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

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

3. - 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. g bounded Result I

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

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

6. g, h bounded Maximum Principle -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 - g, h bounded -If a non-negative maximum M occurs at x=a, then -If a non-negative maximum M occurs at x=b, then - u is not constant - ;.

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

9. - for and, we obtain Remarks - 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 u 1 and u 2 are both solutions of the above Initial Value Problem in [a,b], then u 1 ≡ u 2.

11. Boundary Value Problem - not as straightforward as the Initial Value Problem Uniqueness: - if u 1 and u 2 are both solutions of the above Boundary Value Problem in [a,b] and, then u 1 ≡ u 2.

12. Approximation in Boundary Value Problems - 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. u z2z2 z1z1

13. g, h and f bounded Bounds for the Solution of the BVP Upper Bound z 1 Lower Bound z 2 - z 1 and z 2 are easily constructed. They may be polynomials, rational functions, exponentials…

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

15. Approximation in Initial Value Problems - in most cases we cannot find an explicit solution; - this time we can find not only an approximation for u, but also for u. u z2z2 z1z1 u’ z’ 1 z’ 2

16. g, h and f bounded Upper Bound z 1 Lower Bound z 2 Bounds for the Solution of the IVP Then and

17. Comparison Results If we have then between two consecutive zeros x 0 and x 1 of the function w, u can have at most one zero., u w (If w > 0 in [x 0, x 1 ], 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 andare continuous functions of x, y and z throughout their domains of definition. ifthenis equivalent to

20. Nonlinear Operators andare continuous functions of x, y and z throughout their domains of definition and If,, 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. If u satisfies the inequality at each point of D, then it cannot attain its maximum at any interior point of D Elliptic Equations If the function u, defined on D, has a local maximum at an interior point of D, then: D Maximum Principle: unless if u is constant..

22. Parabolic Equations If on a region E the function u verifies then it cannot have a local maximum at any interior point. t x 0 E If u satisfies the inequality Maximum Principle: in a rectangular region R, then the maximum of u on R  ∂R must occur on S 1, S 2 or S 3. t x 0 RS1S1 S2S2 S3S3

23. is a solution of Example: These equations do not exhibit the same type of Maximum Principle as the Elliptic and Parabolic equations. Hyperbolic Equations 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). x B (Weak) Maximum Principle: t 0 A a b D

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.