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Fourth-order operators with unbounded coefficients

Federica Gregorio, Chiara Spina, Cristian Tacelli

Abstract

We prove that operators of the form $A=-a(x)^2Δ^{2}$, with $|D a(x)|\leq c a(x)^\frac{1}{2}$, generate analytic semigroups in $L^p(\mathbb{R}^N)$ for $1<p\leq\infty$ and in $C_b(\mathbb{R}^N)$. In particular, we deduce generation results for the operator $A :=- (1+|x|^2)^α Δ^{2}$, $0\leqα\leq2$. Moreover, we characterize the maximal domain of such operators in $L^p(\mathbb{R}^N)$ for $1<p<\infty$.

Fourth-order operators with unbounded coefficients

Abstract

We prove that operators of the form , with , generate analytic semigroups in for and in . In particular, we deduce generation results for the operator , . Moreover, we characterize the maximal domain of such operators in for .
Paper Structure (4 sections, 14 theorems, 93 equations)

This paper contains 4 sections, 14 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. A priori estimates and domain characterization
  3. Generation results in $L^p(\mathbb{R}^N)$
  4. Generation results in $C_{b}(\mathbb{R}^N)$ and $L^\infty(\mathbb{R}^N)$

Key Result

Lemma 1

Let $1<p<\infty$ and assume that $a$ satisfies (gradient). Then $C_c^\infty(\mathbb{R}^N)$ is dense in $D(A_p)$ with respect to the norm

Theorems & Definitions (26)

  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1
  • Lemma 2
  • proof
  • Remark 2
  • Proposition 2
  • Proposition 3
  • proof
  • ...and 16 more