Takwon Kim | Searcharxiv

arXiv2026

Liouville theorems for nonlocal Lane--Emden inequalities with degenerate kernels

We establish a Liouville-type theorem for the inequality $\mathcal{L}_K u \ge |u|^q$ in $\mathbb{R}^n$, where $\mathcal{L}_K$ is a translation-invariant, degenerate elliptic integro-differential operator whose kernel is even and satisfies $0\le K(z)\le \Lambda|z|^{-n-2s}$. No lower ellipticity bound is imposed on the kernel, and no sign condition is imposed on the solution. We prove that every such solution is trivial when $1 \le q \le \frac{n}{n-2s}$ if $n > 2s$, and for every $q \ge 1$ if $n \le 2s$. The proof combines a test function method with a dyadic decomposition of the nonlocal tail and uses neither the maximum principle nor a fundamental solution.

math.AP↗

arXiv2023

Generalized Schauder Theory and its Application to Degenerate/Singular Parabolic Equations

In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form $$u_t = a^{i'j'}u_{i'j'} + 2 x_n^{\gamma/2} a^{i'n} u_{i'n} + x_n^{\gamma} a^{nn} u_{nn} + b^{i'} u_{i'} + x_n^{\gamma/2} b^n u_{n} + c u + f \quad (\gamma \leq1).$$ When the equation above is singular, it can be derived from Monge--Amp\`ere equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution $u$, which is called $s$-polynomial. To prove $C_s^{2+\alpha}$-regularity and higher regularity of the solution $u$, we establish generalized Schauder theory which approximates coefficients of the operator with $s$-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap method to obtain higher regularity.

math.AP↗