Although speculation about such connections is interesting, it seems that the so-called link between Batman, The Shadow, and Louis Feuillade's massive 5-hour silent serial, JUDEX, is pretty thin.
First of all, JUDEX was released in France in 1916. There's no evidence that it was ever released in the US, especially as late as the early 1930s (when the Shadow was created), a time in which sound films had all but totally supplanted silents. It's hard to imagine a theater devoting so much time (money) to a French serial that, at the time, would have seemed extremely "old fashioned." There was also a 1934 French feature version of this title, but again, there's no indication that it was ever released in the US.
It's possible that Walter Gibson and/or Bob Kane/Bill Finger read the novelization of the Feuillade film by Feuillade and Arthur Bernede, but as neither of them (as far as I know) ever listed that as one of their inspirations, it doesn't seem likely.
The Shadow began his life as the mysterious announcer on radio's Detective Story Hour in 1930. The show was a hit, and as people were going to newsstands looking for "that Shadow magazine," publisher Street & Smith decided that they should publish such a title. They hired Gibson to bring the character to life, which he did for nearly two decades afterward. Some have linked the Shadow to such filmic forebears as "The Mysterious Shadow," (Chapter 2 of Judex) and The Shielding Shadow, a 1916 US serial directed by Louis J. Gasnier, but in both cases, there is no similarity aside from the name itself. Those that claim that these are forebears to Gibson's Shadow are on thin ice, especially those that claim that "The Mysterious Shadow" was released alone in the US as a feature. It's virtually inconceivable that this would be the case.
As far as tone goes, Gibson would almost certainly have found some inspiration in the new wave of German Expressionism in film that was, at the time, coming over from Germany, as many fled the country to avoid the Nazis. Filmmakers like Karl Freund, Fritz Lang, Peter Lorre, Edgar Ulmer, Robert Siodmak, and others settled in Hollywood, bringing the Expressionist aesthetic with them. This was most fully expressed in the Universal Monster Films, beginning in 1931, and in the rise of Film Noir, beginning in about 1941.
As far as Batman goes, although Kane claims that his inspiration was Zorro, DaVinci's bat-like ornithopter, and the movie The Bat Whispers, it's clear that his leading inspiration was The Shadow himself. IN fact, as Anthony Tollin has pointed out, Gibson's Shadow story Partners in Peril (Nov. 1936) was directly lifted for The Case of the Chemical Syndicate, Batman's debut story in Detective Comics #27 (May 1939). Not just the story, but the interior illustrations as well. Clearly, Batman would not have existed without the Shadow.
So, it doesn't seem that Judex was involved, or even necessary, for the creation of these two pop culture icons. Saying this is not to denigrate Feuillade's film, as it is, by any measure, a masterpiece, and well-deserving of the acclaim it has received over the last century. It is not, however, likely as a contributing influence for either the Batman or the Shadow.
