Quasiminimizing properties of solutions to Riccati type equations

Martio, Olli 

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 4, pp. 823-832

Résumé

Solutions $u$ of the Riccati equation $-\nabla ·A(x,\nabla u)={b\left(x\right)\left|\nabla u\right|}^q$ with $A(x,h)·h\approx {\left|h\right|}^p$ and $b$ a bounded function are studied in an open set $\Omega \subset {\mathbf{R}}^n$. It is shown that the solutions $u$ are local quasiminimizers whenever $p-1\le q\le p$ for $p>n$ and $n-1\le q<n$ for $p=n$. This extends the results in the author’s earlier paper [8] where the case $p<n$ was studied. Continuous solutions in the range $p/n+p-1\le q\le p$ are also local quasiminimizers. Examples show that the results are quite sharp.

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Publié le : 2019-02-21

Classification : 35J60, 35J25