The Dirichlet problem for sub-elliptic second order equations
Let \(\Omega\) be a smooth bounded open subset of \(\mathbb{R}^n\), and let \(L\) be a second-order differential operator: \[ L-\sum_{1\leq i,j\leq n} \partial_{x_i}(a_{ij} \partial_{x_j}u) +cu, \] where the matrix \((a_{ij})\) is positive semidefinite on a neighbourhood of \(\overline\Omega\), \(\partial\Omega\) is noncharacteristic for \(L\), \(c(x)\geq c_0>0\), the coefficients \(a_{ij},c\) are \({\mathcal C}^\infty\) real functions, and \(L\) satisfies the Fefferman-Phong's condition. Then, by using the Dirichlet principle associated with the operator \(L\) and a comparison principle for \(L\) with an operator of the form sum-of-squares, the authors prove that there exists a unique solution \(u\in{\mathcal C}^\infty (\overline\Omega)\) of the Dirichlet problem \[ Lu=f\text{ in }\Omega,\;u|_{\partial\Omega}=g, \quad f\in{\mathcal C}^\infty (\overline\Omega),\quad g\in {\mathcal C}^\infty (\partial\Omega). \]
scientific article; zbMATH DE number 1383226
scientific article