# Why are viscosity solutions useful solutions?

I refer to definition of viscosity solution in user's guide to viscosity solutions of second order partial differential equations by Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions.

Viscosity solutions are generalized solutions which can be implied if the Sobolev theory (or similar) doesn't provide "useful" solutions. A standard example is the problem

$$|u'| = -1, u(-1) = 1, u(1)=1$$

All "zig-zag" functions with appropriate boundary conditions provide a solution, but $$u(x)=|x|$$ is the unique viscosity solution.

But except its formal beauty why do we regard a viscosity solution as useful, and what is the 'physical' or 'intuitive' interpretation of being a viscosity solution?

• Doesn't the first paragraph in the paper you link to give a great way to sell the subject? :) Mar 24, 2011 at 17:23
• Suppose we have got a a unique viscosity solution. Ok, if it is smooth (enough, $C^2$ should suffice), it does indeed solve the problem in a classical sense. How can I imagine a viscosity solution, or why is this generalized notion still useful to concretely handle with it? The problem will likely be embedded in a bigger agenda, so the solution must be more than abstract non-sense. I don't see where authors refer to this. Mar 24, 2011 at 18:44
• Martin, as I read it, Mariano's comment was meant to support your sentiment (not the converse).
– user9072
Mar 24, 2011 at 21:27