Criteria for the LpL^{p}-dissipativity of systems of second order differential equations

We give complete algebraic characterizations of the $L^{p}$-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form $\partial_{h}({\mathscr A}^{hk}(x)\partial_{k})$, were ${\mathscr A}^{hk}(x)$ are $m\times m$ 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 $$({1\over 2}-{1\over p})^{2} \leq {2(\nu-1)(2\nu-1)\over (3-4\nu)^{2}},$$ $\nu$ being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the $L^{p}$-dissipativity of the operator $\partial_{h} ({\mathscr A}^{h}(x)\partial_{h})$, where ${\mathscr A}^{h}(x)$ are $m\times m$ matrices with complex $L^{1}_{\rm loc}$ entries, and we describe the maximum angle of $L^{p}$-dissipativity for this operator.Comment: 42 pages, LaTeX, no figure

Criterion for the LpL^{p}-dissipativity of second order differential operators with complex coefficients

We prove that the algebraic condition $|p-2| |< {\mathscr Im}{\mathscr A}\xi,\xi>| \leq 2 \sqrt{p-1}$ (for any $\xi\in\mathbb{R}^{n}$) is necessary and sufficient for the $L^{p}$-dissipativity of the Dirichlet problem for the differential operator $\nabla^{t}({\mathscr A}\nabla)$, where ${\mathscr A}$ is a matrix whose entries are complex measures and whose imaginary part is symmetric. This result is new even for smooth coefficients, when it implies a criterion for the $L^{p}$-contractivity of the corresponding semigroup. We consider also the operator $\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$, where the coefficients are smooth and ${\mathscr Im}{\mathscr A}$ may be not symmetric. We show that the previous algebraic condition is necessary and sufficient for the $L^{p}$-quasi-dissipativity of this operator. The same condition is necessary and sufficient for the $L^{p}$-quasi-contractivity of the corresponding semigroup. We give a necessary and sufficient condition for the $L^{p}$-dissipativity in $\mathbb{R}^{n}$ of the operator $\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$ with constant coefficients.Comment: 37 pages, LaTeX, no figure

A quasi-commutativity property of the Poisson and composition operators

Let $\Phi$ be a real valued function of one real variable, let $L$ denote an elliptic second order formally self-adjoint differential operator with bounded measurable coefficients, and let $P$ stand for the Poisson operator for $L$. A necessary and sufficient condition on $\Phi ensuring the equivalence of the Dirichlet integrals of$\Phi\circ Ph$and$P(\Phi\circ h)$ is obtained. We illustrate this result by some sharp inequalities for harmonic functions.Comment: 13 pages, no figure

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 $\varphi:R^+ \to R^+$. Under the assumption that the $Im\, A$ is symmetric, we prove that the condition $|s\, \varphi'(s)| \, | \langle Im\, A (x)\, \xi,\xi\rangle |\leq 2\, \sqrt{\varphi(s)\, [s\, \varphi(s)]'}\, \langle Re\, A(x) \, \xi,\xi\rangle$ (for almost every $x\in\Omega\subset R^N$ and for any $s>0$, $\xi\in R^N$) is necessary and sufficient for the functional dissipativity of $E$

COMPLETENESS THEOREMS: FICHERA’S FUNDAMENTAL RESULTS AND SOME NEW CONTRIBUTIONS

After recalling Fichera&rsquo;s fundamental results in the study of the problem of the completeness of particular solutions of a partial differential equation, we give some new completeness theorem. They concern the Dirichlet problem for a general elliptic operator of higher order with real constant coefficients in any number of variables.After recalling Fichera&rsquo;s fundamental results in the study of the problem of the completeness of particular solutions of a partial differential equation, we give some new completeness theorem. They concern theDirichlet problem for a general elliptic operator of higher order with realconstant coefficients in any number of variables

Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains

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A complement to potential theory in the Cosserat elasticity

In this paper we investigate the internal second, third and fourth boundary value problems of the three-dimensional Cosserat elasticity by means of potential theory. The obtained integral representations differ from the classical ones. These results complete the ones related to the first BVP, which have recently been obtained by the authors