B777 FLT CREW IS INFORMED AFTER DEPARTURE BY ATC THAT RUBBER WAS FOUND ON THE RWY. FLT CREW CONTINUES TO DESTINATION FOR UNEVENTFUL LANDING.

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AFTER DEPARTING ZZZZ DURING LAST PHASE OF THE CLBOUT; ATC ADVISED THAT RUBBER WAS FOUND ON THE RWY AND INQUIRED AS TO THE PERFORMANCE/CONDITION OF OUR FLT. WE NOTED NO REAL NOTICEABLE PERFORMANCE; VIBRATION; OR HANDLING ISSUES. AT THIS POINT WE MOST LIKELY WOULD NOT HAVE BEEN AWARE OF ANY CIRCUMSTANCES HAD ATC NOT INQUIRED. THERE WAS A RAIN SHOWER ON THE RWY; THE RWY AT ZZZZ IS SOMEWHAT ROUGH; AND THE AIRPLANE WAS DEPRESSURIZED DUE TO 'PACKS OFF' TKOF (MORE NOISY). AS A RESULT; EXTRA TKOF ROLL VIBRATION AND NOISE WAS DIFFICULT TO DETECT. WE ASKED THE FLT ATTENDANTS IF THEY HAD NOTICED ANY UNUSUALLY HIGH VIBRATIONS; NOISES; ETC. THIS BROUGHT MIXED RESULTS WITH SOME SAYING SLIGHTLY MORE; AND SOME NOT NOTICING ANYTHING. NONE OF THEM INITIATED A CALL TO THE COCKPIT AFTER DEP TO CALL ANYTHING TO OUR ATTN. WE PULLED UP THE 'GEAR' SYNOPTIC; AND FOUND NO UNUSUAL PRESSURES OR TEMPS. WE FURTHER INQUIRED AS TO THE LOCATION OF THE RUBBER ON THE RWY IN ORDER TO DETERMINE IF IT MIGHT HAVE BEEN DUE TO A LNDG ACFT. WE CONTACTED DISPATCH AND TECHNICAL DEPTS IN ORDER TO DISCUSS THE ISSUE. AFTER DETERMINING THAT THERE WAS A CHANCE THAT SOME RUBBER MAY HAVE SEPARATED FROM 1 OR 2 TIRES; DUE TO THE AMOUNT OF RUBBER FOUND; THAT NO PERFORMANCE; STRUCTURAL; SAFETY ISSUES EXISTED; WE DECIDED THAT CONTINUING THE FLT WAS THE PREFERRED ACTION. POSSIBLE ANTICIPATED LNDG ISSUES WERE CONSIDERED. THE OPS MANUAL WAS REVIEWED; AND IT WAS DECIDED THAT LNDG IN GOOD WX; ON A SMOOTHER RWY; AT A LIGHTER WT; WAS PREFERRED TO MARGINAL WX; ROUGH RWY; HVY WT CONDITIONS. NO ISSUES WERE NOTED DURING CRUISE; AND THE TIRE PRESSURES WERE MONITORED DURING THE FLT TO MAKE SURE WE DIDN'T HAVE ANY LATENT FAILURES. THE FLT ATTENDANTS AND PAX WERE PROPERLY BRIEFED AS TO THE SITUATION; AND INTENTIONS. UPON CONTACT WITH APCH; WE DECLARED AN EMER AND ADVISED THE MAINT PERSONNEL; AND EMER PERSONNEL WHERE WE WOULD EXIT THE RWY. WE HAD THEM REVIEW THE GEAR; TIRES; AND STRUCTURE OF THE AIRPLANE TO DETERMINE ANY POSSIBLE ISSUES. A NORMAL APCH AND LNDG WAS MADE BY THE CAPT WITH FLAPS 30 DEGS. WE EXITED RWY AND STOPPED THE AIRPLANE. EMER AND MAINT PERSONNEL INSPECTED THE AREA AND AIRPLANE; AND ADVISED THAT ALL TIRES REMAINED INFLATED AND NO HOT AREAS EXISTED. MAINT INDICATED THAT TIRE #12 HAD SHED RUBBER; BUT REMAINED INTACT AND INFLATED. WE WERE CLRED TO TAXI TO THE GATE BY MAINT. AFTER PARKING AT THE GATE; FURTHER INSPECTION FOUND SLIGHT DAMAGE TO THE R-HAND GEAR DOOR; AND FLAPS. NO OTHER INDICATIONS OR INCIDENTS WERE NOTED.