Superparabolic functions related to second order hypoelliptic operators
Let \(L=\sum_{i,j=1}^na^{ij}(x,t)\partial_{x_ix_j}^2+ \sum_{j=1}^n b^j(x,t)\partial_{x_j}-\partial_t,\) \((x,t)\in{\mathbb R}^n\times{\mathbb R},\) where the functions \(a^{ij},b^j\) are real and smooth, and the matrix \(A:=(a^{ij})_{ij}\) is symmetric and positive-semidefinite. In this paper, the authors consider a class of examples of second order operators \(L\) of the above form such that: \(L\) is hypoelliptic, \(L\) is the heat operator outside a compact subset \(F_0 \subset{\mathbb R}^{n+1}\), \(a^{11}(x,t)\not=0\) for any given \((x,t)\in{\mathbb R}^{n+1}\). (The latter is an utterly non-invariant condition.) They consider \(L\)-superparabolic functions (that is to say, functions \(u:\Omega \rightarrow (-\infty,+\infty],\) \(\Omega\subset{\mathbb R}^{n+1}\) an open set, that are lower semicontinuous, belong to \(L_{loc}^1(\Omega)\), and the inequality \(Lu\leq 0\) holds in the distribution sense), and consider the problems of monotone regularization of \(L\)-superparabolic functions, and of finding representation formulas for \(L\)-superparabolic functions. The basic tool is the use of the detailed description the authors have of the fundamental solution \(\Gamma(x,t;y,s)\) of \(L.\) The main results they prove are the theorems below. Theorem 1. Let \(\Omega\subset{\mathbb R}^{n+1}\) be open, and let \(u:\Omega\longrightarrow(-\infty,+\infty]\) be \(L\)-superparabolic. For any given bounded open set \(V\subset\overline{V}\subset\Omega\) there exists and increasing sequence \(\{u_N\}_{N\in{\mathbb N}}\) of smooth \(L\)-superparabolic functions defined on \({\mathbb R}^{n+1}\) such that \(\lim_{N\to +\infty} u_N(x,t)=u(x,t),\) for every \((x,t)\in V\). A representation formula for \(C^2\) functions and Theorem 1 yield then Theorem 2. For \(r>0\) and \((x,t)\in{\mathbb R}^{n+1}\) let \(\Omega_r(x,t):=\{(y,s)\in{\mathbb R}^{n+1}\); \(\Gamma(x,t;y,s)>1/r\}.\) Let \(\Omega\subset{\mathbb R}^{n+1}\) be open, and let \(u:\Omega\rightarrow(-\infty,+\infty]\) be \(L\)-superparabolic. Put \(\mu=-Lu\) (then \(\mu\) is a non-negative measure in \(\Omega\)). For any given \((x,t)\in{\mathbb R}^{n+1}\) and \(r>0\) such that \(\overline{\Omega_r(x,t)}\subset\Omega,\) one has \[ u(x,t)=\int_{\Omega_r(x,t)}u(y,s)E_r(x,t;y,s)dyds+{{1}\over{r}} \int_0^r \int_{\Omega_\sigma(x,t)}\left(\Gamma(x,t;y,s)-{{1}\over{\sigma}} \right) d\mu(x,t) d\sigma, \] where \[ E_r(x,t;y,s)={{1}\over{r}}\left\langle A(y,s)\nabla_y\Gamma(x,t;y,s), {{\nabla_y\Gamma(x,t;y,s)}\over{\Gamma(x,t;y,s)^2}}\right\rangle+ {{1}\over{r}}\text{ div}Y(y,s) \log\left(r\Gamma(x,t;y,s)\right), \] and the vector field \(Y(y,s)={\sum_{j=1}^nb^j(y,s)\partial_{x_j} -\sum_{i,j=1}^n(\partial_{x_i}a^{ij}(y,s))\partial_{x_j}}\).
monotone regularization of \(L\)-superparabolic functions
scientific article; zbMATH DE number 1385176