Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients

TL;DR

This paper introduces the concept of p-ellipticity for accretive matrix functions and explores its implications for the $L^p$ theory of elliptic PDEs with complex coefficients, including convexity, embeddings, and functional calculus.

Contribution

It establishes a new condition called p-ellipticity and demonstrates its applications to various aspects of elliptic PDEs with complex coefficients, extending previous results.

Findings

Generalized convexity of power functions (Bellman functions)

Dimension-free bilinear embeddings

L^p-contractivity of semigroups

Abstract

We introduce a condition on accretive matrix functions, called $p$-ellipticity, and discuss its applications to the $L^p$ theory of elliptic PDE with complex coefficients. Our examples are: (i) generalized convexity of power functions (Bellman functions), (ii) dimension-free bilinear embeddings, (iii) $L^p$-contractivity of semigroups and (iv) holomorphic functional calculus. Recent work by Dindo\v{s} and Pipher (arXiv:1612.01568v3) established close ties between $p$-ellipticity and (v) regularity theory of elliptic PDE with complex coefficients. The $p$-ellipticity condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on one hand, and the Hessian of a power function on the other. Our results regarding contractivity extend earlier theorems by Cialdea and Maz'ya.