Takwon Kim (KAIST)
Free-boundary analysis in the obstacle problem and its applications (Lecture note)
Free boundary problems are considered today as one of the most important topics in partial differential equations, with abundant applications across various sciences, physics, finance, and biology. This lecture series will focus on exploring the regularity properties of free boundaries, specifically in obstacle-type problems. The primary goal of this series is to provide comprehensive methods for analyzing solutions to obstacle problems and their associated free boundaries. Additionally, we will apply free boundary analysis to problems in financial mathematics.
Junkee Jeon (Kyung Hee University)
Parabolic obstacle problems in mathematical finance
In this talk, I will introduce various forms of parabolic obstacle problems that arise in mathematical finance. I will discuss the existence, uniqueness, and regularity of the strong solutions of the obstacle problem and the free boundary, and how each result can be extended from the perspective of partial differential equations (PDEs).
Minhyun Kim (Hanyang University)
Recent advances in nonlocal nonlinear potential theory
The classical nonlinear potential theory has recently been extended to nonlocal nonlinear potential theory, which studies harmonic functions associated with nonlocal nonlinear operators. In this talk, we focus on the harmonic functions solving the nonlocal Dirichlet problem. As in the study of classical Dirichlet problem, the nonlocal Dirichlet problem can be solved by using Sobolev and Perron solutions. We provide several properties of such solutions. This talk is based on joint works with Anders Björn, Jana Björn, Ki-Ahm Lee and Se-Chan Lee.
Se-Chan Lee (Seoul National University)
Regularity for functions in solution classes
In this talk, we establish the global C1,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.
Taehun Lee (KIAS)
Free boundary problems in curvature flows
Free boundary problems, such as ice melting, involve the study of unknown interfaces and have been intensively researched over the past few decades. These problems can be naturally formulated within a geometric setting. In this talk, I will discuss two types of free boundary problems in geometric flow whose hypersurfaces evolve according to their curvatures: (1) Gauss curvature flow with flat sides, and (2) curvature flows with obstacles. The focus will be on the geometric properties and the development of singularities in these problems.
Minkyu Lim (KAIST)
Very weak solution in mixed local and nonlocal problems
The mixed problem, which is an elliptic equation containing both local and nonlocal terms has been recently highlighted. The mixed problem appears in the superposition of two stochastic processes with different scales such as a classical random walk and a Lévy process. In this talk, we establish a sub-natural gradient estimate of a very weak solution to the mixed problem, when the integrability of the source term is below the natural exponent.
Jehan Oh (Kyungpook National University)
Higher integrability for weak solutions to parabolic multi-phase equations
In this talk, we establish a local higher integrability result for the gradient of a weak solution to a parabolic multi-phase equation. To achieve this, we prove parabolic Poincaré type inequalities and reverse Hölder type inequalities for the gradient of a weak solution in each of the different types of intrinsic cylinders. In particular, we formulate a delicate plan of alternatives and stopping time arguments to address the presence of two different transitions.
Jihoon Ok (Sogang University)
Local regularity results for parabolic systems with general growth
We discuss on regularity theory for parabolic systems of the form
uₜ - div A(Du) = 0 in Ω ×(0,T],
where u: Ω ×(0,T] -> ℝN, u=u(x,t), is a vector valued function and the nonlinearity A: ℝnN -> ℝnN satisfies a general Orlicz growth condition characterized by exponents p and q, subject to the inequality 2n/(n+2)<p<q. It is noteworthy that when if p<2<q, the degeneracy of the system remains indeterminate. This talk focuses on presenting developments in the realm of regularity results concerning the spatial gradient of solutions of the above system, which include the higher higher integrability, Hölder continuity when A(ξ ) satisfies the Uhlenbeck structure, i.e., A(ξ )=φ'(|ξ |)|ξ |-1ξ , and partial Hölder continuity. These results are joint works with Giovanni Scilla and Bianca Stroffolini from University of Naples Federico II, and Peter Hästö from University of Helsinki.
Hyungsung Yun (KIAS)
C1,α-regularity for solutions of degenerate/singular fully nonlinear parabolic equations
This talk presents the C1,α-regularity for viscosity solutions of degenerate/singular fully nonlinear parabolic equations. For this purpose, we develop a new type of Bernstein technique in view of the difference quotient to obtain a priori estimates of the regularized equations. Also, we establish the well-posedness and the uniform C1,α-estimates for the regularized Cauchy–Dirichlet problem.
Jaejin Choi (Pusan National Unverisity)
Regularity for elliptic partial differential equation using Moser iteration
In this talk, we study the regularity of second-order elliptic partial differential equations. This approach was discovered by De Giorgi and John Nash independently for general second-order elliptic and parabolic partial differential equations, in which no differentiability or continuity is assumed of the coefficients in the 1950s. Moser identified a new approach to their basic regularity theory, introducing the technique of Moser iteration and developing it for both elliptic and parabolic problems in the 1960s. Based on this approach, we obtain the boundedness of the supremum of u which is for the weak solution of second-order elliptic partial differential equations.
Gu gyum Ha (Sogang University)
Optimal growth at free boundary points
In this talk we discuss the obstacle problem for the p-laplacian operator i.e. find smallest u such that ∆ₚu≤ f, u ≥ 𝜙 in B1 where ∆ₚ is the p-lapacian operator and 𝜙 is our obstacle. We prove the optimal growth results for the solution u at free boundary points where 𝜙 is given C1,b. This provides regularity results for the solution u.
Aesol Jeon (Seoul National University)
Diffusion-reaction epidemic model with free boundary
This talk is about one of the applications of free boundary problems in mathematical biology, especially epidemiology. The research was conducted by upgrading from the classical epidemiology model by using diffusion terms to consider space variables. We expect that methods of free boundary problems can be useful for many biological or biological problems. I’ll introduce how to set the environment of a problem with brief proof of the theorems and lemmas which we want to show.
Bogi Kim (Kyungpook National University)
Regularity for double phase functionals with two modulating coefficients
In this talk, we deal with local minimizers of functionals with non-standard growth conditions and non-uniform ellipticity properties. Additionally, we establish that a local minimizer is Hölder continuous and that its gradient is also Hölder continuous when additional conditions are provided.