The goal of this paper is to produce a series of counterexamples for the $L^p$ spectral radius conjecture, $1 <p <\infty$, for double-layer potential operators associated to a distinguished class of elliptic systems in polygonal domains in $\mathbb R^2$. More specifically the class under discussion is that of second-order elliptic systems in two dimensions whose coefficient tensor (with constant real entries) is symmetric and strictly positive definite. The general techniques employed are those of the Mellin transform and Calderón-Zygmund theory. For the case $p\in(1,4)$, we construct a computer-aided proof utilizing validated numerics based on interval analysis.

In this paper we study local well-posedness in the energy space for a family of dispersive equations that can be seen as dispersive "interpolations" between the KdV and the Benjamin-Ono equation.

In this paper we use the comparison method to study the generation and metastability of patterns for a class of local and nonlocal nonlinear evolution equations. We show that for a typical initial datum, the pattern generated can last for a very long time, but is eventually destroyed.

In this article, we prove a necessary and sufficient condition for symmetric Palais-Smale conditions, then apply it to assert the existence of three positive solutions of the equation $ (1.1) $ in an axially symmetric domain $D_{R}$ in which one is axially symmetric and the other two are nonaxially symmetric.

We prove the existence of multiple nonnegative nonsemitrivial solutions for a quasilinear elliptic system, defined on an arbitrary domain (bounded or unbounded).