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November  2017, 16(6): 2023-2045. doi: 10.3934/cpaa.2017099

Two-and multi-phase quadrature surfaces

Avetik Arakelyan 1, , Henrik Shahgholian 2,, and  Jyotshana V. Prajapat 3,

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

* Corresponding author

Received  September 2016 Revised  December 2016 Published  July 2017

Fund Project: A. Arakelyan was supported by State Committee of Science MES RA, in frame of the research project No. 16YR-1A017. H. Shahgholian is partially supported by the Swedish Research Council.

In this paper we shall initiate the study of the two-and multi-phase quadrature surfaces (QS), which amounts to a two/multi-phase free boundary problems of Bernoulli type. The problem is studied mostly from a potential theoretic point of view that (for two-phase case) relates to integral representation
$\int_{\partial Ω^+} g h (x) \ dσ_x - \int_{\partial Ω^-} g h (x) \ dσ_x= \int h dμ \ ,$
where
$dσ_x$
is the surface measure,
$μ= μ^+ - μ^-$
is given measure with support in (a priori unknown domain)
$Ω=Ω^+\cupΩ^-$
,
$g$
is a given smooth positive function, and the integral holds for all functions
$h$
, which are harmonic on
$\overline Ω$
.
Our approach is based on minimization of the corresponding two-and multi-phase functional and the use of its one-phase version as a barrier. We prove several results concerning existence, qualitative behavior, and regularity theory for solutions. A central result in our study states that three or more junction points do not appear.
Keywords: Two-phase quadrature surface, free boundary, Bernoulli boundary conditio.
Mathematics Subject Classification: Primary: 35R35, 31A05, 31B05, 31B20.
Citation: Avetik Arakelyan, Henrik Shahgholian, Jyotshana V. Prajapat. Two-and multi-phase quadrature surfaces. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2023-2045. doi: 10.3934/cpaa.2017099
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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

D. Bucur and B. Velichkov, Multiphase shape optimization problems, SIAM Journal on Control and Optimization, 52 (2014), 3556-3591.  doi: 10.1137/130917272.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

A. Henrot, Subsolutions and supersolutions in a free boundary problem, Arkiv för Matematik, 32 (1994), 79-98.  doi: 10.1007/BF02559524.  Google Scholar

[16]

L. Karp, On null quadrature domains, Computational Methods and Function Theory, 8 (2008), 57-72.  doi: 10.1007/BF03321670.  Google Scholar

[17]

J. L. Lewis and A. Vogel, On pseudospheres, Rev. Mat. Iberoamericana, 7 (1991), 25-54.  doi: 10.4171/RMI/104.  Google Scholar

[18]

J. Mossino, Inögalitös isopörimötriques et applications en physique vol. 2, Editions Hermann, 1984.  Google Scholar

[19]

M. Onodera, Geometric flows for quadrature identities, Mathematische Annalen, 361 (2015), 77-106.  doi: 10.1007/s00208-014-1062-2.  Google Scholar

[20]

M. Sakai, Quadrature Domains vol. 934 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1982.  Google Scholar

[21]

H. Shahgholian, Existence of quadrature surfaces for positive measures with finite support, Potential Analysis, 3 (1994), 245-255.  doi: 10.1007/BF01053435.  Google Scholar

[22]

H. Shahgholian, Quadrature surfaces as free boundaries, Arkiv för Matematik, 32 (1994), 475-492.  doi: 10.1007/BF02559582.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

D. Bucur and B. Velichkov, Multiphase shape optimization problems, SIAM Journal on Control and Optimization, 52 (2014), 3556-3591.  doi: 10.1137/130917272.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

A. Henrot, Subsolutions and supersolutions in a free boundary problem, Arkiv för Matematik, 32 (1994), 79-98.  doi: 10.1007/BF02559524.  Google Scholar

[16]

L. Karp, On null quadrature domains, Computational Methods and Function Theory, 8 (2008), 57-72.  doi: 10.1007/BF03321670.  Google Scholar

[17]

J. L. Lewis and A. Vogel, On pseudospheres, Rev. Mat. Iberoamericana, 7 (1991), 25-54.  doi: 10.4171/RMI/104.  Google Scholar

[18]

J. Mossino, Inögalitös isopörimötriques et applications en physique vol. 2, Editions Hermann, 1984.  Google Scholar

[19]

M. Onodera, Geometric flows for quadrature identities, Mathematische Annalen, 361 (2015), 77-106.  doi: 10.1007/s00208-014-1062-2.  Google Scholar

[20]

M. Sakai, Quadrature Domains vol. 934 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1982.  Google Scholar

[21]

H. Shahgholian, Existence of quadrature surfaces for positive measures with finite support, Potential Analysis, 3 (1994), 245-255.  doi: 10.1007/BF01053435.  Google Scholar

[22]

H. Shahgholian, Quadrature surfaces as free boundaries, Arkiv för Matematik, 32 (1994), 475-492.  doi: 10.1007/BF02559582.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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