I need a proof to problem 14 in the attached file. Please.Document Preview:Hwk f14: (Hopfs boundary point lemma) Suppose u is a superharmonic function in a domain fl and u ) 0 in O, and z(16) : 6 1ot sorr?? 116 ?? AQ. Assume 0 satisfies an interior sphere condition a,l rs, i.e., there exists an open batl B c 0 such that B fl EQ : {16}. Show that the exterior normal derivative O,u(rg), if it exists, is strictly negative. Give a counterexample showing that the lemma fails in the absence of the interior sphere condition. Hint: Prove the result for harmonic first, using the Poisson formula on the ball B to estimate (() -u(rs)) llr * rol for r on the same radius as r0. The superharmonic case can be obtained by comparison. -For the counterexample, a prettyy simple function in a quadrant will do. Hwk f 18: (Separation of variables technique) (a) For the 1-dim. heat equatiortu,1: ur, find all those bounded solutions in R. x ]0, -[ that have the special form u(r,t) : A(r)B(l) with C2 functions ,4 and B. (b) Among the solutions in (a), you should have found e-tcos),r. Calculate the following superposition explicitly (for I > 0 only): -) o, ),r d), Hint: Convert the trig into an exponential and evaluate the Gaussian integral. If you have complex contour integration availabie, you may use it; if not, you may instead want to show, as a lemma, by differentiation with respect to a, that /iexp[-(s + ia)2)ds does not actually depend on a. ,lr@,t) ,: lo* Attachments: hw-14-15.pdf