Overview

1. Overview October 13- The Maximum Principle- an Introduction Next talks: - ДраганБeжановић (Dragan) - Gert-Jan Pieters - Kamyar Malakpoor

2. Simplest Case of the Maximum Principle g bounded The maximum of u in [a,b] is attained at one of the endpoints.

3. Result I g bounded - Either the maximum of u in [a,b] is attained at one of the endpoints or u is constant. - The maximum of u in [a,b] is attained at one of the endpoints.

4. Result II • If the maximum occurs at x=a, then - g bounded - u is not constant • If the maximum occurs at x=b, then

5. Conclusions g bounded • u has no relative maximums at interior points; • u has at most one relative minimum; • u has no horizontal points of inflection; • we can obtain analogous results for the solutions of , yielding an associated Minimum Principle.

6. Maximum Principle g, h bounded • If u attains a non-negative maximum M, either it is attained at one of the endpoints or u ≡ M.

7. Behaviour at boundary points • If a non-negative maximum M occurs at x=a, then - g, h bounded ; - • If a non-negative maximum M occurs at x=b, then - u is not constant .

8. Generalized Maximum Principle • There exists w such that - g, h bounded - u is not constant - A possible w: In that interval, satisfies the two last results.

9. Remarks - for and , we obtain - u cannot oscillate too rapidly, because it can have have at most 2 zeros (between which it must be negative) in [a,a+ε], where the Generalized Maximum Principle holds; - if u verifies , then it can have at most one zero in [a,a+ε].

10. Initial Value Problem Uniqueness: - if u1and u2are both solutions of the above Initial Value Problem in [a,b], then u1≡ u2.

11. Boundary Value Problem - not as straightforward as the Initial Value Problem Uniqueness: - if u1and u2are both solutions of the above Boundary Value Problem in [a,b] and , then u1≡ u2.

12. Approximation in Boundary Value Problems z1 u z2 - in most cases we cannot find an explicit solution; - we will approximate a solution in such a way that an explicit bound for the error is known, which is the same as determining both upper and lower bounds for the values of the solution.

13. Bounds for the Solution of the BVP g, h and f bounded Upper Bound z1 Lower Bound z2 - z1and z2 are easily constructed. They may be polynomials, rational functions, exponentials…

14. Example 0 < x < 1 g, h and f bounded Lower Bound Upper Bound Then .

15. Approximation in Initial Value Problems z’1 z1 u z’2 u’ z2 - in most cases we cannot find an explicit solution; - this time we can find not only an approximation for u, but also for u.

16. Bounds for the Solution of the IVP g, h and f bounded Upper Bound z1 Lower Bound z2 Then and

17. Comparison Results If we have u , w then between two consecutive zeros x0 and x1 of the function w, u can have at most one zero. (If w > 0 in [x0, x1], then verifies ).

18. Comparison Results If we have , where , then between two consecutive zeros a and b of the function w, u can have at most one zero in [a,b], unless if u is a constant multiple of

19. Nonlinear Operators and are continuous functions of x, y and z throughout their domains of definition. if then is equivalent to

20. Nonlinear Operators If , and are continuous functions of x, y and z throughout their domains of definition and , then if v(x)-u(x) attains a non-negative maximum M in [a,b], either it is attained at one of the endpoints or v(x)-u(x) is constant.

21. Elliptic Equations D If the function u, defined on D, has a local maximum at an interior point of D, then: Maximum Principle: If u satisfies the inequality at each point of D, then it cannot attain its maximum at any interior point of D . unless if u is constant.

22. Parabolic Equations t If on a region E the function u verifies E 0 x then it cannot have a local maximum at any interior point. t Maximum Principle: If u satisfies the inequality S1 R S3 S2 0 x in a rectangular region R, then the maximum of u on R∂R must occur on S1, S2 or S3.

23. Hyperbolic Equations These equations do not exhibit the same type of Maximum Principle as the Elliptic and Parabolic equations. (Weak) Maximum Principle: Example: If u satisfies and , for , then its maximum on D∂D must occur on the initial line AB (and eventually also in an interior point). is a solution of t D A B a b 0 x

24. Summary • Generalized Maximum Principle; • Uniqueness and approximation in Initial Value Problems and Boundary Value Problems; • Comparison Results; • Nonlinear Operators; • Elliptic, Parabolic and Hyperbolic Equations.