Second order elliptic equations whose coefficients have their first derivatives weakly-\(L^ n\)
type: Item
instance of: scholarly article
title: Second order elliptic equations whose coefficients have their first derivatives weakly-\(L^ n\)
zbMATH Open document ID: 0579.35019
DOI: 10.1007/BF01762551
author: Angelo Alvino
author: Guido Trombetti
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1984-01-01T00:00:00Z
review text: Let \(a_{ij}(x)\) \(W^{1,n}(G)\), G open bounded subset of \({\mathbb{R}}^ n\), be such that \(| \xi |^ 2\leq \sum_{i,j}a_{ij}(x)\xi_ i\xi_ j\leq m| \xi |^ 2\), suppose moreover that \[ \text{[spherical rearrangement of }\sum_{i}(\sum_{j}\frac{\partial a_{ij}(x)}{\partial x_ j})^ 2]<\frac{B^ 2}{| x|^ 2},\quad B<\frac{n-2}{2}, \] \[ \text{[spherical rearrangement of }\sum_{irs}(\sum_{k} \frac{\partial}{\partial x_ k}(a_{ir}a_{ks}- a_{ik}a_{rs}))^ 2]\leq \] \[ \leq D^ 2/| x|^ 2,\quad D<\frac{n-2}{2}2^{-1/n}; \] thenthe problem \[ \sum_{i,j}a_{ij}(x)u_{x_ ix_ j}=f\quad in\quad G,\quad u|_{\partial G}=0 \] has a unique solution in \(W^{2,2}(G)\cap W_ 0^{1,2}(G)\) \(\forall f\in L^ 2(G)\).
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 3927463
zbMATH Keywords: spherical rearrangement
zbMATH Keywords: unique solution
reviewed by: Antonio Bove
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3927463