arXiv subjects
Publications and source records attributed to Ki-Ahm Lee.
We present three equivalent definitions of the fractional $p$-Laplacian $(-Δ_{\mathbb{H}^{n}})^{s}_{p}$, $0 1$, with normalizing constants, on hyperbolic spaces. The explicit values of the constants enable us to study the convergence of the fractional $p$-Laplacian to the $p$-Laplacian as $s \to 1^{-}$.
We establish the Krylov--Safonov theory for a large class of nonlocal operators of order $2s \in (0,2)$ on hyperbolic spaces $\mathbb{H}^{n}_κ$ with curvature $-κ<0$. We prove the Alexandrov--Bakelman--Pucci (ABP) estimates, Krylov--Safonov Harnack inequality, and Hölder estimates. Notably, the Harnack inequality is new even for the fractional Laplacian. The novelty of the results lies in the robustness of the regularity estimates as $s \to 1$ and $κ\to 0$: they recover the classical regularity estimates for second-order operators on $\mathbb{H}^{n}_κ$ as $s \to 1$, and for fractional-order operators on Euclidean spaces as $κ\to 0$. Since the operators on hyperbolic spaces exhibit qualitatively different behavior compared to their Euclidean counterparts, we introduce new scale functions which take the effect of negative curvatures into account.
This study investigates an SEIS PDE model with a free boundary, which captures the dynamics of epidemic transmission, including diseases like COVID-19. This parabolic PDE system is analyzed in a rotationally symmetric domain, and the existence and uniqueness of the local solution are established through the straightening lemma. Furthermore, the existence and uniqueness of the global solution are established under specific conditions on the diffusion coefficients. Then the model introduces the basic reproductive number, $R_0$, which provides sufficient conditions for determining whether the disease will vanish or spread. Notably, when $R_0<1$, the disease-free equilibrium(DFE) is shown to be globally stable, and when $R_0>1$, the DFE is unstable. Lastly, we investigate the convergence speed of solutions by applying nonlinear elliptic eigenvalue techniques to the associated parabolic PDE system.
We study the existence, uniqueness, and regularity of weak solutions to a class of obstacle problems, where the obstacle condition can be imposed on a subset of the domain. In particular, we establish the optimal Hölder regularity for Signorini-type problems, that is, the obstacle condition is imposed only on a subset of codimension one. For this purpose, we employ capacities, Alt--Caffarelli--Friedman-type and Almgren-type monotonicity formulae, and investigate an associated mixed boundary value problem. Further, we apply this problem to study classical obstacle problems for irregular obstacles.
We establish the Alexandroff-Bakelman-Pucci estimate, the Harnack inequality, the Hölder regularity and the Schauder estimates to a class of degenerate parabolic equations of non-divergence form in all dimensions \begin{equation} \mathcal{L}u:= u_t -Lu= u_t -(x a_{11} u_{xx} +2\sqrt{x} \sum_{j=2}^n a_{1j} u_{x y_j} + \sum_{i,j=2}^n a_{ij} u_{y_i y_j} + b_1 u_x +\sum_{j=2}^n b_j u_{y_j} ) =g\ \end{equation} on \(x \geq 0, y=(y_2,\ldots, y_n) \in \mathbb{R}^{n-1}\), with bounded measurable coefficients.
We develop the viscosity method for the homogenization of an obstacle problem with highly oscillating obstacles. The associated operator, in non-divergence form, is linear and elliptic with variable coefficients. We first construct a highly oscillating corrector, which captures the singular behavior of solutions near periodically distributed holes of critical size. We then prove the uniqueness of a critical value that encodes the coupled effects of oscillations in both the coefficients and the obstacles.
We prove the Wolff potential estimates for nonlocal equations with Orlicz growth. As an application, we obtain the Wiener criterion in this framework, which provides a necessary and sufficient condition for boundary points to be regular. Our approach relies on the fine analysis of superharmonic functions in view of nonlocal nonlinear potential theory.
We consider a flow by powers of Gauss curvature under the obstruction that the flow cannot penetrate a prescribed region, so called an obstacle. For all dimensions and positive powers, we prove the optimal curvature bounds of solutions and all time existence with its long time behavior. We also prove the $C^1$ regularity of free boundaries under a uniform thickness assumption.
In this paper, we investigate energy-minimizing curves with fixed endpoints $p$ and $q$ in a constrained space. We prove that when one of the endpoints, say $p$, is fixed, the set of points $q$ for which the energy-minimizing curve is not unique has no interior points.
In this paper, we establish the boundary regularity results for viscosity solutions of fully nonlinear degenerate/singular parabolic equations of the form $$u_t - x_n^γ F(D^2 u,x,t) = f,$$ where $γ<1$. These equations are motivated by the porous media type equations. We show the boundary $C^{1,α}$-regularity of functions in their solutions class and the boundary $C^{2,α}$-regularity of solutions. As an application, we derive the global regularity results and the solvability of the Cauchy-Dirichlet problems.
We prove sharp near-diagonal pointwise bounds for the Green function $G_Ω(x,y)$ for nonlocal operators of fractional order $α\in (0,2)$. The novelty of our results is two-fold: the estimates are robust as $α\to 2-$ and we prove the bounds without making use of the Dirichlet heat kernel $p_Ω(t;x,y)$. In this way we can cover cases, in which the Green function satisfies isotropic bounds but the heat kernel does not.
In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form $$u_t = a^{i'j'}u_{i'j'} + 2 x_n^{γ/2} a^{i'n} u_{i'n} + x_n^γ a^{nn} u_{nn} + b^{i'} u_{i'} + x_n^{γ/2} b^n u_{n} + c u + f \quad (γ\leq1).$$ When the equation above is singular, it can be derived from Monge--Ampère equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution $u$, which is called $s$-polynomial. To prove $C_s^{2+α}$-regularity and higher regularity of the solution $u$, we establish generalized Schauder theory which approximates coefficients of the operator with $s$-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap method to obtain higher regularity.
We establish the interior $C^{1,α}$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t = |Du|^γF(D^2u) + f.$$ For this purpose, we prove the well-posedness of the regularized Dirichlet problem \begin{equation*} \left\{ \begin{aligned} u_t&=(1+|Du|^2)^{γ/2}F(D^2u) &&\text{in $Q_1$} \newline u&=φ&&\text{on $\partial_p Q_1$}. \end{aligned}\right. \end{equation*} Our approach utilizes the Bernstein method with approximations in view of difference quotient.
We study the boundary continuity of solutions to fully nonlinear elliptic equations. We first define a capacity for operators in non-divergence form and derive several capacitary estimates. Secondly, we formulate the Wiener criterion, which characterizes a regular boundary point via potential theory. Our approach utilizes the asymptotic behavior of homogeneous solutions, together with Harnack inequality and the comparison principle.
We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, α}$-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator.
We establish an optimal C^{1,α}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.
We study the boundary behavior of solutions to the Dirichlet problems for integro-differential operators with order of differentiability $s \in (0, 1)$ and summability $p>1$. We establish a nonlocal counterpart of the Wiener criterion, which characterizes a regular boundary point in terms of the nonlocal nonlinear potential theory.
This article is concerned with uniform $C^{1,α}$ and $C^{1,1}$ estimates in periodic homogenization of fully nonlinear elliptic equations. The analysis is based on the compactness method, which involves linearization of the operator at each approximation step. Due to the nonlinearity of the equations, the linearized operators involve the Hessian of correctors, which appear in the previous step. The involvement of the Hessian of the correctors deteriorates the regularity of the linearized operator, and sometimes even changes its oscillating pattern. These issues are resolved with new approximation techniques, which yield a precise decomposition of the regular part and the irregular part of the homogenization process, along with a uniform control of the Hessian of the correctors in an intermediate level. The approximation techniques are even new in the context of linear equations. Our argument can be applied not only to concave operators, but also to certain class of non-concave operators.