# American Institute of Mathematical Sciences

August  2022, 21(8): 2679-2700. doi: 10.3934/cpaa.2022068

## On the solvability of a semilinear higher-order elliptic problem for the vector field method in image registration

 1 Elementary Education School, Hainan Normal University, Haikou 571158, China 2 School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, China

* Corresponding author

Received  August 2021 Revised  February 2022 Published  August 2022 Early access  March 2022

Fund Project: The first author is supported by NSF of Hainan Province of China (Grant No.118QN023)

We study the existence of the solution to a semilinear higher-order elliptic system
 $\mathcal{L}v(t, \cdot) = F_{S, T}\circ G(v)(t, \cdot), \quad \forall t\in [0, \tau],$
with the homogeneous Dirichlet boundary conditions. Here,
 $\mathcal{L} = (-\Delta)^m$
is a harmonic operator of order
 $m$
,
 $v = v(t, x):[0, \tau]\times\Omega\rightarrow \mathbb{R}^n$
is the unknown,
 $t$
is a parameter,
 $F_{S, T}$
is a function related to given functions
 $S$
and
 $T$
, and
 $G(v)(t, x)$
is defined by the solution
 $y^v(s;t, x)$
of an ODE-IVP
 ${\rm d}y/\mathrm{d}s = v(s, y), \quad y(t) = x$
. The elliptic equations is the Euler-Lagrange equation of the vector field regularization model widely used in image registration. Although we have showed the existence of a solution to this BVP by the variational method, we hope to study it by the fixed point method further. This is mainly because the elliptic equations is novel in form, and the method here put more emphasis on the quantitative analysis whereas the variational method focus on the qualitative analysis. Since the system here is a higher order semilinear system, and its nonlinear term is dominated by an exponential function with respect to the unknown, we use an exponential inequality to construct a closed ball, and then apply the Schauder fixed point theorem to show the existence of a solution under some assumptions.
Citation: Xiaojun Zheng, Zhongdan Huan, Jun Liu. On the solvability of a semilinear higher-order elliptic problem for the vector field method in image registration. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2679-2700. doi: 10.3934/cpaa.2022068
##### References:
 [1] S. Arguill$\grave{e}$re and E. Tr$\acute{e}$lat, Sub-Riemannian structures on groups of diffeomorphisms, J. Inst. Math. Jussieu, 2014. [2] V.I. Arnold, Ordinary Differerntial Equations, Springer-Verlag, Berlin, 1992. [3] A. Bahrouni, Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity, Commun. Pure Appl. Anal., 16 (2017), 243-252.  doi: 10.3934/cpaa.2017011. [4] J. Bao, N. Lam and G. Lu, Polyharmonic equations with critical exponential growth in the whole space $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 36 (2016), 577-600.  doi: 10.3934/dcds.2016.36.577. [5] M.F. Beg, M.I. Miller, A. Trouv$\acute{e}$ and L. Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Int. J. Comput. Vision, 61 (2005), 139-157. [6] L.G. Brown, A survey of image registration techniques, ACM Comput. Surv., 24 (1992), 325-376. [7] L. Chen, J. Li, G. Lu and C. Zhang, Sharpened Adams inequlity and ground state solutions to the bi-Laplacian equation in $\mathbb{R}^4$, Adv. Nonlinear Stud., 18 (2018), 429-452.  doi: 10.1515/ans-2018-2020. [8] M. Chuaqui, C. Cortázar, M. Elgueta and J. García-Melián, Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights, Commun. Pure Appl. Anal., 3 (2004), 653-662.  doi: 10.3934/cpaa.2004.3.653. [9] J. Davila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592.  doi: 10.1137/060665579. [10] W. Du, Existence and uniqueness of large solutions for competitive type elliptic systems, Int. Journal of Math. Analysis, 7 (2013), 2878-2884.  doi: 10.12988/ijma.2013.36159. [11] P. Dupuis, U. Grenander and M.I. Miller, Variational problems on flows of diffeomorphisms for image matching, Q. Appl. Math., 56 (1998), 587-600.  doi: 10.1090/qam/1632326. [12] L. C. Evants, Partial Differential Equations, American Mathematical Society, Providence, 2010. doi: 10.1090/gsm/019. [13] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Springer-Verlag, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9. [14] F. Gazzola, H. C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2001. [16] D. L. Hill, P. G. Batchelor, M. Holden and D. J. Hawkes, Medical image registration, Phys. Med. Biol., 46 (2001), R1–R45. [17] A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385.  doi: 10.1016/j.jmaa.2013.12.059. [18] A. Klein, J. Andersson and B. A. Ardekani, Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration, NeuroImage, 46 (2009), 786-802. [19] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in $\mathbb{R}^N$, J. Funct. Anal., 262 (2012), 1132-1165.  doi: 10.1016/j.jfa.2011.10.012. [20] N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth, Discrete Contin. Dyn. Syst., 32 (2012), 2187-2205.  doi: 10.3934/dcds.2012.32.2187. [21] C. Liu and Z. Yang, Boundary blow-up quasilinear elliptic problems of the Bieberbach type with nonlinear gradient terms, Nonlinear Anal., 69 (2008), 4380-4391.  doi: 10.1016/j.na.2007.10.060. [22] J. Moser, A sharp form of an inequality by N.Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101. [23] B. Tr$\acute{e}$lat, Optimal Control, Theory and Applications, Vuibert, Paris, 2005. [24] L. Yin, Y. Guo, J. Yang, B. Lu and Q. Zhang, Existence and asymptotic behavior of boundary blow-up solutions for weighted p(x)-Laplacian equations with exponential nonlinearities, Abstr. Appl. Anal., 2010 (2010), 1-20.  doi: 10.1155/2010/971268.

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##### References:
 [1] S. Arguill$\grave{e}$re and E. Tr$\acute{e}$lat, Sub-Riemannian structures on groups of diffeomorphisms, J. Inst. Math. Jussieu, 2014. [2] V.I. Arnold, Ordinary Differerntial Equations, Springer-Verlag, Berlin, 1992. [3] A. Bahrouni, Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity, Commun. Pure Appl. Anal., 16 (2017), 243-252.  doi: 10.3934/cpaa.2017011. [4] J. Bao, N. Lam and G. Lu, Polyharmonic equations with critical exponential growth in the whole space $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 36 (2016), 577-600.  doi: 10.3934/dcds.2016.36.577. [5] M.F. Beg, M.I. Miller, A. Trouv$\acute{e}$ and L. Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms, Int. J. Comput. Vision, 61 (2005), 139-157. [6] L.G. Brown, A survey of image registration techniques, ACM Comput. Surv., 24 (1992), 325-376. [7] L. Chen, J. Li, G. Lu and C. Zhang, Sharpened Adams inequlity and ground state solutions to the bi-Laplacian equation in $\mathbb{R}^4$, Adv. Nonlinear Stud., 18 (2018), 429-452.  doi: 10.1515/ans-2018-2020. [8] M. Chuaqui, C. Cortázar, M. Elgueta and J. García-Melián, Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights, Commun. Pure Appl. Anal., 3 (2004), 653-662.  doi: 10.3934/cpaa.2004.3.653. [9] J. Davila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592.  doi: 10.1137/060665579. [10] W. Du, Existence and uniqueness of large solutions for competitive type elliptic systems, Int. Journal of Math. Analysis, 7 (2013), 2878-2884.  doi: 10.12988/ijma.2013.36159. [11] P. Dupuis, U. Grenander and M.I. Miller, Variational problems on flows of diffeomorphisms for image matching, Q. Appl. Math., 56 (1998), 587-600.  doi: 10.1090/qam/1632326. [12] L. C. Evants, Partial Differential Equations, American Mathematical Society, Providence, 2010. doi: 10.1090/gsm/019. [13] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Springer-Verlag, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9. [14] F. Gazzola, H. C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2001. [16] D. L. Hill, P. G. Batchelor, M. Holden and D. J. Hawkes, Medical image registration, Phys. Med. Biol., 46 (2001), R1–R45. [17] A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385.  doi: 10.1016/j.jmaa.2013.12.059. [18] A. Klein, J. Andersson and B. A. Ardekani, Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration, NeuroImage, 46 (2009), 786-802. [19] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in $\mathbb{R}^N$, J. Funct. Anal., 262 (2012), 1132-1165.  doi: 10.1016/j.jfa.2011.10.012. [20] N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth, Discrete Contin. Dyn. Syst., 32 (2012), 2187-2205.  doi: 10.3934/dcds.2012.32.2187. [21] C. Liu and Z. Yang, Boundary blow-up quasilinear elliptic problems of the Bieberbach type with nonlinear gradient terms, Nonlinear Anal., 69 (2008), 4380-4391.  doi: 10.1016/j.na.2007.10.060. [22] J. Moser, A sharp form of an inequality by N.Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101. [23] B. Tr$\acute{e}$lat, Optimal Control, Theory and Applications, Vuibert, Paris, 2005. [24] L. Yin, Y. Guo, J. Yang, B. Lu and Q. Zhang, Existence and asymptotic behavior of boundary blow-up solutions for weighted p(x)-Laplacian equations with exponential nonlinearities, Abstr. Appl. Anal., 2010 (2010), 1-20.  doi: 10.1155/2010/971268.
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