Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) * Give appropriate graphs and required codes. * Make use of inequalities if you think that required. * You are supposed to use kreszig for reference. Holder inequality: j=1 m=1 1 where p > 1 and + 1 P q Cauchy-Schwarz inequality: [ {% ≤ (SP) | j=1 k=1 (E)' ΣΙ Minkowski inequality: (+1)* (Σx+r)'s (Σar)² + (Σm²)" where >1 Problem 37: Schauder Estimates in PDEs Problem Statement: Schauder estimates provide bounds on solutions to elliptic partial differential equations. Tasks: a) Schauder Estimate Statement: State the Schauder Estimate for solutions to the Poisson equation -Auf in a bounded domain 2 CR" with smooth boundary. b) Proof of Schauder Estimates: Outline the key steps in proving Schauder Estimates, focusing on the use of Hölder spaces. c) Application to Regularity: Use Schauder Estimates to prove regularity results for solutions to elliptic PDEs. d) Visualization: For = (0, 1) and f(x, y)=sin(x) sin(y), sketch the solution u(x, y) to the Poisson equation and illustrate the bounds provided by Schauder Estimates.