# A potential well theory for the wave equation with nonlinear - download pdf or read online

By Vitillaro E.

Read or Download A potential well theory for the wave equation with nonlinear source and boundary damping terms PDF

Best mathematics books

Get Topos Theory PDF

Put up yr be aware: First released in 1977
------------------------

Focusing on topos theory's integration of geometric and logical rules into the principles of arithmetic and theoretical laptop technology, this quantity explores inner type thought, topologies and sheaves, geometric morphisms, different subjects.

In dieser Arbeit wird die automatische Berechnung von Steigungstupeln zweiter und h? herer Ordnung behandelt. Die entwickelten Techniken werden in einem Algorithmus zur verifizierten globalen Optimierung verwendet. Anhand von Testbeispielen auf einem Rechner wird der neue Algorithmus mit einem Algorithmus aus der Literatur verglichen.

Extra resources for A potential well theory for the wave equation with nonlinear source and boundary damping terms

Example text

Theorem. Let X be reflexive. a) / / the operator A : X -* X* is monotone and hemi-continuous, then A is maximal monotone. b) If A 0 for all x G X. We choose x G X and set xt = txo + (1 — t)x, 0 < t < 1. Then we have 0< {xt- x0, Axt - y0) -(l-t)(xx0, Axt - y0) so that (x - x0, Axt - y0) > 0 for all x £ X, 0 < t < 1. 59) (x - x0, Ax0 - yo) > 0 for all x G X. This implies AXQ = yo> which contradicts the assumption that A is not maximal monotone.

Let ip be a proper convex functional on X. Then the operator dtp C X x X* is monotone. If, in addition,

0 for all Xi G domdtp, x* G dip(xi), i = 1,2, with x\ / x2. Proof. Let x* G dtp(xi) with Xi G domdtp, i = 1,2. Then we have ip(xi)-tp{x2)>{xi-x2,x*2) and y ^ ) - ip{xi) > (x2 - x1,x*1) which implies 0 < (x± — x2, x\ — x2). Assume now that

Let

x. , x G Dc. Assume conversely that the level sets Dc are closed and let {xn)neji be a sequence with x„ —> x. +oo ip(xnk) = d.