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Hongsoo Kim

Publications and source records attributed to Hongsoo Kim.

10 recordsLinked to original sources
arXiv2026

A priori estimates for solutions of degenerate fully nonlinear elliptic equations with $L^p$ data

We establish a priori regularity estimates for viscosity solutions of degenerate fully nonlinear elliptic equations with integrable right-hand sides. When the nonhomogeneous term belongs to $L^p$ with $p>n$, we prove optimal interior $C^{1,\alpha}$ estimates. In the critical case, we obtain a log-Lipschitz modulus of continuity under the Lorentz condition $f\in L^{n,1}$. We utilize sliding paraboloid or cusp methods to develop uniform H\"older estimates for equations that are elliptic only in suitable gradient regimes. Finally, we establish an approximation lemma for integrable right-hand sides via a corrector argument, which allows us to deduce the corresponding Schauder-type estimates.

math.AP↗
arXiv2026

Harnack inequality for anisotropic fully nonlinear equations with nonstandard growth

We establish Harnack inequalities for viscosity solutions of a class of degenerate fully nonlinear anisotropic elliptic equations exhibiting non-standard growth conditions. A primary example of such operators is the degenerate anisotropic $(p_i)$-Laplacian. Our approach relies on the sliding paraboloid method, adapted with suitably chosen anisotropic functions to derive the basic measure estimates. A central contribution of this work is the development of a doubling property, achieved through the explicit construction of a novel barrier function. By combining these tools with the intrinsic geometry techniques introduced in [DGV08, VV25], we prove the intrinsic Harnack inequality for this class of operators under appropriate conditions on the exponents $(p_i)$.

math.AP↗
arXiv2026

Calderon-Zygmund estimates for generalized double phase equations with matrix weights

We prove Calderon-Zygmund estimates for generalized double phase equations with Orlicz growth and variable matrix weights. The operator combines a non-uniformly elliptic double phase structure with a degenerate or singular matrix weight satisfying a small log-BMO condition. Under appropriate structural assumptions, we show that higher integrability of the weighted datum yields higher integrability of the weighted gradient of weak solutions. Our results extend the existing Calderon-Zygmund theory for double phase problems and weighted elliptic equations to a unified framework capturing the interaction between Orlicz growth and matrix-weighted structures, thereby building upon and unifying the results in [BBO20] and [BCR26].

math.AP↗
arXiv2025

Lipschitz regularity for anisotropic fully nonlinear equations with nonstandard growth

We establish interior Lipschitz regularity for solutions to anisotropic fully nonlinear equations with nonstandard growth, without imposing any restriction on the gap between the highest and lowest growth exponents. Our proof is based on an anisotropic variant of the seminal Ishii Lions method. Our result furnishes a viscosity analogue of the divergence-form theory in [Bousquet20], adapted to the non-divergence setting.

math.AP↗
arXiv2025

Lipschitz regularity for fully nonlinear elliptic equations with $(p,q)$-growth

We prove the interior and global Lipschitz regularity results for a solution of fully nonlinear equations with $(p,q)$-growth. We prove that for a small gap $q-p$, a solution is locally or globally Lipschitz continuous. We also prove that a given H\"older continuous solution is Lipschitz continuous under improved bounds for the gap. These gap conditions are similar to those required for the regularity of double phase problems in divergence form.

math.AP↗
arXiv2024

Interior $W^{2,\delta}$ type estimates for degenerate fully nonlinear elliptic equations with $L^n$ data

We establish interior $W^{2,\delta}$ type estimates for a class of degenerate fully nonlinear elliptic equations with $L^n$ data. The main idea of our approach is to slide $C^{1,\alpha}$ cones, instead of paraboloids, vertically to touch the solution, and estimate the contact set in terms of the measure of the vertex set. This shows that the solution has tangent $C^{1,\alpha}$ cones almost everywhere, which leads to the desired Hessian estimates. Accordingly, we are able to develop a kind of counterpart to the estimates for divergent structure quasilinear elliptic problems.

math.AP↗
arXiv2024

$C^{1,\alpha}$ regularity for degenerate fully nonlinear elliptic equations with oblique boundary conditions on $C^1$ domains

We provide a sharp $C^{1,\alpha}$ estimate up to the boundary for a viscosity solution of a degenerate fully nonlinear elliptic equation with the oblique boundary condition on a $C^1$ domain. To this end, we first obtain a uniform boundary H{\"o}lder estimate with the oblique boundary condition in an "almost $C^1$-flat" domain for the equations which is uniformly elliptic only where the gradient is far from some point, and then we establish a desired $C^{1,\alpha}$ regularity based on perturbation and compactness arguments.

math.AP↗
arXiv2023

Bandit-supported care planning for older people with complex health and care needs

Long-term care service for old people is in great demand in most of the aging societies. The number of nursing homes residents is increasing while the number of care providers is limited. Due to the care worker shortage, care to vulnerable older residents cannot be fully tailored to the unique needs and preference of each individual. This may bring negative impacts on health outcomes and quality of life among institutionalized older people. To improve care quality through personalized care planning and delivery with limited care workforce, we propose a new care planning model assisted by artificial intelligence. We apply bandit algorithms which optimize the clinical decision for care planning by adapting to the sequential feedback from the past decisions. We evaluate the proposed model on empirical data acquired from the Systems for Person-centered Elder Care (SPEC) study, a ICT-enhanced care management program.

stat.ML↗