Fourth-order operators with unbounded coefficients in $L^1$ spaces

TL;DR

The paper addresses the generation of analytic semigroups for fourth-order elliptic operators with unbounded coefficients in $L^1(\mathbb{R}^N)$, focusing on $A=-a(x)^2\Delta^{2}$ under growth conditions on $a$. The authors combine a duality-based sectoriality approach, resolvent estimates in a complex sector, and perturbation/approximation techniques to extend known results from bounded coefficients to unbounded ones. They establish $L^1$-generation for the model $A=-(1+|x|^2)^{\alpha}\Delta^{2}$ with $0\leq\alpha\leq 2$ and fully characterize the maximal domain $D(A_1)$, with complementary results in $L^p$ for $1<p<\infty$ and a priori estimates linking operator and function norms. These findings advance the theory of higher-order elliptic operators with unbounded coefficients in $L^1$, providing tools for parabolic problems in unbounded domains and precise domain and resolvent structure.

Abstract

We prove that operators of the form $A=-a(x)^2Δ^{2}$, with suitable growth conditions on the coefficient $a(x)$, generate analytic semigroups in $L^1(\mathbb{R}^N)$. In particular, we deduce generation results for the operator $A :=- (1+|x|^2)^α Δ^{2}$, $0\leqα\leq2$. Moreover, we characterise the maximal domain of $A$ in $L^1(\mathbb{R}^N)$.

Theorems & Definitions (23)

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