Research

🍎 Variational problems for geometric functionals

One of famous variational problems for geometric functionals is Euler's elastica problem: minimize the total squared curvature among smooth curves with fixed length and subject to a certain boundary condition. I am interested in variational problems for the following geometric functionals defined among curves: the total squared curvature, so-called p-bending energy, and O’hara energy. In particular, I often try to deduce qualitative properties of minimizers (or critical points) related to ``shapes’’ such as the number of intersections, locally-convexity, and so on. 

Here are some of the papers which inspired me (not all!):   

• J. Arroyo, Ó. Garay, and Á. Pampano, Boundary value problems for Euler-Bernoulli planar elastica. A solution construction procedure (2020).

• K. Deckelnick and H.-C. Grunau, Boundary value problems for the one-dimensional Will-more equation (2007).

• T. Miura, Elastic curves and phase transitions (2020).

• T. Nagasawa, On Möbius invariant decomposition of the Möbius energy (2018).

• K. Watanabe, Planar p-elastic curves and related generalized complete elliptic integrals (2014).

These are just a few examples, and indeed there are many other interesting papers related to this field. In addition, I am also interested in uniqueness of minimizers, and to this end I applied so-called shooting method to the Euler-Lagrange equation of the bending energy (e.g. [Y. (2022)]). 

🍎 Obstacle problems for elliptic and parabolic equations

There are many contexts in obstacle problems, and in particular I am interested in unilateral obstacle problems for (higher order) elliptic and parabolic equations: the solutions are subject to the constraints that solution must lie above a given function. One of interesting points in this field is regularity of the solutions. In fact, obstacle (function which characterizes the constraint) often prevents from applying a standard regularity theory, and in general it is not a standard matter to discuss the optimal regularity of the solutions to obstacle problems. 

Here I collect some of the papers which are close to my interests (not all!): 

• L. A. Caffarelli and A. Friedman, The obstacle problem for the biharmonic operator (1979).

• A. Dall'Acqua and K. Deckelnick, An obstacle problem for elastic graphs (2018)

• T. Miura, Polar tangential angles and free elasticae (2021).

• M. Müller, On gradient flows with obstacles and Euler's elastica (2021).

• M. Novaga, S. Okabe, Regularity of the obstacle problem for the parabolic biharmonic equation (2015).

I have also been considered the obstacle problems related to the (generalized) bending energy, in particular the optimal regularity of the solutions (see [DMOY (2022)], [Y. (2021)]). 

🍎 Gradient flows: its (full-)convergence V.S. singularity in finite time

Various parabolic equations can be characterized as a gradient flow, also known as steepest descent flow, such as the heat equation, the porous medium equation, the Chan-Hilliard equation, and so on. The solutions to some gradient flows have two (possibly more than two) possibilities, either global-in-time existence (and convergence to equilibrium) or a singularity in finite time (e.g. blow-up of some norm), and which occurs in general depends on the initial data. For instance, for the curve diffusion flow (gradient flow for the length functional), solutions with certain initial data close to a circle exist global-in-time and converge to a circle; on the other hand there is a solution with winding number zero that shrinks to a point in a finite time. I am interested in such a behavior of gradient flows and some of papers close to my interest are as follows (not all!)

• S. Blatt, C. Hopper, and N. Vorderobermeier, A regularized gradient flow for the p-elastic energy (2022).

• M. Edwards, A. Gerhardt-Bourke, J. McCoy, G. Wheeler, and V.-M. Wheeler, The shrinking figure eight and other solitons for the curve diffusion flow (2015).

• M. Müller, On gradient flows with obstacles and Euler’s elastica (2020).

• S. Okabe, and P. Schrader, Convergence of Sobolev gradient trajectories to elastica (preprint).

🍎 C oadient flows: its (full-)convergence V.S. singularity in finite time

Various parabolic equations can be characterized as a gradient flow, also known as steepest descent flow, such as the heat equation, the porous medium equation, the Chan-Hilliard equation, and so on. The solutions to some gradient flows have two (possibly more than two) possibilities, either global-in-time existence (and convergence to equilibrium) or a singularity in finite time (e.g. blow-up of some norm), and which occurs in general depends on the initial data. For instance, for the curve diffusion flow (gradient flow for the length functional), solutions with certain initial data close to a circle exist global-in-time and converge to a circle; on the other hand there is a solution with winding number zero that shrinks to a point in a finite time. I am interested in such a behavior of gradient flows and some of papers close to my interest are as follows (not all!)

• S. Blatt, C. Hopper, and N. Vorderobermeier,