Local estimates for some fully nonlinear elliptic equations. (English) Zbl 1159.35343
From the text: We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. All these equations are of the form
\[
F(g^ {-1}W)=f(x,u)h(x,\nabla u),
\]
where \(g^ {-1}\) is the inverse of the metric tensor \(g\) on the manifold, \(W\) is a \((0,2)\) tensor given by
\[
W=\nabla^ 2 u+a(x)du\otimes du+b(x)| \nabla u| ^ 2g + B(x),
\]
\(a\), \(b\), \(B\), \(f\), \(h\) are functions of the corresponding variables and \(F\) is a homogeneous symmetric function of degree one defined on an open convex cone \(\Gamma\) with \(\{\lambda\,\colon \,\lambda_ i>0,\,\forall i\}\subset\Gamma\subset\{\lambda\,\colon \, \sum_ i\lambda_ i>0\}\) normalized so that \(F(1,\ldots,1)=1\) and satisfying the following structure conditions: \(F\) is positive; \(F\) is concave; \(F\) is monotone.
The advantage of our method is that we derive Hessian estimates directly from \(C^0\) estimates. Also, the method is flexible and can be applied to a large class of equations.
The advantage of our method is that we derive Hessian estimates directly from \(C^0\) estimates. Also, the method is flexible and can be applied to a large class of equations.
MSC:
35J60 | Nonlinear elliptic equations |
53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
35B45 | A priori estimates in context of PDEs |
58J05 | Elliptic equations on manifolds, general theory |