Dissipative operator

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In mathematics, a dissipative operator is a linear operator A defined on a dense linear subspace D(A) of a real or complex Hilbert space (H, 〈 , 〉), taking values in H, for which

\mathrm{Re} \langle x, A x \rangle \leq 0

for every x in D(A). If there exists a constant ω ≥ 0 such that

\mathrm{Re} \langle x, A x \rangle \leq \omega \langle x, x \rangle \equiv \omega \| x \|^{2}

for every x in D(A), then A is said to be a quasidisspative operator.

[edit] Examples

x \cdot A x = x \cdot (-x) = - \| x \|^{2} \leq 0,
so A is a dissipative (and hence quasidissipative) operator.
  • Consider H = L2([0, 1]; R) with its usual inner product, and let Au = u′ with domain D(A) equal to those functions u in the Sobolev space H1([0, 1]; R) with u(1) = 0. D(A) is dense in L2([0, 1]; R). Moreover, for every u in D(A), using integration by parts,
\langle u, A u \rangle = \int_{0}^{1} u(x) u'(x) \, \mathrm{d} x = - \frac1{2} u(0)^{2} \leq 0.
Hence, A is a dissipative (and hence quasidissipative) operator.
\langle u, \Delta u \rangle = \int_{\Omega} u(x) \Delta u(x) \, \mathrm{d} x = - \int_{\Omega} \big| \nabla u(x) \big|^{2} \, \mathrm{d} x = - \| \nabla u \|_{L^{2} (\Omega; \mathbf{R})} \leq 0,
so the Laplacian is a dissipative (and hence quasidissipative) operator.

[edit] References

  • Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations, Second edition, Texts in Applied Mathematics 13, New York: Springer-Verlag, 356. ISBN 0-387-00444-0.  (Definition 11.25)