Criteria for the L p -dissipativity of systems of second order differential equations

TLDR: In this paper, the authors give complete algebraic characterizations of the Dirichlet dissipativity of partial differential operators of the form ∆ ∆( ∆) ( ∆ + ∆+ ∆ (∆+∆(∆) ∆)) for a general scalar operator with complex coefficients, where ∆ is the sharp angle of dissipativity.

Abstract: We give complete algebraic characterizations of the L p -dissipativity of the Dirichlet problem for some systems of partial differential operators of the form $\partial_{h}({\mathop{\cal A}\nolimits}^{hk}(x)\partial_{k})$ , where ${\mathop{\cal A}\nolimits}^{hk}(x)$ are m× m matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is L p -dissipative if and only if $$ \left({1\over 2}-{1\over p}\right)^{2} \leq {2(\nu-1)(2\nu-1)\over (3-4\nu)^{2}}, $$ ν being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the L p -dissipativity of the operator $\partial_{h} ({\mathop{\cal A}\nolimits}^{h}(x)\partial_{h})$ , where ${\mathop{\cal A}\nolimits}^{h}(x)$ are m× m matrices with complex L1loc entries, and we describe the maximum angle of L p -dissipativity for this operator. Keywords: L p -dissipativity, Algebraic conditions, Elasticity system Mathematics Subject Classification (2000): 47D03, 47D06, 47B44, 74B05