This is a great example showing that practice at the piano to address a pianistic problem is only half the answer. Start working on this AWAY from the piano. Specifically, use your weak hand as much as you can during the day. Use your weak hand for whatever you would usually do with your strong hand, e.g. combing hair, brushing teeth, opening doors, pushing button, eating, writing, etc. If you have problems, carefully look at how you do things with your strong hand. Analyze the motions. Before you do anything, take a second to THINK about how you want to do it. Then try. Over time, your weak hand will become as strong as the strong hand. You will also find situations where your "weak" hand is actually the strong hand, because you have been predominantly using it for that particular activity. It's a matter of coordination, and coordination is a general issue that is not restricted to the piano.
Yes it is a matter of coordination, but movements that need to be coordinated are controlled or manged by the BRAIN...so you may need to do a little mental preparation as you do these things, so you don't get frustrated when at first you don't succeed. I also feel it's not a matter of the hand being STRONG, but being equally ABLE or ADEPT. I felt the same way once, so I started practicing with my LH alone, and a dislocated right shoulder aquired in a basketball game in the 10th grade pushed me along too; I had to take notes and tests and quizzes for two months with my LH, which made me very aware of how much I depended on my RH. During this period (incidentally this pre-empted my first piano competition

and was the end of my contact sport career (very undistinguished at any rate )

) I also played at the Scriabin Nocturne for the LH, which helped a lot too.
One of the strategies for quitting smoking is to hold the cigarette with your LH or the hand you don't hold the cigarette with. This supposedly starts the dissacociation process, because the habit is programmed by the BRAIN to be carried out by the RH. It seems unnatural to hold the cigarette with the "wrong" hand. This is in line with what xvimbi is talking about...just start to do stuff with your LH, be aware of it, and practice at the piano with it only. We're not clear what level you're at, maybe the Scriabin Prelude for the LH op.9 #? would be a good exercise in addition to practicing your scales and technical exercises with the LH alone.
Here's a fun one, I make my advanced students do it;
1. Sign your name with both hands at the same time
2. Next write "My name is...."
3. Now write "My name is...." in English AND French! ("Je m'appelle...") or Spanish "Me llamo..."
You can evantually carry this to comlex extremes if you desire. I learned about this from a biography of American Presidents; evidently James Garfield the 20th President was a professor of Greek and Latin before entering politics; to impress his students he would write the same quotations on the classroom board simeltaneously in Latin and Greek, then in English and French...
And so you see it's not your HAND, it's your BRAIN. The brain controls all physical motions. Your hand will do what your brain tells it too. It may take some time before your hand is properly conditioned to do what it's supposed to at the keyboard. Be patient, impatience is an emotional reaction it can only interfere and impede your progress....
Hope this helps!
