Criteria of solvability for multidimensional Riccati equations

Kurt Hansson Department of Mathematics, Linköping University Vladimir G. Maz'ya Department of Mathematics, Linköping University Igor E. Verbitsky Department of Mathematics, University of Missouri

TBD mathscidoc:1701.332907

Arkiv for Matematik, 37, (1), 87-120, 1997.6
We study the solvability problem for the multidimensional Riccati equation −∇$u$=|∇$u$|^{q}+ω, where$q$>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation −Δ$u$−ω$u$=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions on$R$^{n}in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type$−Lu=f(x, u, ∇u)$+ω where, and$L$is a uniformly elliptic operator.
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@inproceedings{kurt1997criteria,
  title={Criteria of solvability for multidimensional Riccati equations},
  author={Kurt Hansson, Vladimir G. Maz'ya, and Igor E. Verbitsky},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203554941814716},
  booktitle={Arkiv for Matematik},
  volume={37},
  number={1},
  pages={87-120},
  year={1997},
}
Kurt Hansson, Vladimir G. Maz'ya, and Igor E. Verbitsky. Criteria of solvability for multidimensional Riccati equations. 1997. Vol. 37. In Arkiv for Matematik. pp.87-120. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203554941814716.
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