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Seick Kim

Publications and source records attributed to Seick Kim.

At least 19 recordsLinked to original sources
arXiv2026

Boundary blow-up solutions: gradient asymptotics and uniqueness

Let $\Omega\subset\mathbb R^n$ be a bounded domain, and let $f$ be a nonnegative, nondecreasing function satisfying the Keller-Osserman condition. We study boundary blow-up solutions of $\Delta u=f(u)$ in $\Omega$. Although existence is classical, uniqueness under these assumptions is known in balls but remains open even for smooth convex domains. We identify the normalized gradient $Q_u=|\nabla u|^2/(2F(u))$, $F'=f$, as a quantity governing uniqueness. Under a structural condition on $f$, a boundary blow-up solution $u$ is unique if $\limsup_{x\to\partial\Omega}Q_u(x)\le 1$, without any regularity assumption on $\partial \Omega$. For $C^{1,1}$ domains, assuming a growth condition on $f$, we prove $Q_u(x)\to 1$ for every boundary blow-up solution and hence obtain uniqueness under the structural condition. For convex domains, we prove $Q_u\le 1$ for the minimal boundary blow-up solution and obtain uniqueness when $\sqrt F$ is eventually convex, without imposing any additional boundary regularity.

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arXiv2026

A sign-changing Poisson kernel for a non-symmetric elliptic operator in a bounded domain

We construct an explicit sign-changing Poisson kernel for a uniformly elliptic divergence form operator in the unit disk. The coefficient matrix is non-symmetric, and its skew-symmetric part has a jump discontinuity across a fixed diameter, with the size of the jump determined by $k\in\mathbb R$. For each $1<p<\infty$, the associated $L^p$ Dirichlet problem undergoes a sharp transition as $k$ crosses a $p$-dependent threshold. In the unique solvability regime, the solution operator preserves positivity and is represented by a nonnegative Poisson kernel. Beyond the threshold, the homogeneous problem admits a nontrivial solution, and for real-valued boundary data, every bounded linear selection from this nonunique solution family is represented by a sign-changing kernel. Thus, even in bounded domains, uniform ellipticity and the maximum principle in the energy class do not prevent Poisson kernels from changing sign. Using a Riemann-Hilbert formulation, we obtain explicit solution formulas and show that the factorization index governs both nonuniqueness and sign change. To our knowledge, this is the first explicit example of this counterintuitive phenomenon in a bounded domain.

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arXiv2025

Refined regularity at critical points for linear elliptic equations

We investigate the regularity of solutions to linear elliptic equations in both divergence and non-divergence forms, particularly when the principal coefficients have Dini mean oscillation. We show that if a solution $u$ to a divergence-form equation satisfies $Du(x^o)=0$ at a point, then the second derivative $D^2u(x^o)$ exists and satisfies sharp continuity estimates. As a consequence, we obtain ``$C^{2,\alpha}$ regularity'' at critical points when the coefficients of $L$ are $C^\alpha$. This result refines a theorem of Teixeira (Math. Ann. 358 (2014), no. 1--2, 241--256) in the linear setting, where both linear and nonlinear equations were considered. We also establish an analogous result for equations in non-divergence form.

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arXiv2025

Two-sided Gaussian estimates for fundamental solutions of second-order parabolic equations in non-divergence form

We establish two-sided Gaussian bounds for the fundamental solution of second-order parabolic operators in non-divergence form under minimal regularity assumptions. Specifically, we show that the upper and lower bounds follow from the local boundedness property and the weak Harnack inequality for the adjoint operator $P^*$, respectively. This provides a simpler and more direct proof of the Gaussian estimates when the coefficients have Dini mean oscillation in $x$, avoiding the use of normalized adjoint solutions required in previous works.

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arXiv2025

Regular boundary points and the Dirichlet problem for elliptic equations in double divergence form

We study the Dirichlet problem for second-order elliptic operators in double divergence form, which arise as formal adjoints of non-divergence form operators and include the stationary Fokker-Planck-Kolmogorov equation. Assuming that the leading coefficients have Dini mean oscillation and that the lower-order coefficients satisfy natural integrability conditions, we construct the Perron solution in arbitrary bounded domains. We prove that a boundary point is regular with respect to the operator if and only if it satisfies the classical Wiener criterion for the Laplacian. In particular, the Dirichlet problem is uniquely solvable in every bounded domain that is regular for the Laplacian.

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arXiv2025

The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients: The two-dimensional case

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^2$. Assuming that the principal coefficients satisfy the Dini mean oscillation condition, we establish the equivalence between regular points for $L$ and those for the Laplace operator. This result closes a gap left in the authors' recent work on higher-dimensional cases (Math. Ann. 392(1): 573--618, 2025). Furthermore, we construct the Green's function for $L$ in regular two-dimensional domains, extending a result by Dong and Kim (SIAM J. Math. Anal. 53(4): 4637--4656, 2021).

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arXiv2025

Hopf-Oleinik lemma for elliptic equations in double divergence form

We establish, for the first time, a Zaremba-Hopf-Oleinik type boundary point lemma for uniformly elliptic partial differential equations in double divergence form, also known as stationary Fokker-Planck-Kolmogorov equations. As an application, we derive sharp two-sided estimates for the Green's function associated with second-order elliptic equations in non-divergence form in $C^{1,\alpha}$ domains.

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arXiv2024

Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients

This paper investigates the Harnack inequality for nonnegative solutions to second-order parabolic equations in double divergence form. We impose conditions where the principal coefficients satisfy the Dini mean oscillation condition in $x$, while the drift and zeroth-order coefficients belong to specific Morrey classes. Our analysis contributes to advancing the theoretical foundations of parabolic equations in double divergence form, including Fokker-Planck-Kolmogorov equations for probability densities.

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arXiv2024

The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^d$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's function in a ball and derive two-sided pointwise estimates for it. Utilizing these results, we demonstrate the equivalence of regular points for $L$ and those for the Laplace operator, characterized via the Wiener test. This equivalence facilitates the unique solvability of the Dirichlet problem with continuous boundary data in regular domains. Furthermore, we construct the Green's function for $L$ in regular domains and establish pointwise bounds for it. This advancement is significant, as it extends the scope of existing estimates to domains beyond $C^{1,1}$, contributing to our understanding of elliptic operators in non-divergence form.

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arXiv2024

Regularity of elliptic equations in double divergence form and applications to Green's function estimates

We investigate the regularity of elliptic equations in double divergence form, where the leading coefficients satisfying the Dini mean oscillation condition. We prove that the solutions are differentiable on the zero level set and derive a pointwise bound for the derivative, which substantially improve a recent result by Leit\~ao, Pimentel, and Santos (Anal. PDE 13(4):1129--1144, 2020). As an application, we establish global pointwise estimates for the Green's function of second-order uniformly elliptic operators in non-divergence form, considering Dini mean oscillation coefficients in bounded $C^{1,\alpha}$ domains. This result extends a recent work by Chen and Wang (Electron. J. Probab. 28(36):54 pp, 2023).

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arXiv2023

The Neumann Green function and scale invariant regularity estimates for elliptic equations with Neumann data in Lipschitz domains

We construct the Neumann Green function and establish scale invariant regularity estimates for solutions to the Neumann problem for the elliptic operator $Lu=-{\rm div}({\bf A} \nabla u+ \boldsymbol{b}u)+ \boldsymbol{c} \cdot \nabla u+du$ in a Lipschitz domain $\Omega$. We assume that ${\bf A}$ is elliptic and bounded, that the lower order coefficients belong to scale invariant Lebesgue spaces, and that either $d\geq{\rm div}\boldsymbol{b}$ in $\Omega$ and $\boldsymbol{b}\cdot\nu\geq 0$ on $\partial\Omega$ in the sense of distributions, or the analogous condition for $\boldsymbol{c}$ holds. We develop the $L^2$ theory, construct the Neumann Green function and show estimates in the respective optimal spaces, and show local and global pointwise estimates for solutions. The main novelty is that our estimates are scale invariant, since our constants depend on the lower order coefficients only via their norms, and on the Lipschitz domain only via its Lipschitz character. Moreover, our pointwise estimates are shown in the optimal scale invariant setting for the inhomogeneous terms and the Neumann data.

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arXiv2022

Learning Green's functions associated with time-dependent partial differential equations

Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are especially useful to learn solution operators associated with partial differential equations (PDEs) from pairs of forcing functions and solutions when numerical solvers are not available or the underlying physics is poorly understood. In this work, we attempt to provide theoretical foundations to understand the amount of training data needed to learn time-dependent PDEs. Given input-output pairs from a parabolic PDE in any spatial dimension $n\geq 1$, we derive the first theoretically rigorous scheme for learning the associated solution operator, which takes the form of a convolution with a Green's function $G$. Until now, rigorously learning Green's functions associated with time-dependent PDEs has been a major challenge in the field of scientific machine learning because $G$ may not be square-integrable when $n>1$, and time-dependent PDEs have transient dynamics. By combining the hierarchical low-rank structure of $G$ together with randomized numerical linear algebra, we construct an approximant to $G$ that achieves a relative error of $\smash{\mathcal{O}(\Gamma_\epsilon^{-1/2}\epsilon)}$ in the $L^1$-norm with high probability by using at most $\smash{\mathcal{O}(\epsilon^{-\frac{n+2}{2}}\log(1/\epsilon))}$ input-output training pairs, where $\Gamma_\epsilon$ is a measure of the quality of the training dataset for learning $G$, and $\epsilon>0$ is sufficiently small.

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arXiv2022

Estimates for fundamental solutions of parabolic equations in non-divergence form

We construct the fundamental solution of second order parabolic equations in non-divergence form under the assumption that the coefficients are of Dini mean oscillation in the spatial variables. We also prove that the fundamental solution satisfies a sub-Gaussian estimate. In the case when the coefficients are Dini continuous in the spatial variables and measurable in the time variable, we establish the Gaussian bounds for the fundamental solutions. We present a method that works equally for second order parabolic systems in non-divergence form.

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arXiv2022

Note on Green's functions of non-divergence elliptic operators with continuous coefficients

We improve a result in Kim and Lee (Ann. Appl. Math. 37(2):111--130, 2021): showing that if the coefficients of an elliptic operator in non-divergence form are of Dini mean oscillation, then its Green's function has the same asymptotic behavior near the pole $x_0$ as that of the corresponding Green's function for the elliptic equation with constant coefficients frozen at $x_0$.

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arXiv2020

Green's function for second order parabolic equations with singular lower order coefficients

We construct Green's functions for second order parabolic operators of the form $Pu=\partial_t u-{\rm div}({\bf A} \nabla u+ \boldsymbol{b}u)+ \boldsymbol{c} \cdot \nabla u+du$ in $(-\infty, \infty) \times \Omega$, where $\Omega$ is an open connected set in $\mathbb{R}^n$. It is not necessary that $\Omega$ to be bounded and $\Omega = \mathbb{R}^n$ is not excluded. We assume that the leading coefficients $\bf A$ are bounded and measurable and the lower order coefficients $\boldsymbol{b}$, $\boldsymbol{c}$, and $d$ belong to critical mixed norm Lebesgue spaces and satisfy the conditions $d-{\rm div} \boldsymbol{b} \ge 0$ and ${\rm div}(\boldsymbol{b}-\boldsymbol{c}) \ge 0$. We show that the Green's function has the Gaussian bound in the entire $(-\infty, \infty) \times \Omega$.

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arXiv2020

Green's function for nondivergence elliptic operators in two dimensions

We construct the Green function for second-order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition. We show that the Green's function is BMO in the domain and establish logarithmic pointwise bounds. We also obtain pointwise bounds for first and second derivatives of the Green's function.

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