We introduce a condition on accretive matrix functions, called p-ellipticity, and discuss its applications to the Lp theory of elliptic PDEs with complex coefficients. Our examples are: (i) generalized convexity of power functions (Bellman functions), (ii) dimension-free bilinear embeddings, (iii) Lp-contractivity of semigroups, and (iv) holomorphic functional calculus. Recent work by Dindoš and Pipher established close ties between p-ellipticity and (v) regularity theory of elliptic PDEs with complex coefficients. The p-ellipticity condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on the one hand, and the Hessian of a power function on the other. Our results regarding contractivity extend earlier theorems by Cialdea and Maz’ya.
Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients
A. Carbonaro;
2020-01-01
Abstract
We introduce a condition on accretive matrix functions, called p-ellipticity, and discuss its applications to the Lp theory of elliptic PDEs with complex coefficients. Our examples are: (i) generalized convexity of power functions (Bellman functions), (ii) dimension-free bilinear embeddings, (iii) Lp-contractivity of semigroups, and (iv) holomorphic functional calculus. Recent work by Dindoš and Pipher established close ties between p-ellipticity and (v) regularity theory of elliptic PDEs with complex coefficients. The p-ellipticity condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on the one hand, and the Hessian of a power function on the other. Our results regarding contractivity extend earlier theorems by Cialdea and Maz’ya.File | Dimensione | Formato | |
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