Reseach Statement

Reseach Statement
KiHyun Yun
kyun@math.msu.edu
Research Statement
http://www.math.msu.edu/∼ kyun
Overview
My principal interest lies in studies strongly stimulated by practical applications. I am particularly attracted
to interdisciplinary projects which require knowledge of many fields in order to produce results that can be
applied in real life. I have endeavored to obtain meaningful information from mathematical models motivated
by the increased importance of interdisciplinary research. My current research can be summarized in three
themes as follows:
• Identification of the unknown coefficients in partial differential equations:
The main portion of my research involves in the inverse problems to identify unknown coefficients in
partial differential equations via boundary measurements. Collaborating with Professor Gang Bao, I
recently have obtained an improved stability of an inverse wave problem in [1] and continue my efforts
to attack another crucial inverse wave problem to identify the unknown wave speed coefficient. During
my Ph.D. period, several remarkable results on various inverse problems could also be obtained under
the supervision of Professor Hyeonbae Kang in [2, 3, 4].
• High stress concentration occurring in stiff fiber reinforced materials:
The electric field increases toward infinity in the narrow region between closely adjacent perfect conductors as they approach each other. Much attention has been devoted to the blow-up estimate, especially
in two dimensions, for the practical relevance to high stress concentration in fiber-reinforced elastic
composites. In my recent papers [5, 6], I have established optimal estimates for the electric field in
terms of the distance between a pair of conductors in two dimensions. More recently, in cooperation
with Professor Mikyoung Lim [7, 8], two advanced estimates have been obtained to explicitly describe
what role the geometric parameters of conductors play in blow-up.
• Evolution of the grain boundaries in three dimensions:
In work with Professor David Kinderlehrer, the primary investigations concern mathematical issues of
interfaces in polycrystals, a subject of great historical importance. The main objective is to show the
existence of the solution to the mathematical system to describe the evolution of granular network in
three dimensions. We have established a short time existence of the solution in three dimensions under
the assumption that the system is initially close to the equilibrium configuration in [9].
In what follows, I will describe each of these areas and also my future plans in details.
1
Identification of the unknown coefficients in partial differential
equations via boundary measurements
The inverse problems to identify unknown objects by boundary measurements have several potential applications such as reconstruction of the interior of the human body from exterior electrical and magnetic
measurements, mine and rock detection, the search for underground water from exterior measurements in
geophysics and the locating of flying objects from their acoustic or electromagnetic fields.
These works related to inverse problems are the longest continuous subjects that I have been working
since my graduate study at Seoul National University. These research topics are not only interesting to me
but have also provided me with valuable background that I need as a mathematician.
1
KiHyun Yun
Research Statement
http://www.math.msu.edu/∼ kyun
kyun@math.msu.edu
In studying these topics for a long time, I have been exposed to a wide range of inverse problems. This
section focuses on the recent results and the ongoing research projects among my research findings on inverse
problems.
1.1
Stabilities of inverse problems for wave equations
In my recent work with Professor Gang Bao [1], I have considered the following hyperbolic problem


(x, t) ∈ Ω × (0, T )

utt − ∆u + qu = 0
u = ut = 0


 ∂u = f
∂ν
(1.1)
x ∈ Ω, t = 0
(x, t) ∈ ∂Ω × (0, T )
Here, the potential function q = q(x) above is unknown. The Neumann to Dirichlet map Λq is associated
¯
with (1.1) as Λq : f → u¯Γ .
This work is mainly concerned with a stability of an inverse problem inverting the map q 7→ Λq , which was
initiated by Rakesh and Symes. In their paper [10], the technique inspired by Sylverster and Uhlmann [11]
was used to prove the uniqueness theorem that q can be uniquely determined by Λq . Furthermore, Sun [12]
modified their method to get a Hölder continuity with exponent 1/3 − ² for the inverse problem as follows:
For any small ² > 0, there is β0 > 0 such that
1
kq1 − q2 kL2 (Ω) ≤ CkΛq1 − Λq2 k∗3
−²
when kq1 − q2 kH β (Rn ) ≤ M for some β > β0 . Here, the norm k · k∗ represents the operator norm k ·
kB(L2 (Γ),H 1 (Γ)) and H β (Rn ) is the Sobolev Space of order β.
In our work [1], an effective way to use the same condition as Sun has been obtained, which has been
motivated by the optimal mass transport. Based on this, a Hölder stability with exponent 1 − ² is established
under essentially the same assumption as Sun [12]: For any small ² > 0, there is β0 > 0 such that
kq1 − q2 kL2 (Ω) ≤ CkΛq1 − Λq2 k1−²
∗
when kq1 − q2 kH β (Rn ) ≤ M for some β > β0 .
After a successful completion of the work above, I have started to investigate another inverse problem to
determine an unknown coefficient ρ(x) in the following hyperbolic problem
ρutt − ∆u = 0 (x, t) ∈ Ω × (0, T ).
In this work, the main objective is to establish the stability estimate for the unknown coefficient ρ(x) by
the Dirichlet to Neumann map. Some initial progress has recently been made in stability by Stefanov and
Uhlmann.
In both of the inverse problems above, the unknown Fourier transformed coefficients pb(η) and ρb(η) have
been considered. The previous studies provided the good estimates for pb(η) and ρb(η) only in a bounded
area |η| ¿ ∞. This is the reason why we could not have the Lipschitz type stability. Therefore, my current
investigation concerns the behavior of pb(η) and ρb(η) near infinity to make improvement.
During the visiting period at Michigan State University, I have developed a strong interest in the inverse
scattering problems. There are many interesting questions that remain open about these subjects with
significant applications. Another current research direction is to study the dependency of the solution with
respect to high wave number in a Helmholtz type problem with applications to EM cavities. I intend to
continue attacking those crucial questions in inverse problems and the related subjects.
2
KiHyun Yun
Research Statement
http://www.math.msu.edu/∼ kyun
kyun@math.msu.edu
1.2
Early works on inverse problems for elliptic equations
This subsection provides brief descriptions of my early works I have done under the supervision of Professor
Hyeonbae Kang during my Ph.D. period.
Identification of sound hard obstacles in scattering problems
In this work, I have solved a challenging problem in the inverse scattering problem. The interest of this
subject lies in the detection of an object (an obstacle D) from the pattern (or the scattering amplitude) of
the field generated by (plane) incident waves of frequency k. The field in the simplest situation of scattering
by an obstacle D is a solution to a Helmholtz equation with the homogeneous Dirichlet (or Neumann)
boundary data a sound-soft obstacle (or a sound-hard obstacle) respectively. For details, refer to [14].
In the case of sound-soft obstacles, owing to Colton’s theorem in [15], C. Liu proved the uniqueness
within the class of sound-soft balls by a scattering amplitude corresponding to a single incident direction
and one fixed wave number k in [16].
In the case of sound-hard obstacles, the uniqueness within the class of sound-hard balls had been a
challenging problem for a long time. However, after twenty four years from Colton [15] and ten years from
Liu [16], I have established the analogue of Colton for Neumann data. This immediately implies the desirable
uniqueness within the class of sound-hard balls by a scattering amplitude corresponding to a single incident
direction and one fixed wave number k in [2]. This remarkable result was cited by R. Kress in [17] et al.
Identification of a source term in elliptic equations
Collaborating with Professor Hyeonbae Kang and Doctor Kiwoon Kwon, I have studied identification of a
source term in elliptic equations appearing in the determination of the metal-to-semiconductor contact and
its resistivity of electric devices. Current density g is applied to a side of semiconductor layer Ω (called the
diffusion layer) and the corresponding electric potential u satisfies an elliptic equation
4u − pχ(D)u = 0
in Ω
Rs
and ∂u
∂ν = g on ∂Ω where p = ρc (Rs : the sheet resistance of the semiconductor layer and ρc : the contact
resistivity), D is the metal-to-semiconductor contact, ν is the unit outward normal to ∂Ω. There have been
several studies of this inverse problem (e.g. [18, 19]).
We have proved a uniqueness of D within the class of two- or three-dimensional balls from a single
boundary measurement in [3]. This result have laid the foundation for further research by S. Kim and M.
Yamamoto [20, 21].
Boundary determination of anisotropic conductivities
The purpose of inverse conductivity problems is mainly to identify an interior conductivity or to detect
inclusions in a conductor from the Dirichlet-to-Neumann map (or the voltage-to-current map). Those problems are strongly motivated by electrical impedance tomography, which are new and promising methods to
prospect the interior of the human body by surface electromagnetic measurements.
In cooperation with Professor Hyeonbae Kang, I established an explicit reconstruction of boundary
values of less regular anisotropic (possibly non-analytic) conductivities and their derivatives in [4], while the
previous results are associated with the analytic regularity. This subject had been brought by J. Lee and G.
Uhlmann, who proved the uniqueness in identifying an anisotropic conductivity for three dimension under
3
KiHyun Yun
kyun@math.msu.edu
Research Statement
http://www.math.msu.edu/∼ kyun
some restrictions when conductivities are real analytic in a domain (see [23]). Moreover, M. Lassas and
G. Uhlmann [22] extended the result to the case when g is real analytic up to a portion of the boundary
and removed some restrictions of [23]. Furthermore, we have obtained the improved identification and
reconstruction results with the relaxed regularity restrictions by a new method motivated by G. Nakamura
and K. Tanuma [24].
2
High stress concentration in fiber reinforced materials
In fiber-reinforced compositions, even though stiff fibers are present for reinforcement, the stress blows up
as the fibers approximate to each other. These high shear stress concentrations cause low strengths of
fiber-reinforced composites. Much attention has been devoted to the blow-up estimate for stress due to the
practical significance.
In my recent work [5, 6], I have established the optimal growth rate on the stresses occurring between a
pair of closely spaced stiff fibers with arbitrary shaped cross-sections. Furthermore, in the collaboration with
Lim, two significant improvements [7, 8] also have been made to describe explicitly what role the geometric
parameters play in the blow-up phenomenon. In particular, the estimate presented in [8] reveals a surprising
fact that the high stress concentration can be significantly increased by a small geometrical change of a fiber.
Referring to a model of anti-plane shear, the estimate for stress is reduced to the gradient estimate
for the solution u to conductivity problems as follows: for a given entire harmonic function H in R2 , the
out-of-displacement u is satisfied with

´
o
n³
 ∇· 1+P
(a
−
1)χ(D
)
∇u
=0
i
i
i=1,2
 u(x) − H(x) = O(|x|−1 ) as |x| → ∞
where D1 and D2 are closely spaced inclusions ² apart in R2 , representing the cross-sections of fibers.
For the case that the inclusions have finite conductivities (shear moduli), with the effort of E. Bonnetier
and M. Vogelius [25] as the beginning point, Y.Y. Li and M. Vogelius [26] derived the uniform bound of the
stress that are independent of the distance ². Moreover, this result has been extended to elliptic systems
by Y.Y. Li and L. Nirenberg in [27]. These results point out that the extremely high conductivity (or the
stiffness of fibers) is indispensable to the blow-up phenomena. In this respect, much attention has been
focused on the model of a pair of perfect conductors which are ² apart. H. Ammari, H. Kang, H. Lee, J.
Lee and M. Lim [28, 29] have established the optimal blow-up rate ²−1/2 as the distance ² goes to zero, only
when conductors are of circular shape in two dimensions.
In my recent papers [5, 6], the above mentioned result restricted only to circular inclusions was extended
to a sufficiently general class of the conductors’ shapes in two dimensions. Furthermore, the technique I used
is completely different from the previous one restricted to the case of circular shape. My result has been
extended by S. Bao, Y.Y. Li and B. Yin [30] to higher dimensions. However, their estimates are only given
by the distance between two conductors and geometric information of conductors are not incorporated into
the estimates.
In collaboration with Professor Mikyoung Lim, under the assumption that the perfect conductors
have spherical shapes in three and higher dimensions, the gradient estimates have been established in
terms of the radii as well as the distance ² between inclusions in a way different from Bao et al. [30].
4
KiHyun Yun
kyun@math.msu.edu
Research Statement
http://www.math.msu.edu/∼ kyun
Furthermore, we have recently obtained another crucial result
in [8]. This work focuses on the case when
a smaller fiber is located in-between area of two fibers, see Figure
1. This paper reveals that, with the addition of a smaller fiber,
possibly attached on a fiber, the growth of stress is significantly
increased: if the diameter d of the fiber in the middle is sufficiently small and the distance between adjoining fibers is ², then
the stress blows up at the rate of √1d² in the narrow region, even
though the blow-up rate has been known as √1² as in the case of
a pair of fibers. This means that the defect of fiber as a protru- Figure 1: The electric field increased by the
sion causes much lower strengths in composites than had been small inclusion between a pair of inclusions.
thought. This subject still has many crucial questions which I
shall continue.
3
Evolution of grain boundaries in three dimensions
In work with Professor David Kinderleher, our investigation concerns the evolution of granular network
governed by the Mullins Equations of a curvature driven growth supplemented by the Herring Condition
force balanced by the junctions. In particular, the main challenge is to show the existence of the solution to
the mathematical systems of the evolution. Including triple junctions in rigorous mathematical models had
very limited success [31], even though it has been successfully modeled via simulation [32].
In our recent work [9], we consider the soap bubble with six facets in the tetrahedron as a simplified
model in three dimensions. The motion of each surface is determined by the curvature driven growth, so called
the Mullins Equation:
(Xt · n) − κ = 0 on the facets
with the Herring Condition for the interior angle 2π
3 at the triple junction lines, where
κ is the mean curvature. We have obtained a short time existence of the solution X
satisfying the growth above under the assumption that the system is initially close to
the equilibrium configuration in [9].
In the case of grain boundaries in the plane, L. Bronsard and F. Reitich derived
the short time evolution in [33]. From the viewpoint of methodology, the governing
nonlinear equation of a motion by curvature can be separated into a linear parabolic equation and a higher
order term as a source term as follows: Let X = I + Z where I is the identity function. Then
µ
¶
∂
− ∆ Z = F (DZ, D2 Z)
∂t
with |F | ≤ C|DZ||D2 Z| for small |DX|. The result can be derived by an iteration of solving the linear
equation and substituting the solution for the higher order term repeatedly. Based on Solonnikov-type
estimates, D. Kinderlehrer and C. Liu [34] established the long time evolution by similar argument to [33]
under the assumption that the system is initially closed to some equilibrium.
However, in the case of the soap bubble in three dimensions, the previous straightforward iteration by
Bronsard et.al.
µ
¶
∂
− ∆ Zn+1 = F (DZn , D2 Zn )
∂t
5
KiHyun Yun
kyun@math.msu.edu
Research Statement
http://www.math.msu.edu/∼ kyun
is not available due to the cornered shape of six surfaces. According to Grisvard’s result on the angular
domains [35], a singular term with an unexpectedly low regularity can be often contained in the solution
defined in the angled domains. Thus, a solution Zn+1 with a low regularity may not be available for the
recursive process.
In our work, a new iteration has been established to solve the difficulties occurring in the previous
process. Based on this, a short time existence of X above has also been obtained under the assumption that
the system is initially close to the equilibrium configuration. Moreover, I would like to investigate the long
time existence with the convergence of the surface to the stable bubble.
References
[1] G. Bao and K. Yun, On the stability of an inverse problem for the wave equation, Inverse Problems 25, no 4 (2009), 045003.
[2] K. Yun, The refection of solutions of Helmhotz equation and an applicaion, Comm. Korean Math, Soc., 16, No 3 (2001),
pp. 427–436.
[3] H. Kang, K. Kwon, and K. Yun, Recovery of an inhomogeneity in an elliptic equation, Inverse Problems, 17 (2001), pp.
25–44.
[4] H. Kang and K. Yun, Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann
operator, SIAM Jour. Math. Anal., Vol. 34, No. 3 (2003), pp. 719–735.
[5] K. Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl.
Math., 67, No 3 (2007), pp. 714–730.
[6] K. Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal.
Appl., 350 (2009), pp. 306-312.
[7] M. Lim and K. Yun, Blow-up of Electric Fields between Closely Spaced Spherical Perfect Conductors, Communications in
P.D.E. 34, no 10 (2009), pp. 1287 - 1315.
[8] M. Lim and K. Yun, Strong Influence of a Small Fiber on Shear Stress in Fiber-Reinforced Composites, submitted.
[9] D. Kinderlehrer and K. Yun, An approach to local evolution of cellular networks in higher dimension, in preparation.
[10] Rakesh and W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. P.D.E., 13, No 1 (1988), pp.
87–96.
[11] J. Sylverster and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann of Math., 125
(1987), pp. 153–169.
[12] Z. Sun, On continuous dependence for an inverse initial boundary value problem for the wave equation, J. Math. Anal.
Appl., 150 (1990), pp. 188–204.
[13] R.V. Kohn, H. Shen, M.S. Vogelius, and M.I. Weinstein, Cloaking via change of variables in electric impedance tomography,
Inverse Problems, 24 (2008), 015016 (21pp).
[14] D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences vol 93,
Berlin, Springer, 1992.
[15] D. Colton, A reflection principle for solutions to the Helmholtz equation and an application to the inverse scattering
problems, Glasgow Math J., 18 (1977), pp. 125–130.
[16] C. Liu, An inverse obstacle problem: a uniqueness theorem for balls, inverse problems in wave propagations, IMA volume
in Math. and its Appl., 90 (1991), pp. 347–355.
[17] R. Kress, Uniqueness in inverse obstacle scattering for electromagnetic waves, Proceedings of the URSI General Assembly
2002, Maastricht.
[18] W. Fang and E. Cumberbatch, Inverse problems for metal oxide semiconductor field-effect transistor contact resistivity,
SIAM J. Appl. Math., 52 (1992), pp. 699–709.
[19] F. Hettlich and W. Rundell, Recovery of the support of a source term in an elliptic differential equation, Inverse Problems,
13 (1997), pp. 959–976.
[20] S. Kim, Recovery of an unknown support of a source term in an elliptic equation, Inverse Problems, 20 no. 2 (2004), pp
565–574.
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KiHyun Yun
kyun@math.msu.edu
Research Statement
http://www.math.msu.edu/∼ kyun
[21] S. Kim and M. Yamamoto, Uniqueness in identification of the support of a source term in an elliptic equation, SIAM J.
Math. Anal., 35, No. 1 (2003), pp. 148–159.
[22] M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. Ecole
Norm. Sup. (4), 34 (2001), pp. 771–787.
[23] J. Lee and G. Uhlmann, Determining anistropic real-analytic conductivities by boundary measurements, Comm. Pure
Appl. Math., 42 (1989), pp. 1097–1112.
[24] G. Nakamura and K. Tanuma, Local determination of conductivity at the boundary from Dirichlet to Neumann map,
Inverse Problems, 17 (2001), pp. 405–419.
[25] E. Bonnetier and M. Vogelius, An elliptic regularity result for a composite medium with ”touching” fibers of circular
cross-section, SIAM J. Math. Anal., 31, No 3 (2000), pp. 651–677.
[26] Y.Y. Li and M. Vogelius, Gradient estimates for solution to divergence form elliptic equation with discontinuous coefficients, Arch. Rational Mech. Anal., 153 (2000), pp. 91–151.
[27] Y.Y. Li and M. Nirenberg, Estimates for ellliptic system from composite material, Comm. Pure Appl. Math., LVI (2003),
pp. 892–925.
[28] H. Ammari, H. Kang, and M. Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann., 332(2)
(2005), pp. 277–286.
[29] H. Ammari, H. Kang, H. Lee, J. Lee, and M. Lim, Optimal bounds on the gradient of solutions to conductivity problems,
J. Math. Pures Appl., 88 (2007), pp. 307–324.
[30] E. Bao, Y.Y. Li and B. Yin, Gredient estimats for the conductivity problem, Arch. Rational Mech. Anal, to appear.
[31] J. E. Taylor, Mathematical models of triple junctions, Interface Science, 7 (1999), pp. 243–250
[32] D. Kinderlehrer, I., Livshits, G.S. Rohrer, S. Ta’asan, and P. Yu, Mesoscale evolution of the grain boundary character
distribution, Recrystallization and Grain Growth, Materials Science Forum, Vols 467-470, (2004), pp 1063–1068
[33] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau
equation, Arch. Rational Mech. Anal. 124 (1993), no. 4, pp. 355–379.
[34] D. Kinderlehrer and C. Liu, Evolution of grain boundaries, Math. Models Methods Appl. Sci., 11, No. 4 (2001), pp.
713–729.
[35] P. Grisvard, Elliptic problems in nonsmooth domains., Monographs and Studies in Mathematics, 24, Pitman (Advanced
Publishing Program), Boston, MA, 1985.
7
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