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Hyungsung Yun

Publications and source records attributed to Hyungsung Yun.

10 recordsLinked to original sources
arXiv2026

Time derivative estimates for normalized parabolic $p$-Laplace equations

We establish the local boundedness of time derivatives of bounded viscosity solutions to the normalized parabolic $p$-Laplace equation for every $p\in(1,\infty)$. The proof combines a logarithmic Bernstein argument with quantitative stability to derive a recursive estimate between different regularizations. For $p=1$ and $n\geq2$, we construct a bounded viscosity solution which is not Hölder continuous in time with any positive exponent.

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arXiv2025

Boundary Hölder gradient estimates for fully nonlinear degenerate or singular parabolic equations

We study boundary regularity of viscosity solutions to fully nonlinear degenerate or singular parabolic equations. The gradient-dependent degeneracy or singularity, along with the time derivative, introduces significant challenges beyond the elliptic case. By combining compactness methods, barrier constructions, regularization techniques, and the boundary regularity of small perturbation solutions, we establish boundary Hölder gradient estimates that unify and extend previous results.

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arXiv2025

Optimal regularity for degenerate parabolic equations on a flat boundary

We establish the optimal regularity of viscosity solutions to \begin{equation*} u_t - x_n^γΔu = f, \end{equation*} which arises in the regularity theory for the porous medium equation. Specifically, we prove that under the zero Dirichlet boundary condition on $\{x_n=0\}$, the optimal regularity of $u$ up to the flat boundary $\{x_n=0\}$ is $C^{1,1-γ}$. Moreover, for the homogeneous equations, we establish that the optimal regularity of $u$ is $C^{2,1-γ}$ in the spatial variables, and that $x_n^{-γ}u$ is smooth in the variables $x'$ and $t$.

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arXiv2025

Time derivative estimates for parabolic $p$-Laplace equations and applications to optimal regularity

We establish the boundedness of time derivatives of solutions to parabolic $p$-Laplace equations. Our approach relies on the Bernstein technique combined with a suitable approximation method. As a consequence, we obtain an optimal regularity result with a connection to the well-known $C^{p'}$-conjecture in the elliptic setting. Finally, we extend our method to treat global regularity results for both fully nonlinear and general quasilinear degenerate parabolic problems.

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arXiv2024

Regularity theory for fully nonlinear equations of porous medium-type

In this paper, we establish the regularity results for nonnegative viscosity solutions to fully nonlinear equations of porous medium-type in bounded domains with the zero Dirichlet boundary condition, to be precise, we prove the global $C^{2,α}$-estimates of viscosity solutions. In many PDE problems, the $C^{2,α}$-estimates have been obtained through Schauder-type estimates. However, the Schauder-type estimates are not applicable to the porous medium-type equations. We provide techniques for handling porous medium-type equations so that the global $C^{2,α}$-estimates can be established.

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arXiv2023

Boundary Regularity for viscosity solutions of Fully nonlinear degenerate/singular parabolic equations

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.

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arXiv2023

Generalized Schauder Theory and its Application to Degenerate/Singular Parabolic Equations

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.

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arXiv2023

$C^{1, α}$-regularity for functions in solution classes and its application to parabolic normalized $p$-Laplace equations

We establish the global $C^{1, α}$-regularity for functions in solution classes, whenever ellipticity constants are sufficiently close. As an application, we derive the global regularity result concerning the parabolic normalized $p$-Laplace equations, provided that $p$ is close to 2. Our analysis relies on the compactness argument with the iteration procedure.

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arXiv2023

$C^{1, α}$-regularity for solutions of degenerate/singular fully nonlinear parabolic equations

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.

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