BARBIE ISN'T JUST A DOLL, she's a trademark Mattel is fighting for on- and offline. The battle continued last month, when the company filed a copyright- infringement lawsuit against Miller's magazine, which caters to collectors of anything and everything Barbie. ""[She] is one of our most valuable assets,'' says Lisa McKendall of Mattel, which sells more than a billion dollars' worth of Barbies each year. ""We will protect our trademark.'' That may be bad news for doll fans like Dean Brown, who created the Barbie Chronicles (www.erols.com/browndk) Web site for fun, not profit. His photographs place Barbie in famous works of art, like Whistler's painting of his mother and Botticelli's ""Birth of Venus.'' ""I don't think I'm decreasing her value in the marketplace,'' he says. Neither does the creator of the Plastic Princess Page (http://d.armory.com/~zenugirl/barbie.html), which offers up a complete history of Barbie's origins, descriptions of her many outfits and links to other sites devoted to doll collecting. But Mattel insists its site (www.barbie.com) is the official source of information on Barbie. Will the company go after its online competition? ""If it's unauthorized in print or on the Web, we will act,'' says McKendall.