• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.747-775
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• 2020

In this work we investigate the nonlocal elliptic equation with critical Hardy-Sobolev exponents as follows, $$(P)\;\{(-{\Delta}_p)^su={\lambda}{\mid}u{\mid}^{q-2}u+{\frac{{\mid}u{\mid}^{p{^*_s}(t)-2}u}{{\mid}x{\mid}^t}}{\hspace{10}}in\;{\Omega},\\u=0{\hspace{217}}in\;{\mathbb{R}}^N{\backslash}{\Omega},$$ where Ω ⊂ ℝ^N is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < t < sp < N, 1 < q < p < p^∗_s where $p^*_s={\frac{N_p}{N-sp}}$, $p^*_s(t)={\frac{p(N-t)}{N-sp}}$, are the fractional critical Sobolev and Hardy-Sobolev exponents respectively. The fractional p-laplacian (-∆_p)^su with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by $\displaystyle(-{\Delta}_p)^su(x)=2{\lim_{{\epsilon}{\searrow}0}}\int{_{{\mathbb{R}}^N{\backslash}{B_{\epsilon}}}}\;\frac{{\mid}u(x)-u(y){\mid}^{p-2}(u(x)-u(y))}{{\mid}x-y{\mid}^{N+ps}}dy$, x ∈ ℝ^N. The main goal of this work is to show how the usual variational methods and some analysis techniques can be extended to deal with nonlocal problems involving Sobolev and Hardy nonlinearities. We also prove that for some α ∈ (0, 1), the weak solution to the problem (P) is in C^1,α(${\bar{\Omega}}$).

• Korean Journal of Radiology
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• v.22 no.5
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• pp.759-769
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• 2021

Objective: To evaluate the application of laplacian-regularized mean apparent propagator (MAPL)-MRI to brain glioma-induced corticospinal tract (CST) injury. Materials and Methods: This study included 20 patients with glioma adjacent to the CST pathway who had undergone structural and diffusion MRI. The entire CSTs of the affected and healthy sides were reconstructed, and the peritumoral CSTs were manually segmented. The morphological characteristics of the CST (track number, average length, volume, displacement of the affected CST) were examined and the diffusion parameter values, including fractional anisotropy (FA), mean diffusivity (MD), axial diffusivity (AD), radial diffusivity (RD), mean squared displacement (MSD), q-space inverse variance (QIV), return-to-origin probability (RTOP), return-to-axis probabilities (RTAP), and return-to-plane probabilities (RTPP) along the entire and peritumoral CSTs, were calculated. The entire and peritumoral CST characteristics of the affected and healthy sides as well as those relative CST characteristics of the patients with motor weakness and normal motor function were compared. Results: The track number, volume, MD, RD, MSD, QIV, RTAP, RTOP, and RTPP of the entire and peritumoral CSTs changed significantly for the affected side, whereas the AD and FA changed significantly only in the peritumoral CST (p < 0.05). In patients with motor weakness, the relative MSD of the entire CST, QIV of the entire and peritumoral CSTs, and the AD, MD, RD of the peritumoral CST were significantly higher, whereas the RTPP of the entire and peritumoral CSTs and the RTOP of the peritumoral CST were significantly lower than those in patients with normal motor function (p < 0.05 for all). In contrast, no significant changes were found in the CST morphological characteristics, FA, or RTAP (p > 0.05 for all). Conclusion: MAPL-MRI is an effective approach for evaluating microstructural changes after CST injury. Its sensitivity may improve when using the peritumoral CST features.

• Honam Mathematical Journal
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• v.43 no.2
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• pp.289-304
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• 2021

In this paper, we establish several geometric characterizations focusing on the relationship between the squared norm of the second fundamental form and the warping function of SCR-lightlike warped product submanifolds in an indefinite Kaehler manifold. In particular, we find an estimate for the squared norm of the second fundamental form h in terms of the Hessian of the warping function λ for SCR-lightlike warped product submanifolds of an indefinite complex space form. Consequently, we derive an optimal inequality, namely $${\parallel}h{\parallel}^2{\geq}2q\{{\Delta}(ln{\lambda})+{\parallel}{\nabla}(ln{\lambda}){\parallel}^2+\frac{c}{2}p\}$$, for SCR-lightlike warped product submanifolds in an indefinite complex space form. We also provide one non-trivial example for this class of warped products in an indefinite Kaehler manifold.