Title & Abstract

Intensive Lectures 

• Se-Chan Lee (KIAS)
  
  Regularity for viscosity solutions of degenerate/singular equations (Lecture notes)
  1. Classical theory for equations in non-divergence form: viscosity solutions, Harnack inequality, compactness, comparison principle. 

2. Regularity for viscosity solutions of degenerate/singular elliptic equations (by Imbert and Silvestre): compactness method, Ishii-Lions method, equations that hold in "large gradient”.

3.  Regularity for viscosity solutions of degenerate/singular parabolic equations (by Jin and Silvestre): approximation method, two alternatives, Bernstein method.

• Kyeong Song (KIAS)
  
  A basic regularity theory for nonlocal equations (Lecture notes)
  This series of lectures are concerned with the De Giorgi-Nash-Moser theory for nonlocal equations with measurable coefficients. 

In the first lecture, we will introduce basic concepts of nonlocal operators and related function spaces, and then derive a Caccioppoli estimate. 

In the second lecture, we will prove the local boundedness of weak solutions by using a De Giorgi type argument. We will also derive a logarithmic estimate. 

In the third lecture, we will derive an expansion of positivity lemma to prove the local Hölder continuity of weak solutions. If time permits, we will also briefly discuss nonlocal Harnack inequalities for weak solutions. 

Invited Talks

• Junkee Jeon (Kyung Hee University)
  
  Parabolic Double Obstacle Problems in Stochastic Control
  In this talk, we discuss parabolic double obstacle problems that arise from stochastic control problems. In particular, when the state process follows a Markov process under a finite horizon, we examine how the solutions to two-state optimal switching problems and two-sided singular control problems are related to parabolic obstacle problems. We illustrate these connections with examples including a firm’s reversible investment problem, a job switching problem, and a consumption adjustment cost problem.

• Takwon Kim (Sungshin Women's University)
  
  Characterization of self-similar solutions to semilinear heat equations in the supercritical case
      We characterize self-similar solutions to semilinear heat equations in the supercritical region. First of all, we prove that vanishing-tail solutions must be radially symmetric with a particular decay rate at the infinity. The radial symmetry allows us to investigate the asymptotic behavior of self-similar solutions by utilizing various ODE-type approaches. In particular, we show the energy gap results among suitable weak solutions, and provide several corollaries concerning the classification and the existence of vanishing-tail solutions.

• Mikyoung Lee (Pusan National University)
  
  L^q-regularity for nonlinear elliptic equations with lower order terms
  We study nonlinear elliptic equations of p-Laplacian type that include lower order terms involving nonnegative potentials satisfying a reverse Hölder-type condition. Our focus lies in establishing sharp interior and boundary L^q estimates for both the gradient of weak solutions and the lower order terms—separately and independently—under optimal regularity assumptions on the coefficients and domain boundaries. A key feature of our approach is that it avoids the use of Fefferman–Phong-type inequalities, which are fundamental in the analysis of linear equations. This enables a new framework for handling nonlinear problems with singular potentials beyond the classical linear theory.

• Taehun Lee (Konkuk University)
  
  Ancient mean curvature flows with finite total curvature
  Ancient flows have been intensively studied over the past decade as singularity models for the mean curvature flow. In the spirit of the parabolic Liouville-type theorem for the non-compact case, flows with prescribed asymptotic behavior play an important role. In this context, we present family of ancient mean curvature flows that converge to a given two-sided complete embedded minimal hypersurface in Euclidean space. We establish that these flows possess geometric properties such as finite total curvature, finite mass drop, and mean convexity for one family of these flows. This work is joint with Kyeongsu Choi (KIAS) and Jiuzhou Huang (KIAS).

• Jehan Oh (Kyungpook National University)
  
  Gradient higher integrability for degenerate parabolic double phase systems with two modulating coefficients
  In this talk, we present an interior gradient higher integrability result for weak solutions to degenerate parabolic double phase systems involving two modulating coefficients. To prevent the simultaneous vanishing of these coefficients, we assume that their sum is bounded from below by a positive constant. Although the presence of two modulating coefficients increases mathematical complexity, it provides a more suitable framework for modeling strongly anisotropic materials. To establish the higher integrability result, we introduce a suitable intrinsic geometry and develop a refined analytical approach to separate and investigate the distinct phases--namely, the p-phase, q-phase, and (p,q)-phase.

• Jihoon Ok (Sogang University)
  
  Nonlocal equations with degenerate weights
  We discuss on fractional weighted Sobolev spaces with degenerate weights and related weighted nonlocal integrodifferential equations. We provide embeddings and Poincare inequalities for these spaces and show robust convergence when the parameter of fractional differentiability goes to $1$. Moreover, we prove local H\"older continuity and Harnack inequalities for solutions to the corresponding nonlocal equations. The regularity results naturally extend those for degenerate linear elliptic equations presented in [Comm. Partial Differential Equations 7 (1982); no. 1; 77--116] by Fabes, Kenig, and Serapioni to the nonlocal setting. This is a joint work with Linus Behn, Lars Diening and Julian Rolfes from Bielefeld.