A B777-200 ON APCH AT 4500 FT ENCOUNTERED 2 LIGHTNING STRIKES. FLT DIRECTORS DISAPPEARED BRIEFLY. FLAP FAIRING WAS DAMAGED.

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DISPATCH HAD GIVEN US A HEADS UP ABOUT 1 HR OUT OF LAX THAT A HVY RAIN CELL WAS OVER BUR AND MAY BE IN THE LAX AREA ON OUR ARR. WE WERE NOTIFIED BY SOCAL APCH AFTER COMMENCING OUR APCH THAT A PRECEDING GULFSTREAM HAD RPTED A LIGHTNING STRIKE AND SEVERE TURB EARLIER ON THE APCH. CAPT TRIED TO DETERMINE FROM APCH HOW LONG AGO THE RPT WAS MADE. IT SOUNDED LIKE THERE WERE SEVERAL ACFT BTWN US AND THE ACFT MAKING THE RPT. RADAR SHOWED AN AREA OF YELLOW RETURNS AHEAD AND TO OUR L. ALTHOUGH WE CONSIDERED BREAKING OFF THE APCH; BY THE TIME DETAILS WERE CLARIFIED AS TO HOW LONG AGO THE RPT WAS MADE AND WHAT FOLLOWING ACFT WERE RPTING; THE OPTION TO CONTINUE THE APCH SEEMED TO BE THE SAFEST OPTION WITH TFC ON OUR R FOR RWY 24R AND THE SAME WX CONDITIONS ON POSSIBLE EXIT RTES. WE ENTERED THE AREA OF HVY RAIN 10-15 NM ON FINAL AND WE ENCOUNTERED 2 LIGHTNING STRIKES ON OR NEAR OUR ACFT. THE FLT DIRECTORS BRIEFLY DISAPPEARED; BUT RETURNED AFTER RECYCLING THE FLT DIRECTOR SWITCHES. THE REMAINDER OF THE APCH AND LNDG WERE UNEVENTFUL. DAMAGE TO A FLAP FAIRING ON THE L WING WAS DISCOVERED AFTER PARKING. A SMALL HOLE WAS ALSO NOTED. DISPATCH AND MAINT WERE CALLED BY THE CAPT AFTER LNDG. SUPPLEMENTAL INFO FROM ACN 613084: ON APCH; CTLR ADVISED THAT ACFT AHEAD RPTED TURB AND LIGHTNING. OUR RADAR SHOWED WX AHEAD AND TO L. I ASKED TIME AND LOCATION OF RPT; AND WAS ADVISED IT WAS FROM AN ACFT 25 MI AHEAD. WE WERE LESS THAN 25 MI FROM THE ARPT AT THE TIME. WE WERE ADVISED THAT RPTING ACFT HAD CONTINUED APCH AND LANDED. THE 5+ ACFT AHEAD AND TO OUR R FOR RWY 24R RPTED LIGHTNING IN THE AREA; BUT WERE UNABLE TO CONTINUE AND LAND. WE ENCOUNTERED A BRIEF PERIOD OF TURB AND EXPERIENCED 2 LIGHTNING EVENTS; WHICH WE THOUGHT WERE STRIKES. WE BROKE OUT AT APPROX 3800 FT; AND LANDED WITHOUT FURTHER INCIDENT. WE ADVISED MAINT; AND LOGGED THE EVENT. ON THE GND; INSPECTION SHOWED THAT FIBERGLASS FLAP DRIVEN FAIRING ON L TRAILING EDGE OF WING HAD DELAMINATED AND FRAYED; AND THE NOSE HAD 2 SMALL MARKS; WHICH WERE PROBABLY THE CONTACT POINTS OF THE STRIKES. I CALLED DISPATCH AND DUTY MGR WITH RPT.