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De Giorgi’s theorem for a class of strongly degenerate elliptic equations. (English) Zbl 0543.35041
Let \(L=\sum^{n}_{i,j=1}\partial_ i(a_{ij}\partial_ j)\) be a second order degenerate non uniformly elliptic operator, where \(a_{ij}=a_{ji}\in L^{\infty}(\Omega), i,j=1,...,n\) (\(\Omega\) is a bounded open subset of \({\mathbb{R}}^ n)\) and let us suppose that there exists \(c_ 0>0\) such that \[ c_ 0\sum^{n}_{j=1}\lambda^ 2_ j(x)\xi^ 2_ j\leq \sum^{n}_{i,j=1}a_{ij}(x)\xi_ i\xi_ j\leq c_ 0^{-1}\sum^{n}_{j=1}\lambda^ 2_ j(x)\xi^ 2_ j \] where \(\lambda_ j(x)=\lambda_ j^{(1)}(x_ 1)...\lambda_ j^{(n)}(x_ n)\) are nonnegative continuous functions such that: i) \(\lambda_ j^{(k)}\) is an even \(C^ 1\)-function outside of the origin, \(j,k=1,...,n\), \(j\neq k\); ii) \(\lambda_ j^{(j)}\) is Lipschitz continuous, \(j=1,...,n\); iii) there exists \(\rho_{j,k}>0\) such that \(0\leq t(\lambda_ j^{(k)})'(t)\leq \rho_{j,k}\lambda_ j^{(k)}(t), \forall t>0\), \(j,k=1,...,n\), \(j\neq k\); iv) \(\Omega\) is locally \((\lambda_ 1\partial_ 1,...,\lambda_ n\partial_ n)\)-connected, i.e., \(\forall x\in \Omega\) and for every neighbourhood W of x there exists a neighbourhood V of x such that, \(\forall y\in V\) it is possible to connect x and y by a continuous curve \(\gamma\) which is a chain of a finite number of integral curves of the vector fields \(\pm \lambda_ 1\partial_ 1,...,\pm \lambda_ n\partial_ n\), \(\gamma\) lying in V.
The hypothesis of \(\lambda\)-connectedness can be viewed, in a suitable sense, as a ”weak formulation” of Hörmander’s hypothesis on the Lie algebra generated by \(\lambda_ 1\partial_ 1,...,\lambda_ n\partial_ n\). Due to iv), we can define a suitable metric d in \(\Omega\) closely fitting the operator L; so we can adapt the classical Moser’s technique to the metric space of homogeneous tye (\(\Omega\),d). If i)-iv) are satisfied, the weak solutions of \(Lu=0\) are locally Hölder continuous. The proof is only sketched; a detailed version of it appeared in the authors’ article in Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. 10, 523-541 (1983).

35J70 Degenerate elliptic equations
35J25 Boundary value problems for second-order elliptic equations