We give complete algebraic characterizations of the $L^{p}$-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form $\de_{h}(A^{hk}(x)\de_{k})$, were $A^{hk}(x)$ 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 $$\left({1\over 2}-{1\over p}\right)^{2} \leq {2(\nu-1)(2\nu-1)\over (3-4\nu)^{2}},$$ $\nu$ being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the $L^{p}$-dissipativity of the operator $\de_{h} (\A^{h}(x)\de_{h})$, where $\A^{h}(x)$ 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.

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

CIALDEA, Alberto;
2006

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 $\de_{h}(A^{hk}(x)\de_{k})$, were $A^{hk}(x)$ 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 $$\left({1\over 2}-{1\over p}\right)^{2} \leq {2(\nu-1)(2\nu-1)\over (3-4\nu)^{2}},$$ $\nu$ being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the $L^{p}$-dissipativity of the operator $\de_{h} (\A^{h}(x)\de_{h})$, where $\A^{h}(x)$ 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.
File in questo prodotto:
File Dimensione Formato  
cialmaz2.pdf

non disponibili

Tipologia: Documento in Post-print
Licenza: DRM non definito
Dimensione 311.24 kB
Formato Adobe PDF
311.24 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11563/3608
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact