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Two-and multi-phase quadrature surfaces
1. | Institute of Mathematics National Academy of Sciences of Armenia, 0019 Yerevan, Armenia |
2. | Department of Mathematics Royal Institute of Technology, 100 44 Stockholm, Sweden |
3. | Department of Mathematics, University of Mumbai Vidyanagari, Santacruz (east), 400 097 Mumbai, India |
$\int_{\partial Ω^+} g h (x) \ dσ_x - \int_{\partial Ω^-} g h (x) \ dσ_x= \int h dμ \ ,$ |
$dσ_x$ |
$μ= μ^+ - μ^-$ |
$Ω=Ω^+\cupΩ^-$ |
$g$ |
$h$ |
$\overline Ω$ |
References:
[1] |
H. W. Alt, L. A. Caffarelli and A. Friedman,
Variational problems with two phases and their free boundaries, Transactions of the American Mathematical Society, 282 (1984), 431-461.
doi: 10.2307/1999245. |
[2] |
H. Alt and L. A. Caffarelli,
Existence and regularity for a minimum problem with free boundary., J. Reine Angew. Math., 325 (1981), 105-144.
|
[3] |
J. Andersson, H. Shahgholian and G. S. Weiss, In preparation. Google Scholar |
[4] |
A. Arakelyan and H. Shahgholian,
Multi-phase quadrature domains and a related minimization problem, Potential Analysis, 45 (2016), 135-155.
doi: 10.1007/s11118-016-9539-0. |
[5] |
F. Bahrami and A. Chademan,
Existence of unbounded quadrature domains for the p-laplace operator, Bulletin of Iranian Mathematical Society, 24 (1998), 1-13.
|
[6] |
D. Bucur and B. Velichkov,
Multiphase shape optimization problems, SIAM Journal on Control and Optimization, 52 (2014), 3556-3591.
doi: 10.1137/130917272. |
[7] |
L. A. Caffarelli, D. Jerison and C. E. Kenig,
Some new monotonicity theorems with applications to free boundary problems, Annals of Mathematics, 155 (2002), 369-404.
doi: 10.2307/3062121. |
[8] |
M. Conti, S. Terracini and G. Verzini,
A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J., 54 (2005), 779-815.
doi: 10.1512/iumj.2005.54.2506. |
[9] |
G. David, M. Filoche, D. Jerison and S. Mayboroda, A free boundary problem for the localization of eigenfunctions, arXiv preprint arXiv: 1406.6596. Google Scholar |
[10] |
P. J. Davis,
The Schwarz Function and Its Applications Carus mathematical monographs, Mathematical Association of America, 1974, |
[11] |
B. Emamizadeh, J. V. Prajapat and H. Shahgholian,
A two phase free boundary problem related to quadrature domains, Potential Analysis, 34 (2011), 119-138.
doi: 10.1007/s11118-010-9184-y. |
[12] |
A. Friedman and D. Phillips,
The free boundary of a semilinear elliptic equation, Transactions of the American Mathematical Society, 282 (1984), 153-182.
doi: 10.2307/1999583. |
[13] |
B. Gustafsson and H. Shahgholian,
Existence and geometric properties of solutions of a free boundary problem in potential theory, J. Reine Angew. Math., 473 (1996), 137-179.
|
[14] |
L. Hauswirth, F. Hélein and F. Pacard,
On an overdetermined elliptic problem, Pacific Journal of Mathematics, 250 (2011), 319-334.
doi: 10.2140/pjm.2011.250.319. |
[15] |
A. Henrot,
Subsolutions and supersolutions in a free boundary problem, Arkiv för Matematik, 32 (1994), 79-98.
doi: 10.1007/BF02559524. |
[16] |
L. Karp,
On null quadrature domains, Computational Methods and Function Theory, 8 (2008), 57-72.
doi: 10.1007/BF03321670. |
[17] |
J. L. Lewis and A. Vogel,
On pseudospheres, Rev. Mat. Iberoamericana, 7 (1991), 25-54.
doi: 10.4171/RMI/104. |
[18] |
J. Mossino,
Inögalitös isopörimötriques et applications en physique vol. 2, Editions Hermann, 1984. |
[19] |
M. Onodera,
Geometric flows for quadrature identities, Mathematische Annalen, 361 (2015), 77-106.
doi: 10.1007/s00208-014-1062-2. |
[20] |
M. Sakai,
Quadrature Domains vol. 934 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1982. |
[21] |
H. Shahgholian,
Existence of quadrature surfaces for positive measures with finite support, Potential Analysis, 3 (1994), 245-255.
doi: 10.1007/BF01053435. |
[22] |
H. Shahgholian,
Quadrature surfaces as free boundaries, Arkiv för Matematik, 32 (1994), 475-492.
doi: 10.1007/BF02559582. |
[23] |
M. Traizet,
Classification of the solutions to an overdetermined elliptic problem in the plane, Geometric and Functional Analysis, 24 (2014), 690-720.
doi: 10.1007/s00039-014-0268-5. |
[24] |
B. Velichkov, A note on the monotonicity formula of caffarelli-jerison-kenig Preprint available at: http://cvgmt.sns.it/paper/2266.
doi: 10.4171/RLM/673. |
show all references
References:
[1] |
H. W. Alt, L. A. Caffarelli and A. Friedman,
Variational problems with two phases and their free boundaries, Transactions of the American Mathematical Society, 282 (1984), 431-461.
doi: 10.2307/1999245. |
[2] |
H. Alt and L. A. Caffarelli,
Existence and regularity for a minimum problem with free boundary., J. Reine Angew. Math., 325 (1981), 105-144.
|
[3] |
J. Andersson, H. Shahgholian and G. S. Weiss, In preparation. Google Scholar |
[4] |
A. Arakelyan and H. Shahgholian,
Multi-phase quadrature domains and a related minimization problem, Potential Analysis, 45 (2016), 135-155.
doi: 10.1007/s11118-016-9539-0. |
[5] |
F. Bahrami and A. Chademan,
Existence of unbounded quadrature domains for the p-laplace operator, Bulletin of Iranian Mathematical Society, 24 (1998), 1-13.
|
[6] |
D. Bucur and B. Velichkov,
Multiphase shape optimization problems, SIAM Journal on Control and Optimization, 52 (2014), 3556-3591.
doi: 10.1137/130917272. |
[7] |
L. A. Caffarelli, D. Jerison and C. E. Kenig,
Some new monotonicity theorems with applications to free boundary problems, Annals of Mathematics, 155 (2002), 369-404.
doi: 10.2307/3062121. |
[8] |
M. Conti, S. Terracini and G. Verzini,
A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J., 54 (2005), 779-815.
doi: 10.1512/iumj.2005.54.2506. |
[9] |
G. David, M. Filoche, D. Jerison and S. Mayboroda, A free boundary problem for the localization of eigenfunctions, arXiv preprint arXiv: 1406.6596. Google Scholar |
[10] |
P. J. Davis,
The Schwarz Function and Its Applications Carus mathematical monographs, Mathematical Association of America, 1974, |
[11] |
B. Emamizadeh, J. V. Prajapat and H. Shahgholian,
A two phase free boundary problem related to quadrature domains, Potential Analysis, 34 (2011), 119-138.
doi: 10.1007/s11118-010-9184-y. |
[12] |
A. Friedman and D. Phillips,
The free boundary of a semilinear elliptic equation, Transactions of the American Mathematical Society, 282 (1984), 153-182.
doi: 10.2307/1999583. |
[13] |
B. Gustafsson and H. Shahgholian,
Existence and geometric properties of solutions of a free boundary problem in potential theory, J. Reine Angew. Math., 473 (1996), 137-179.
|
[14] |
L. Hauswirth, F. Hélein and F. Pacard,
On an overdetermined elliptic problem, Pacific Journal of Mathematics, 250 (2011), 319-334.
doi: 10.2140/pjm.2011.250.319. |
[15] |
A. Henrot,
Subsolutions and supersolutions in a free boundary problem, Arkiv för Matematik, 32 (1994), 79-98.
doi: 10.1007/BF02559524. |
[16] |
L. Karp,
On null quadrature domains, Computational Methods and Function Theory, 8 (2008), 57-72.
doi: 10.1007/BF03321670. |
[17] |
J. L. Lewis and A. Vogel,
On pseudospheres, Rev. Mat. Iberoamericana, 7 (1991), 25-54.
doi: 10.4171/RMI/104. |
[18] |
J. Mossino,
Inögalitös isopörimötriques et applications en physique vol. 2, Editions Hermann, 1984. |
[19] |
M. Onodera,
Geometric flows for quadrature identities, Mathematische Annalen, 361 (2015), 77-106.
doi: 10.1007/s00208-014-1062-2. |
[20] |
M. Sakai,
Quadrature Domains vol. 934 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1982. |
[21] |
H. Shahgholian,
Existence of quadrature surfaces for positive measures with finite support, Potential Analysis, 3 (1994), 245-255.
doi: 10.1007/BF01053435. |
[22] |
H. Shahgholian,
Quadrature surfaces as free boundaries, Arkiv för Matematik, 32 (1994), 475-492.
doi: 10.1007/BF02559582. |
[23] |
M. Traizet,
Classification of the solutions to an overdetermined elliptic problem in the plane, Geometric and Functional Analysis, 24 (2014), 690-720.
doi: 10.1007/s00039-014-0268-5. |
[24] |
B. Velichkov, A note on the monotonicity formula of caffarelli-jerison-kenig Preprint available at: http://cvgmt.sns.it/paper/2266.
doi: 10.4171/RLM/673. |
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