Nonlinear pseudo-differential equations defined by elliptic symbols on ${\lp}$ and the fractional Laplacian

TL;DR

This paper develops an $L^p$-functional calculus for non-classical symbols of pseudo-differential operators like $a(- riangle)$, proving existence of solutions to nonlinear equations involving fractional Laplacians, with applications to physical models.

Contribution

It introduces a new $L^p$-functional calculus for non-classical symbols and establishes existence results for nonlinear pseudo-differential equations involving fractional Laplacians.

Findings

Constructed suitable domains for the operators using Fourier multiplier theory.

Proved existence of solutions in these domains for nonlinear equations.

Applied the theory to physically relevant equations like the Allen-Cahn equation.

Abstract

We develop an $L^p(\mathbb{R}^n)$-functional calculus appropriated for interpreting "non-classical symbols" of the form $a(-\Delta)$, and for proving existence in $L^q(\mathbb{R}^n)$, some $q > p$, of solutions to nonlinear pseudo-differential equations of the form $[1 + a(-\Delta)]^{s/2} (u) = V(\cdot, u)$. More precisely, we use the theory of Fourier multipliers for constructing suitable domains on which the formal operator appearing in the above equation can be rigorously defined, and we prove existence of solutions belonging to these domains. We also include applications of the theory to equations of physical interest involving the fractional Laplace operator such as the Allen-Cahn equation.