Sunhi Choi

Interests

Teaching

mathematics courses in all levels

Research

partial differential equations, free boundary problems, fully nonlinear operators

Courses

Scholarly Contributions

Chapters

• Choi, S., & Kim, I. (2013). Homogenization with oscillatory Neumann boundary data in general domain. In Geometric partial differential equations(pp 105-118). Edizioni della Normale, Pisa. doi:10.1007/978-88-7642-473-1_5
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  In this article we summarize recent progress on understanding averaging properties of fully nonlinear PDEs in bounded domains, when the boundary data is oscillatory. Our result on the Neumann problem is the nonlinear version of the classical result in [4] for divergence-form operators with co-normal boundary data. We also discuss the Dirichlet boundary problem.

Journals/Publications

• Choi, S., & Kim, I. (2023). Homogenization of oblique boundary value problems. Advanced Nonlinear Studies, a Special Issue on D. Jerison's 70th Birthday, 23, 20220051. doi:https://doi.org/10.1515/ans-2022-0051
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  We consider a nonlinear Neumann problem, with periodic oscillation in the elliptic operator and on the boundary condition. Our focus is on problems posed in half-spaces, but with general normal directions that may not be parallel to the directions of periodicity. As the frequency of the oscillation grows, quantitative homogenization results are derived. When the homogenized operator is rotation-invariant, we prove the H\"{o}lder continuity of the homogenized boundary data.
• Choi, S., & Kim, I. C. (2014). Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data.. J. Math. Pures Appl., 102(2), 419-448.
• Choi, S., & Kim, I. (2013). Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data. Analysis and PDE, 951-972. doi:10.48550/arxiv.1302.5386
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  In this article we investigate averaging properties of fully nonlinear PDEs in bounded domains with oscillatory Neumann boundary data. The oscillation is periodic and is present both in the operator and in the Neumann data. Our main result states that, when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant, the solutions uniformly converge to the homogenized solution solving a Neumann boundary problem. Furthermore we show that the homogenized Neumann data is continuous with respect to the normal direction of the boundary.
• Choi, S., & Kim, I. (2012). The two-phase stefan problem: Regularization near lipschitz initial data by phase dynamics. Analysis and PDE, 5(5), 1063-1103.
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  Abstract: In this paper we investigate the regularizing behavior of two-phase Stefan problem near initial Lipschitz data. A description of the regularizing phenomena is given in termsof the corresponding space-time scale. © 2012 by Mathematical Sciences Publishers.
• Choi, S., & Kim, I. C. (2010). Regularity of one-phase stefan problem near lipschitz initial data. American Journal of Mathematics, 132(6), 1693-1727.
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  Abstract: In this paper we show that, starting from Lipschitz initial free boundary with small Lipschitz constant, the solution of the one-phase Stefan problem (ST) immediately regularizes and is smooth in space and time, for a small positive time. ©2010 by The JohnsHopkinsUniversity Press.
• Choi, S. (2009). Regularity near a contact point for flame propagation. Communications in Partial Differential Equations, 34(5), 457-474.
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  Abstract: In this paper we study a free boundary problem, arising from a model for the propagation of laminar flames. Consider a cylindrical region S in IRn, and the following free boundary problem with Dirichlet data on ∂S: ut=Δu in {u>0} ∩S, |Δu|=1 on ∂ {u>0} ∩S and u=0 on ∂S. We show that if there is a contact point of the free boundary {u=0, |Δu|=1} with ∂S, then the free boundary approaches ∂S tangentially and it turns out to be a graph of C1+α,α function near the contact point. In particular, the space normal is Hölder continuous. © Taylor & Francis Group, LLC.
• Choi, S., Jerison, D., & Inwon, K. I. (2009). Local regularization of the one-phase hele-shaw flow. Indiana University Mathematics Journal, 58(6), 2765-2804.
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  Abstract: This article presents a local regularity theorem for the one-phase Hele-Shaw flow. We prove that if the Lipschitz constant of the initial free boundary in a unit ball is small, then for small uniform positive time the solution is smooth. This result improves on our earlier results in [4] because it is scaleinvariant. As a consequence we obtain existence, uniqueness and regularity properties of global solutions with Lipschitz initial free boundary. © Indiana University Mathematics Journal.
• Choi, S., Jerison, D., & Kim, I. (2009). Locating the first nodal set in higher dimensions. Transactions of the American Mathematical Society, 361(10), 5111-5137.
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  Abstract: We extend the two-dimensional results of Jerison (2000) on the location of the nodal set of the first Neumann eigenfunction of a convex domain to higher dimensions. If a convex domain Ω in ℝn is contained in a long and thin cylinder [0, N] × B∈(0) with nonempty intersections with {x1 = 0} and {x1 = N}, then the first nonzero eigenvalue is well approximated by the eigenvalue of an ordinary differential equation, by a bound proportional to ∈, whose coefficients are expressed in terms of the volume of the cross sections of the domain. Also, the first nodal set is located within a distance comparable to ∈ near the zero of the corresponding ordinary differential equation. © 2009 American Mathematical Society.
• Choi, S., Kim, I., & Jerison, D. (2009). Local regularization of the one-phase Hele-Shaw flow. Indiana University Mathematics Journal. doi:10.1512/iumj.2009.58.3802
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  This article presents a local regularity theorem for the one-phase Hele-Shaw flow. We prove that if the Lipschitz constant of the initial free boundary in a unit ball is small, then for small uniform positive time the solution is smooth. This result improves on our earlier results in (4) because it is scale- invariant. As a consequence we obtain existence, uniqueness and regularity properties of global solutions with Lipschitz ini- tial free boundary.
• Choi, S., Jerison, D., & Kim, I. (2007). Regularity for the one-phase hele-shaw problem from a lipschitz initial surface. American Journal of Mathematics, 129(2), 527-582.
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  Abstract: In this paper we show that if the Lipschitz constant of the initial free boundary is small, then for small positive time the solution is smooth and satisfies the Hele-Shaw equation in the classical sense. A key ingredient in the proof which is of independent interest is an estimate up to order of magnitude of the speed of the free boundary in terms of initial data. © 2007 by The Johns Hopkins University Press.
• Choi, S., & Kim, I. (2006). Waiting time phenomena of the Hele-Shaw and the Stefan problem. Indiana University Mathematics Journal, 55(2), 525-551.
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  Abstract: In this paper we investigate the waiting time phenomena for the one-phase Hele-Shaw and Stefan problems. For the Hele-Shaw problem we identify a general criterion on the growth rate of the initial data, which determines the occurrence of a waiting time. For the Stefan problem we show that the waiting time phenomena depend on the balance between the initial data and the geometry of the initial positive phase. Indiana University Mathematics Journal ©.
• Choi, S. (2004). The lower density conjecture for harmonic measure. Journal d'Analyse Mathematique, 93, 237-269.
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  Abstract: In this paper, we establish the lower density conjecture for harmonic measure in simply connected plane domains.