mu is an element of L-infinity, and Omega boolean AND {t = tau} is bounded for tau > 0. The problem has applications in shape-recognition in underground water/oil recovery, subject to shape-change during time intervals. The particular case u greater than or equal to 0, D(t)u greater than or equal to 0, and p = 2, is an example of the well-known Stefan.
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| 9. |
- Henrot, A., et al.
(författare)
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Existence of classical solutions to a free boundary problem for the p-Laplace operator : (II) The interior convex case
- 2000
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Ingår i: Indiana University Mathematics Journal. - 0022-2518 .- 1943-5258. ; 49:1, s. 311-323
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we prove the existence of convex classical solutions for a Bernoulli-type free boundary problem, in the interior of a convex domain. The governing operator considered is the p-Laplacian. This work is inspired by the pioneering work of A. Beurling where he proves the existence for the harmonic case in the plane, using the notion of sub- and super-solutions in a geometrical sense.
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| 10. |
- Henrot, A., et al.
(författare)
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Existence of classical solutions to a free boundary problem for the p-Laplace operator : (I) the exterior convex case
- 2000
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Ingår i: Journal für die Reine und Angewandte Mathematik. - 0075-4102 .- 1435-5345. ; 521, s. 85-97
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we prove, under convexity assumptions for the data, the existence of classical solutions for a Bernoulli-type free boundary problem, with the p-Laplacian as the governing operator. The method employed here originates from a pioneering work of A. Beurling where he proves the existence for the harmonic case in the plane, though with no geometrical restrictions.
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