# zbMATH — the first resource for mathematics

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).

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