Criterion for the functional dissipativity of second order differential operators with complex coefficients

In the present paper we consider the Dirichlet problem for the second order differential operator $E=\nabla(A \nabla)$,where $A$ is a matrix with complex valued $L^\infty$ entries. We introduce the concept of dissipativity of $E$ with respect to a given function $φ:R^+ \to R^+$. Under the assumption that the $Im\, A$ is symmetric, we prove that the condition $|s\, φ'(s)| \, | \langle Im\, A (x)\, ξ,ξ\rangle |\leq 2\, \sqrt{φ(s)\, [s\, φ(s)]'}\, \langle Re\, A(x) \, ξ,ξ\rangle$ (for almost every $x\inΩ\subset R^N$ and for any $s>0$, $ξ\in R^N$) is necessary and sufficient for the functional dissipativity of $E$.

No reviews yet.

No comments yet.