Research

Preprints

1. Kunyi (Mark) Ma. The Alt-Phillips Multiple Membrane Free Boundary Problem.
   arXiv:2611.11736 [math.AP]. Preprint, 2026.
   Abstract

   Motivated by diffusion–reaction processes in porous catalysts, we study an Alt–Phillips model for ordered membranes with weighted gap interactions. The model connects adhesive harmonic junctions [DSS24] and forced multiple membranes [SY19] through the exponent $0<\gamma <1$. For arbitrary positive weights, we prove optimal $C^{1,\beta -1}$ regularity, where $\beta =2/(2-\gamma )$, and obtain nontrivial β-homogeneous blowup limits through Weiss almost monotonicity. Under strict discrete concavity of the weights, we establish a gluing principle, minimality of flat single-gap cones, and $C^{1,\alpha }$ free boundary regularity at points admitting such blowups. We also classify all one-dimensional vertically critical cones. For three membranes, we identify cone families that are non-minimizing for small γ and minimizing near $\gamma =1$.

2. Kunyi (Mark) Ma. A Short Proof of Optimal Regularity for minimizers of the Alt-Phillips Problem.
   arXiv:2607.08685 [math.AP]. Preprint, 2026.
   Abstract

   We give a self-contained short proof of optimal regularity for minimizers of the Alt-Phillips Free Boundary Problem for $\gamma \in (0,1)$. We adopt a dichotomy argument that originates from [DS20].

3. Nicholas Gismondi, Kunyi (Mark) Ma, Mandon Pathak, Alexandru F. Radu. Non-unique solutions to the periodic gKdV equation.
   arXiv:2606.06916 [math.AP]. Preprint, 2026.
   Abstract

   In this paper we utilize a convex integration scheme to construct non-trivial weak solutions to the $k$-generalized KdV equation which lie in $$\bigcap _{\epsilon >0}{C}_t^0{L}_x^{k-\epsilon }\left(\right[0,1]\times 𝕋)$$ and, when $k\ge 3$, it may also be chosen in $$\bigcap _{\epsilon >0}{C}_t^0{H}_x^{\frac{1}{2}-\frac{1}{k}-\epsilon }\left(\right[0,1]\times 𝕋)$$ attaining identically 0 initial data. Since our solutions do not lie in $C_t^0{L}_x^k$, this requires introducing a new notion of weak solution, which is in fact stronger than the classical notion of a weak solution when the nonlinearity is integrable. This result shows that a necessary condition for unconditional uniqueness for $k$-gKdV is that the nonlinearity lies in $C_t^0{L}_x^1$. In the case of KdV this is in fact also sufficient.