Pde Evans Solutions Chapter 6 -

Weak lower semicontinuity of the Dirichlet integral gives $|Du^ |_L^2^2 \le m$. Since $|u^ |_L^2=1$, we have equality.

Problems here often ask you to prove uniqueness given a specific condition, or to construct a solution using a limiting argument. Understanding spectral theory helps immensely here. pde evans solutions chapter 6

Forgetting to verify that $H^1_0$ is the correct space (zero trace conditions). Evans is explicit about this. Weak lower semicontinuity of the Dirichlet integral gives

. The bilinear form must be bounded and coercive (energy-minimizing). Mixed Boundary Conditions pde evans solutions chapter 6