Past exam of the fixed point methods for nonlinear partial differential equations course of the University of Oxford / 2023 / 3 / b / ii

For the spaces in part (b)(i) and fixed $f\in Y$, define $A:X\to X^{\ast }$ byFor each $u$, this is a linear functional in $v$, and the estimates in part (b)(i) showHence $A(u)$ belongs to the continuous dual space $X^{\ast }$, and $A$ sends bounded sets to bounded sets. The weak formulation is exactlyThe forcing term is included in $A$ so that its right side is zero, as requested.