O Costin | Searcharxiv

arXiv2002

Failure of analytic hypoellipticity in a class of PDOs

For the hypoelliptic differential operators $P={\partial^2_ x}+(x^k\partial_ y -x^l{\partial_t})^2$ introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of $k$ and $l$ left open in the analysis, the operators $P$ also fail to be {\em{analytic}} hypoelliptic (except for $(k,l)=(0,1)$), in accordance with Treves' conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator.

math.CA↗

arXiv2002

Asymptotic properties of a family of solutions of the Painleve equation PVI

In this paper we study the asymptotic behavior for large argument of a family of solutions of the Painlevé equation P$_{\rm VI} arising in the context of Random Matrix Theory [1]. We show this family of solutions are uniquely determined by their asymptotic properties.

math.CA↗

arXiv2002

Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane

We prove existence and uniqueness results for nonlinear third order partial differential equations of the form $$ f_t - f_{yyy} = \sum_{j=0}^3 b_j (y, t; f) ~f^{(j)} + r(y, t) $$ where superscript $j$ denotes the $j$-th partial derivative with respect to $y$. The inhomogeneous term $r$, the coefficients $b_j$ and the initial condition $f(y,0)$ are required to vanish algebraically for large $|y|$ in a wide enough sector in the complex $y$-plane. Using methods related to Borel summation, a unique solution is shown to exist that is analytic in $y$ for all large $|y|$ in a sector. Three partial differential equations arising in the context of Hele-Shaw fingering and dendritic crystal growth are shown to be of this form after appropriate transformation, and then precise results are obtained for them. The implications of the rigorous analysis on some similarity solutions, formerly hypothesized in two of these examples, are examined.

math.AP↗