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By Vitillaro E.

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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.

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A potential well theory for the wave equation with nonlinear source and boundary damping terms by Vitillaro E.


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