Section 3.2 Maximum Principle and Uniqueness

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1 Section 3. Mximum Principle nd Uniqueness Let u (x; y) e smooth solution in. Then the mximum vlue exists nd is nite. (x ; y ) ; i.e., M mx fu (x; y) j (x; y) in g Furthermore, this vlue cn e otined y point M u (x ; y ) : This point (x ; y ) ; clled mximum point, my e either inside or on the oundry of : Exmple () u (x; y) x + y ; fx + y < g : Then, oviously, u (x; y) x + y < nd u (x; y) x + y So u reches the mximum vlue for (x; y) in for (x; y) only on the oundry x + y : () fx + y < g ; M mx fu (x; y) j (x; y) in g : u (x; y) + x + y : Then, since + x + y ;

2 nd So u (x; y) is otined y u t (; ) : for (x; y) in ; + x + y u (; ) : M mx + x + y j x + y < In generl, there is no gurntee tht (x ; y ) is on the However, for certin functions, mximum vlues cn only occur on oundry points. Exmple Consider one-vrile convex functions: y f (x) ; f (x) > : Then mximum points of f (x) in ny intervl [; ] ; for instnce [; ] [ ; ] ; re lwys on oundries. y x The reson this is lwys true is tht, y the second-order derivtive test for locl mximum, if x x is mximum point inside ( ; ) ; then f (x ) (otherwise, it would e minimum point).

3 Let u (x; y) e two-vrile function, nd let uxx u J (u (x; y)) xy e the Hessin Mtrix of u (x; y) (lso clled Jcoin mtrix). This is symmetric mtrix. A symmetric mtrix J is clled positive de nite, denoted s J > if J u yx u yy > for ny non-zero vector : If the inequlity sign ">" is chnged to " "; then we cll J semi-positive de nite nd denote y J. We cll J is negtive de nite or semi-negtive de nite, denoting s J < or J ; respectively, if J is positive de nite, or semi-negtive de nite; i.e., J < for ny non-zero vector or J Exmple 3. for ny non-zero vector J > 5 This cn e veri ed s follows. For ny vector J ( + ) + > :, 3

4 . In fct, J J < ( ) + < : 3. J 5 is neither positive de nite nor negtive de nite. This is ecuse we cn select so tht J 5 5 On the other hnd, if we select then J ; 5 5 < : > : 4

5 4. In generl, if J is digonl J ; then J > i oth > nd > : Proposition 4 Let e symmetric mtrix J c c. All eigenvlues of J re positive i J > : This is equivlent to > ; > ; det J c > :. All eigenvlues of J re positive or zero i J : This is equivlent to ; ; det J c : 3. All eigenvlues re negtive i J < : This is equivlent to < ; < ; det J c > : 4. All eigenvlues re negtive or zero i J : This is equivlent to ; ; det J c : Proof: We know tht eigenvlues re solutions of c det (J I) det c ( ) ( ) c ( + ) + c : 5

6 If > ; > ; nd det (J) c > ; then y the qudrtic formul, we nd q ( + ) ( + ) 4 ( c ) ( + ) p c ( + ) p ( + ) + 4c q ( + ) ( ) + 4c : This expression shows tht there re two distinct rel eigenvlues. Moreover, since det (J) c > q q ( + ) 4 ( c ) ( + ) ( + ) + > : Therefore, q ( + ) ( + ) 4 ( c ) : We hve just estlished the fct tht if > ; > ; det (J) c > then ll eigenvlues re positive ) J > : The rest cn e proved similrly. Second-order derivtive test (for locl extreme points of two vrile functions): Let (x ; y ) e criticl point, i.e., u x (x ; y ) u y (x ; y ) ; nd let J (x ; y ) uxx (x ; y ) u xy (x ; y ) : u yx (x ; y ) u yy (x ; y ). If J (x ; y ) > ; then (x ; y ) is locl minimum point. (i.e., u xx (x ; y ) ; u yy (x ; y ) > nd det (J (x ; y )) u xx u yy (u xy ) > ) 6

7 . If J (x ; y ) < ; then (x ; y ) is locl mximum point. (i.e., u xx (x ; y ) ; u yy (x ; y ) < nd det (J (x ; y )) u xx u yy (u xy ) > ) 3. If J (x ; y ) hs two eigenvlues with the opposite signs, then we cll (x ; y ) sddle point. A sddle point is neither mximum nor minimum point. Mximum Principle for Lplce eqution Let s now consider solution u of Lplce s eqution in smooth domin u F (x; in nd ssume tht F (x; y) > : Suppose tht solution u reches the mximum vlue inside the domin, i.e., M mx fu (x; y) j (x; y) in g M u (x ; y ) for some (x ; y ) inside : Then ccording to #3 of the second derivtive test, (x ; y ) cnnot e sddle. This mens tht either J or J : Now since it is mximum point, we must hve J (otherwise it would e minimum point). Thus, u xx (x ; y ) ; u yy (x ; y ) : () Otherwise, it would e minimum point. Consequently < F (x ; y ) u xx (x ; y ) + u yy (x ; y ) ; 7

8 which is contrdiction. Note tht () my e justi ed directly s follows. Consider one vrile function y u (x; y ) : This function reches locl mximum M t x x : Thus y u xx (x ; y ) ; since otherwise it would e locl minimum if u xx (x ; y ) > : We conclude tht, in generl, ny solution u (x; y) of Lplce eqution u (x; y) F (x; y) in oeys the Mximum Principle: u (x; y) reches its mximum vlue somewhere on the of the domin : Similrly, for ny solution u (x; y) of Lplce eqution u (x; y) G (x; y) in ; the function is solution of w u w (x; y) G (x; y) in : Therefore w follows oeys the Mximum Principle. Consequently, u w oeys the Minimum Principle: u (x; y) reches its minimum vlue somewhere on the of the domin : For Poisson eqution u (x; y) ; ny solution u reches oth its mximum vlue nd minimum vlue on the of the domin : In other words, Poisson eqution oeys oth Mximum Principle nd Minimum Principle. Uniqueness for irichlet prolems of Lplce Equtions: 8

9 There is t most one solution to irichlet prolem of the Lplce eqution u (x; y) F (x; y) u (x; y) g (x; y) in for (x; Indeed, suppose there re two solutions u (x; y) nd u (x; y) to the sme prolem, i.e., u (x; y) F (x; y) u (x; y) g (x; y) in for (x; nd u (x; y) F (x; y) u (x; y) g (x; y) in for (x; Let Then we cn verify tht this u solve u u u : u (x; y) in u (x; y) for (x; According to the Mximum Principle for the Poisson eqution, u (x; y) reches the mximum vlue on the oundry, i.e., for ny (x; y) in ; u (x; y) mx fu (x; y) j (x; y) g u (x ; y ) (for some points (x ; y ) (since u nywhere ). On the other hnd, ccording to the Minimum Principle for the Poisson eqution, for ny (x; y) in u (x; y) min fu (x; y) j (x; y) g u (x ; y ) (for some points (x ; y ) (since u nywhere ). Thus u (x; y) ) u u : 9

10 ivergence Theorem For ny vector-vlued function ~v (x; y) hv ; v i ; the divergence theorem sys I (r ~v) dxdy ~n ~vds where ~n is the outwrd norml direction. The rst integrl is doule integrl, nd the second integrl is line integrl of rst kind de ned s follows. Let the oundry hve : x x (t) ; y y (t) ; t : Then g (x; y) ds Z q g (x (t) ; y (t)) x (t) + y (t) dt: Green s Identity Note tht for ny function, r (uru) ru ru + ur (ru) jruj + uu: Now we pply the divergence theorem with ~v uru: It follows (r ~v) dxdy jruj + uu I dxdy ~n uruds; or I jruj dxdy + This lst formul is clled Green s Identity. (Green s Identity) Uniqueness for Neumnn prolem of the Lplce eqution

11 The di erence of ny two solutions to the Neumnn prolem of the Lplce eqution is constnt. In other words, solutions to u (x; y) F (x; y) (x; y) g (x; y) for (x; is uniqueness up to ritrry constnts. Indeed, suppose there re two solutions u (x; y) nd u (x; y) to the sme prolem, i.e., u (x; y) F (x; y) in (x; y) g (x; y) u (x; y) F (x; y) for (x; in (x; y) g (x; y) Then we cn verify tht this u solve u u u : for (x; u (x; y) (x; y) for (x; According to Green s Identity, I jruj dxdy + uudxdy ds; since u in we nd jruj dxdy ) jruj ) ru ) u (x; y) is constnt.

12 . Using Green s Identity (Green s Identity), show tht () The irichlet prolem, with k ; u ku F (x; y) in u g (x; y) hs unique solution. () The Neumnn prolem, with k ; u hs unique solution. ku F (x; y) g (x; y) (c) The Roin prolem, with k ; h > : hs unique solution.. Consider the irichlet prolem u ku F (x; y) + hu g (x; y) u ku F (x; y) in u g (x; y) where k ; F ; g : Using the rgument leding to the mximum principle, show the solution stis es u (x; y) in : 3. Consider the irichlet prolem u F (x; y) in u g (x; y)

13 In ppliction, g (x; y) is often the oundry dt otin from mesurement or smpling. For instnce, if u models the temperture of conducting ody, then g (x; y) is the surrounding temperture (e.g., room temperture). It is quite often tht there is lwys n error in mesurement. So the oundry dt g (x; y) one uses in solving the irichlet prolem is inccurte. Show tht the error to the solution will not exceed the mximum error of the oundry dt. More precisely, show tht if u " is the solution of u " F (x; y) in u " g " (x; y) nd if then jg " (x; y) g (x; y)j " for (x; y) ju (x; y)j " for ll (x; y) in : (Hint: pply oth mximum principle nd minimum principle to the new irichlet prolem in which w u u " is solution.) 3

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