"...there is an existant application."
Back when I was in undergraduate school, I sometimes proved math theorems in a fashion different from that in my textbooks. For instance, as a young wannabe-mathematician, I was not in love with proofs by inductive reasoning (I outgrew that - kind o' like recursion in programming...once you REALLY get it, you love it). When only an inductive argument was provided for some theorem, I'd construct a deductive alternative. I discovered that textbook editors pick proofs to minimize page-space. Induction is usually smaller than deduction.
Back when I was merely an applied mathematician, doin' math-for-food in the insurance sector, I still enjoyed generating the occasional proof. What to prove was often new-to-me - not necessarily new-to-the-world. However, I had that old habit from school of proving theorems, lemmas, assertions, etc. just to reassure my confidence in my understanding (OCD much?). With a more infant Internet ('80's) not yet providing ALL existing math knowledge, I often didn't know whether some conjecture that arose in the context of my field had already been proven, but I needed a convincing demonstration...time to sharpen a pencil (?!?) and think.
My point is that whether some formulation of a solution (read app) to a problem already exists should not preclude revisiting the issue (read new app). The new solution may be superior or may simply offer a new perspective.