arXiv subjects

Seongmin Jeon

Publications and source records attributed to Seongmin Jeon.

At least 19 recordsLinked to original sources

arXiv2026

Existence theory for non-variational systems with free boundaries

We study the existence of solutions for systems of both elliptic and parabolic partial differential equations with potentially singular right-hand sides and free boundaries, posed in general smooth domains. Our results are established within a framework of "meta-theorems." This approach hinges on specific strong properties of the operators and their solutions (in approximate smooth settings) to guarantee the existence of a limit as the approximation parameter tends to zero. The primary challenge lies in applying these meta-theorems to prototype cases, which requires verifying that the necessary strong properties hold. For our analysis, we focus on fully nonlinear and $p$-Laplacian operators and a mixing of these operators, in both elliptic and parabolic contexts. While we focus on these specific cases, the meta-theorems remain valid for any other operators that satisfy the required properties. Beyond the complex proofs of our meta-theorems, and their applications to specific operators, a major challenge is the technical handling of the $p$-parabolic case, which requires proving the regularity of solutions of the $p$-parabolic equation with singular or degenerate right-hand side--addressed in the Appendix--along with several (new) properties, which is missing in the literature.

math.AP↗

arXiv2026

6d Supergravity Blocks

We propose a systematic framework for constructing six-dimensional supergravity theories with eight supercharges that respect all known consistency constraints, including anomaly cancellation and the non-perturbative ${\it H}$-string constraints recently discovered by Kim, Vafa, and Xu. The basic objects in this framework are ${\it supergravity\,blocks}$, which are minimal collections of tensor multiplets consisting of a single little string theory sharing the ${\it H}$-string charge together with additional tensors whose string charges intersect it positively. A characteristic feature of each supergravity block is that its Gram matrix has exactly one positive eigenvalue, and therefore it necessarily contains gravitational BPS strings that cannot become tensionless anywhere in tensor moduli space. Any consistent 6d $(1,0)$ supergravity theory can then be obtained by gluing compatible blocks and subsequently enhancing the gauge algebras and matter content. As a first step toward establishing this framework concretely, we provide a complete classification of the ${\it non\text{-}Higgsable\,supergravity\,blocks}$, (or ${\it non\text{-}Higgsable\,gravity\,blocks}$ for short) namely those built from tensor multiplets that support only non-Higgsable gauge algebras.

hep-th↗

arXiv2026

Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

The recent work of Morini-Oronzio-Spadaro and the third author shows that, in three dimensions, a flat-flow solution of the volume-preserving mean curvature flow that converges to a single ball, which is the case for instance when the initial perimeter is less than that of two disjoint balls, converges exponentially fast in Hausdorff distance. In this paper we strengthen this result by proving that after a finite time the flow becomes smooth, satisfies the equation in the classical sense and converges exponentially fast to the limiting ball in every C^k-norm. In the proof we develop a version of Brakke's epsilon regularity theorem adapted to our setting and derive the necessary nonlinear PDE estimates directly at the level of the discrete minimizing-movement scheme. The same result holds in the planar case.

math.AP↗

arXiv2025

Schauder type estimates for degenerate or singular parabolic systems with partially DMO coefficients

We study elliptic and parabolic systems in divergence form with degenerate or singular coefficients. Under the conormal boundary condition on the flat boundary, we establish boundary Schauder type estimates when the coefficients have partially Dini mean oscillation. Moreover, as an application, we achieve $k^{\text{th}}$ higher-order boundary Harnack principles for uniformly parabolic equations with Hölder coefficients, extending a recent result in [Audrito-Fioravanti-Vita 25] from $k\ge2$ to any $k\ge1$.

math.AP↗

arXiv2025

Degenerate or singular parabolic systems with partially DMO coefficients: the Dirichlet problem

In this paper, we study solutions $u$ of parabolic systems in divergence form with zero Dirichlet boundary conditions in the upper-half cylinder $Q_1^+\subset \mathbb{R}^{n+1}$, where the coefficients are weighted by $x_n^α$, $α\in(-\infty,1)$. We establish higher-order boundary Schauder type estimates of $x_n^αu$ under the assumption that the coefficients have partially Dini mean oscillation. As an application, we also achieve higher-order boundary Harnack principles for degenerate or singular equations with Hölder continuous coefficients.

math.AP↗

arXiv2025

Remarks on the fine structure of the free boundary (the no-sign obstacle problem)

We present a number of results inspired by the approach developed in a recent work by A. Figalli and J. Serra on the fine structure of the obstacle problem, which turns out to be partially effective in addressing the no-sign obstacle problem. However, we also highlight one or two central questions that remain open and appear to require different techniques not presently available to us.

math.AP↗

arXiv2025

The free boundary for a superlinear system

In this paper, we study superlinear systems that give rise to free boundaries. Such systems appear for example from the minimization of the energy functional $$ \int_Ω\left(|\nabla\mathbf{u}|^2+\frac2p|\mathbf{u}|^p\right),\quad 0<p<1, $$ but solutions can be also understood in an ad hoc viscosity way. First, we prove the optimal regularity of minimizers using a variational approach. Then, we apply a linearization technique to establish the $C^{1,α}$-regularity of the ``flat'' part of the free boundary via a viscosity method. Finally, for minimizing free boundaries, we extend this result to analyticity.

math.AP↗

arXiv2025

Boundary estimates for elliptic operators in divergence form with VMO coefficients

We establish boundary regularity estimates for elliptic systems in divergence form with VMO coefficients. Additionally, we obtain nondegeneracy estimates of the Hopf-Oleinik type lemma for elliptic equations. In both cases, the moduli of continuity are expressed in terms of the $L^p$-mean oscillations of the coefficients and data.

math.AP↗

arXiv2024

Free boundary regularity for almost minimizers of the parabolic Signorini problem

In this paper, we study the regularity of the "regular" part of the free boundary for almost minimizers in the parabolic Signorini problem with zero thin obstacle. This work is a continuation of our earlier research on the regularity of almost minimizers. We first establish the Weiss-type monotonicity formula by comparing almost minimizers with parabolically homogeneous replacements and utilizing conformal self-similar coordinates. Subsequently, by deriving the Almgren-type frequency formula and applying the epiperimetric inequality, we obtain the optimal growth near regular free boundary points and achieve the regularity of the regular set.

math.AP↗

arXiv2024

Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients

In this paper, we study degenerate or singular elliptic equations in divergence form $$-\text{div}(x_n^αA\nabla u)=\text{div}(x_n^α\mathbf{g})\quad\text{in }B_1\cap\{x_n>0\}.$$ When $α>-1$, we establish boundary Schauder type estimates under the conormal boundary condition on the flat boundary, provided that the coefficients satisfy Dini mean oscillation (DMO) type conditions. Additionally, as an application, we derive higher-order boundary Harnack principles for uniformly elliptic equations in divergence form with DMO coefficients.

math.AP↗

arXiv2024

Convexity for a parabolic fully nonlinear free boundary problem with singular term

In this paper, we study a parabolic free boundary problem in an exterior domain $$\begin{cases} F(D^2u)-\partial_tu=u^aχ_{\{u>0\}}&\text{in }(\mathbb R^n\setminus K)\times(0,\infty),\\ u=u_0&\text{on }\{t=0\},\\ |\nabla u|=u=0&\text{on }\partialΩ\cap(\mathbb R^n\times(0,\infty)),\\ u=1&\text{in }K\times[0,\infty).\end{cases}$$ Here, $a$ belongs to the interval $(-1,0)$, $K$ is a (given) convex compact set in $\mathbb R^n$, $Ω=\{u>0\}\supset K\times(0,\infty)$ is an unknown set, and $F$ denotes a fully nonlinear operator. Assuming a suitable condition on the initial value $u_0$, we prove the existence of a nonnegative quasiconcave solution to the aforementioned problem, which exhibits monotone non-decreasing behavior over time.

math.AP↗

arXiv2023

Higher order boundary Harnack principles in Dini type domains

Aim of this paper is to provide higher order boundary Harnack principles [De Silva-Savin 15] for elliptic equations in divergence form under Dini type regularity assumptions on boundaries, coefficients and forcing terms. As it was proven in [Terracini-Tortone-Vita 22], the ratio $v/u$ of two solutions vanishing on a common portion $Γ$ of a regular boundary solves a degenerate elliptic equation whose coefficients behave as $u^2$ at $Γ$. Hence, for any $k\geq 1$ we provide $C^k$ estimates for solutions to the auxiliary degenerate equation under double Dini conditions, actually for general powers of the weight $a>-1$, and we imply $C^k$ estimates for the ratio $v/u$ under triple Dini conditions, as a corollary in the case $a=2$.

math.AP↗

arXiv2022

Convexity for free boundaries with singular term (nonlinear elliptic case)

We consider a free boundary problem in an exterior domain \begin{cases}\begin{array}{cc} Lu=g(u) & \text{in }Ω\setminus K, \\ u=1 & \text{on }\partial K,\\ |\nabla u|=0 &\text{on }\partial Ω, \end{array}\end{cases} where $K$ is a (given) convex and compact set in $\mathbb{R}^n$ ($n\ge2$), $Ω=\{u>0\}\supset K$ is an unknown set, and $L$ is either a fully nonlinear or the $p$-Laplace operator. Under suitable assumptions on $K$ and $g$, we prove the existence of a nonnegative quasi-concave solution to the above problem. We also consider the cases when the set $K$ is contained in $\{x_n=0\}$, and obtain similar results.

math.AP↗

arXiv2022

Regularity of almost minimizers for the parabolic thin obstacle problem

In this paper, we study almost minimizers for the parabolic thin obstacle (or Signorini) problem with zero obstacle. We establish their $H^{σ,σ/2}$-regularity for every $0<σ<1$, as well as $H^{β,β/2}$-regularity of their spatial gradients on the either side of the thin space for some $0<β<1$. A similar result is also obtained for almost minimizers for the Signorini problem with variable Hölder coefficients.

math.AP↗

arXiv2022

Almost minimizers for a sublinear system with free boundary

We study vector-valued almost minimizers of the energy functional $$\int_D\left(|\nabla\mathbf{u}|^2+\frac2{1+q}\left(λ_+(x)|\mathbf{u}^+|^{q+1}+λ_-(x)|\mathbf{u}^-|^{q+1}\right)\right)dx,\quad0 0$, we take the epiperimetric inequality approach and prove the regularity for both almost minimizers and the set of "regular" free boundary points.

math.AP↗

arXiv2022

Symmetry for a fully nonlinear free boundary problem with highly singular term

In this paper we prove radial symmetry for solutions to a free boundary problem with a singular right hand side, in both elliptic and parabolic regime. More exactly, in the unit ball $B_1$ we consider a solution to the fully nonlinear elliptic problem $$ \begin{cases} F(D^2u)=f(u)&\text{in }B_1 \cap \{u >0 \},\\ u=M&\text{on }\partial B_1,\\ 0\le u 0\}$, we cannot apply the well-known Serrin-type boundary point lemma. We circumvent this by an exact assumption on a first order expansion and the decay on the second order, along with an ad-hoc comparison principle. We treat equally the parabolic case of the problem, and state a corresponding result.

math.AP↗

arXiv2021

Almost minimizers for a singular system with free boundary

In this paper we study vector-valued almost minimizers of the energy functional $$ \int_D\left(|\nabla\mathbf{u}|^2+2|\mathbf{u}|\right)\,dx . $$ We establish the regularity for both minimizers and the "regular" part of the free boundary. The analysis of the free boundary is based on Weiss-type monotonicity formula and the epiperimetric inequality for the energy minimizers.

math.AP↗

arXiv2020

Almost minimizers for the thin obstacle problem with variable coefficients

We study almost minimizers for the thin obstacle problem with variable Hölder continuous coefficients and zero thin obstacle and establish their $C^{1,β}$ regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.

math.AP↗