• Journal of applied mathematics & informatics
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• 제33권5_6호
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• pp.723-738
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• 2015

In this paper, the prescribed mean curvature Rayleigh p-Laplacian equation with a deviating argument

• 대한수학회지
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• 제53권2호
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• pp.287-304
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• 2016

In this paper, we consider the problem of existence of conformal metrics with prescribed mean curvature on the unit ball of ${\mathbb{R}}^n$, $n{\geq}3$. Under the assumption that the order of flatness at critical points of prescribed mean curvature function H(x) is ${\beta}{\in}[1,n-2]$, we give precise estimates on the losses of the compactness and we prove new existence result through an Euler-Hopf type formula.

• 대한수학회보
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• 제61권1호
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• pp.29-44
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• 2024

In this paper we consider the existence of rotationally symmetric entire solutions for the prescribed higher mean curvature spacelike equations in Minkowski spacetime. As a first step, we study the associated 0-Dirichlet problems on a ball, and then we prove that all possible solutions can be extended to + ∞. The proof of our main results are based upon the topological degree methods and the standard prolongability theorem of ordinary differential equations.

• 호남수학학술지
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• 제44권1호
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• pp.36-51
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• 2022

We study the translation surfaces in the pseudo-Galilean space with the condition that one of generating curves is planar. We classify these surfaces whose mean and Gaussian curvatures are functions of one variable.

• 대한수학회보
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• 제60권4호
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• pp.1085-1100
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• 2023

For any given function f, we focus on the so-called prescribed mean curvature problem for the measure e^-f(|x|2)dx provided thate^-f(|x|2) ∈ L¹(ℝⁿ⁺¹). More precisely, we prove that there exists a smooth hypersurface M whose metric is ds² = dρ² + ρ²d𝜉² and whose mean curvature function is ${\frac{1}{n}}(\frac{u^p}{{\rho}^{\beta}})e^{f({\rho}^2)}{\psi}(\xi)$ for any given real constants p, β and functions f and ψ where u and ρ are the support function and radial function of M, respectively. Equivalently, we get the existence of a smooth solution to the following quasilinear equation on the unit sphere 𝕊ⁿ, $${\sum_{i,j}}({{\delta}_{ij}-{\frac{{\rho}_i{\rho}_j}{{\rho}^2+|{\nabla}{\rho}|^2}})(-{\rho}ji+{\frac{2}{{\rho}}}{\rho}j{\rho}i+{\rho}{\delta}_{ji})={\psi}{\frac{{\rho}^{2p+2-n-{\beta}}e^{f({\rho}^2)}}{({\rho}^2+|{\nabla}{\rho}|^2)^{\frac{p}{2}}}}$$ under some conditions. Our proof is based on the powerful method of continuity. In particular, if we take $f(t)={\frac{t}{2}}$, this may be prescribed mean curvature problem in Gauss measure space and it can be seen as an embedded result in Gauss measure space which will be needed in our forthcoming papers on the differential geometric analysis in Gauss measure space, such as Gauss-Bonnet-Chern theorem and its application on positive mass theorem and the Steiner-Weyl type formula, the Plateau problem and so on.

• Annals of Rehabilitation Medicine
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• 제42권6호
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• pp.863-871
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• 2018

Objective To evaluate the association between progression of curvature of scoliosis, and correction for functional component in patients with juvenile idiopathic scoliosis (JIS). Methods We retrospectively reviewed medical data of patients prescribed custom molded foot orthosis (FO) to correct inequality of RCSPA (resting calcaneal stance position angle), and chose 52 patients (26 females, 26 males) with Cobb angle ${\geq}10^{\circ}$ in radiology and uneven pelvic level at iliac crest by different RCSPA (${\geq}3^{\circ}$) as a factor of functional scoliosis. They had different hump angle ${\geq}5^{\circ}$ in forward bending test, for idiopathic scoliosis component. Their mean age and mean period of wearing FO were $79.5{\pm}10.6months$ and $18.6{\pm}0.70months$. Results Cobb angle was reduced from $22.03^{\circ}{\pm}4.39^{\circ}$ initially to $18.86^{\circ}{\pm}7.53^{\circ}$ after wearing FO. Pelvis height difference and RCSPA difference, were reduced from $1.07{\pm}0.25cm$ initially to $0.60{\pm}0.36$, and from $4.25^{\circ}{\pm}0.71^{\circ}$ initially to $1.71^{\circ}{\pm}0.75^{\circ}$ (p<0.01). Cobb angle improved most in 9 months. However, there was no significant improvement for those with more than $25^{\circ}$ of Cobb angle initially. Mean Cobb angle improved in all age groups, but patients less than 6 years had clinically significant improvement of more than $5^{\circ}$. Conclusion JIS can have functional components, which should be identified and managed. Foot orthosis is useful in correcting functional factors, in the case of pelvic inequality caused by different RCSPA, for patients with juvenile idiopathic scoliosis.