First Order PDEs

Abstract

1. Systems of 1st order ordinary differential equations 2 1.1. Existence of solutions 2 1.2. Uniqueness 4 1.3. Differentiability of solutions with respect to a parameter 6 2. Quasi-linear pdes 10 2.1. Step 1 10 2.2. Step 2 11 2.3. Cauchy problem 11 3. Degenerate case 12 4. Examples 14 4.1. Example 1 14 4.2. Example 2 14 4.3. Example 3 14 4.4. A maximum principle for a first order pde 15 5. Fully nonlinear case 15 5.1. Cauchy problem 17 5.2. Uniqueness 21 5.3. Solution of the eikonal equation 22 5.4. Higher dimensional case 23 6. A transversality problem 24 7. The Legendre transform 25 References 27