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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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