Existence and properties of the Green function for a class of degenerate parabolic equations
We consider the following Dirichlet problem \[ \partial_tu-\sum_{i,j}\partial_i(a_{ij} \partial_ju)=g+\text{div }\vec f,\quad\text{in } \Omega\times (0,T],\tag{P\(_0\)} \] \(u=0\), on the parabolic boundary of \(\Omega\times (0,T]\), where \(a_{ij}=a_{ji}\) are measurable functions such that \[ \nu\omega(x)|\xi|^2\leq\sum_{i,j} a_{ij}(x,t)\xi_i\xi_j\leq {1\over \nu} \omega(x)|\xi|^2, \] almost everywhere in \(\Omega\times (0,T]\). Here \(\Omega\subset\mathbb{R}^n\) is a bounded domain, \(\nu\in (0,1)\), and \(\omega\) is a weight function belonging to the Muckenhoupt class \(A_{1+2/n}\). We prove that problem \((\text{P}_0)\) admits a unique Green function \(\gamma(x,t,\xi,\tau)\) belonging to the natural function space associated with the differential operator in \((\text{P}_0)\). Moreover, the Green function \(\gamma\) satisfies the classical properties concerning the adjoint equations and the representation formula for the solution of general Cauchy problems. Our technique is inspired by \textit{D. G. Aronson's} paper [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 22, 607-694 (1969; Zbl 0182.13802)], where an analogous result is proved for non-degenerate parabolic equations by approaching them with a sequence of parabolic equations with smooth coefficients. In our case, we approximate our degenerate equation by means of a sequence of non-degenerate problems, for which Aronson's results are true.
scientific article; zbMATH DE number 933293