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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.

##### 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
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