## Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients.(English)Zbl 0552.35032

The authors extend the classical De Giorgi theorem by proving the Hölder regularity of the weak solutions of $$Lu=0$$ where $$L=\sum^{n}_{i,j=1}\partial_ i(a_{i,j}\partial_ j)$$ is a degenerate ellipic operator in divergence form; precisely they prove the Theorem: Let $$\Omega$$ be a $$\lambda$$-connected (i.e. for every $$x,y\in R^ n$$ it is possible to join x and y by a continuous curve which is a piecewise integral curve of the vector fields $$\pm \lambda_ 1\partial_ 1,...,\pm \lambda_ n\partial_ n)$$ open subset of $$R^ n$$. If $$u\in W_{\lambda}^{loc}(\Omega)$$ and $$L(u)=0$$ in $$\Omega$$ then u is locally Hölder-continuous in $$\Omega$$. $$(\lambda_ 1,\lambda_ 2,...,\lambda_ n$$ are real continuous nonnegative functions such that the quadratic form $$\sum^{n}_{j=1}\lambda^ 2_ j(x)\xi^ 2_ j$$ is equivalent to $$\sum^{n}_{i,j=1}a_{i,j}n(x)\xi_ i\xi_ j$$ and, in addition, satisfy suitable conditions.)
Reviewer: R.Salvi

### MSC:

 35J70 Degenerate elliptic equations 35J15 Second-order elliptic equations 35B65 Smoothness and regularity of solutions to PDEs
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### References:

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