Vorlesung: Hamilton-Jacobi Equations (SoSe 2016/17)Content of the lecture: [E] L. C. Evans, Partial Differential Equations: Second Edition, AMS (Graduate Studies in Mathematics), 2010. 18 October: Introduction; practical information; requirements for the grade (see main page). Chapter 0: Introduction and motivation. Some notation. (A) What are PDEs? Definition of general 1st order nonlinear PDE and of _classical_ solutions. Special case: Hamilton-Jacobi equations (H-J eqs). Discussion and remarks. Addtional conditions: Boundary Value Problems (BVP) and Initial Value Problems (IVP) / Cauchy Problem. Discussion of 'Well-posedness in the sense of Hadamard'. [E] pp. 1-9 (!!). 25 October: Recall: Will study well-posedness of the Cauchy problem (on R^n) of H-J eq u_t + H(x,t,Du) = 0 (for classical and _generalised_ solutions, with various conditions on H). (B) Where do H-J eq's come from? (i) Classical Mechanics (crash course!): Newtonian, Lagrangian, Hamiltonian mechanics. H-J eq: Condition on the generating function S(t,x,P) of a canonical transformation (x,p) to (X,P) for it to trivialise the Hamilton equations (ie, new Hamiltonian is _zero_, and so X and P are constant, P = α). Theorem (no proof): Let H=H(t,x,p) be C^1, time-dependent Hamiltonian, and assume S=S(t,x) is C^2 solution to H-J eq S_t + H(t,x,D_xS) = 0. Assume X is solution of X' = D_pH(t,X,D_xS), and define P=D_xS(t,X) (MISPRINT/discussion in lecture! D_xS, not D_xH). Then (X,P) solves the Hamilton eq's (X',P') = J DH(t,X,P). (J^2 = - I). Moreover, if S=S(t,x,α) is n-parameter family of solutions to the H-J eq above, then _any_ solution X of X' = D_pH(t,X,D_xS) satisfies that D_αS is constant along X (ie, are first integrals). For proof, see [D] B.Dacorogna, Introduction to the Calculus of Variations (2.Ed.), World Scientific Publishing, 2008, pp. 77-81 (1-dim-case) and pp. 232-234 (N-dim case; see also Gelfand and Fomin). [GF] I.M.Gelfand and S.V.Fomin, Calculus of Variations, Dover Publications, 2000, pp. 90-93. For more on Classical Mechanics in _Mathematics_, see for example: [A] V.I.Arnold, Mathematical Methods of Classical Mechanics, Springer (GTM), 1989. (H-J: pp. 255-270). [AM] R.Abraham and J.E.Marsden, Foundations of Mechanics: Second Edition, AMS Chelsea Publishing, 1978. [DB1] E.DiBenedetto, Classical Mechanics, Springer (Cornerstones), 2011 (!!!!) (H-J: pp. 236, 263, 283, 293, 294, 296, 297). For the mathematical use of symplectic geometry in Physics, see [GS] V.Guillemin and S.Sternberg, Symplectic Techniques in Physics, Cambridge University Press (CUP), 1990. For Classical Mechanics in _Physics_, see for example (in German): [Alt] A.Altland, Theoretische Mechanik, Lecture Notes 'Theoretische Physik I (Klassische Mechanik)', Universität zu Köln, WS2002/03 (!!!!) (H-J: pp. 182-187). [Sch] F.Scheck, Theoretische Physik 1, Springer-Lehrbuch, 2007. (H-J: pp. 139-154). [G] W.Greiner, Klassische Mechanik II, Verlag Harri Deutsch, 2008. (H-J: pp. 356-386). [BFKLRW] M.Bartelmann, B.Feuerbacher, T.Krüger, D.Lüst, A.Rebhan, A.Wipf, Theoretische Physik, Springer Spektrum, 2015 (H-J: pp. 254-258). Here is a list of literature. 02 November: Theorem (Jacobi) (no proof): Let H=H(t,x,p) be C^1, time-dependent Hamiltonian, and assume S=S(t,x,α) is C^2 n-parameter family of solutions to H-J eq S_t + H(t,x,D_xS) = 0, with det(S_xα) not 0. Assume X in C^1(I,R^n) satisfies that D_αS is constant along X, and define P=D_xS(t,X,α). Then (X,P) solves the Hamilton eq's (X',P') = J DH(t,X,P). (J^2 = - I). For proof, see [D] pp. 77-81 (1-dim-case) and pp. 232-234 (N-dim case; see also [GF] pp. 90-93). (ii) From Quantum Mechanics to Classical Mechanics: A WKB-Ansatz of solution to the time-dependent Schrödinger equation (ie constructing high frequency solutions) leads to the phase-function φ having to satisfy the H-J eq φ_t + |Dφ|^2/2 + V φ=0. (iii) Optimal Control Theory: The 'terminal cost' problem consists in minimizing the 'cost functional' (sum of 'running cost' and 'terminal cost') over controls driving the state of the system. The corresponding 'value function' turns out to solve a H-J(-Bellman) eq. Overview over the course: Method of chacarteristics. Convex analysis. Hopf-Lax formula. Viscosity solutions. Chapter 1: Method of characteristics. Definition: Complete integral of F(x,u,Du) = 0: 'proper' n - parameter (C^2) family of solutions. Remarks. Example: ax - t H(a) + b is complete integral of u_t + H(Du) = 0. [E] pp. 91-94. 08 November: Recall: Definition of complete integral, and example (see above). Example: Complete integral (or not?!?) of Eikonal equation |Du| = 1. Definition: Envelope v(x) of m-parameter family (in a) {u(x;a)}_a of functions. Example: Envelope of ax - t |a|^2 is |x|^2/4t. Remarks. Theorem (Construction of new solutions via envelopes): If u(x;a) is m-parameter family (in a) of solutions of F(x,u,Du) = 0, and if the envelope v(x) exists, then v also solves F(x,v,Dv) = 0. Example (cont'd from above): The function |x|^2/4t (being the envelope of the solutions ax - t |a|^2) is solution to u_t + |Du|^2 = 0. Example (cont'd from above/last time): The complete integral ax - t H(a) + b of equation u_t + H(Du) = 0 gives solution to Cauchy problem for this equation with initial value the _affine_ function ax + b. Re-write the complete integral above as: u(x,t;y) = g(y) + (x-y)Dg(y) - t H(Dg(y)). Now, for _general_ g, define n-parameter family (in y) {u(x,t;y)}_y by this formula; compute envelope in y, IF possible. IF possible, the envelope solves Cauchy problem for u_t + H(Du) = 0, with g as initial data. Question (to find envelope): For which (x,t) can solve equation: x = y + t DH(Dg(y)) for y (as C^1 function of (x,t)) ? (see also later). Example: Solution of Cauchy problem for H(p)=p^2, g(x)=x^2. (Exercise: And if change _sign_ of one or both of H & g ?). Motivation: 'Method of Characteristic' for (Cauchy problem for) linear transport equation (without force): Solution, and derivative, is constant along lines. Idea general case: Find lines for which derivative is constant (to be continued next time). [E] pp. 91-96. [Ev] L.C. Evans, Envelopes and nonconvex Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 2014, 50, pp. 257-282 (see pp. 257-258). [L] P.-L. Lions, Generalized solutions of Hamilton-Jacobi equations, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982, 69, iv+317 (see pp. 12-13). (See also review.) 15 November: Recall: Aim is to study local & global in time existence & uniqueness of Cauchy pb for u_t + H(x,Du) = 0 (or, for H = H(Du)). Plan course: (1) Method of Characteristics, (2) Convex Analysis, (3) Hopf-Lax Formula, (4) Viscosity solutions. Seen: By envelopes, existence of classical solutions in case H = H(Du) reduces to inverting (for y) x = y + t DH(Dg(y)). Example (last): H(p) = +/- p^2, g(x) = +/- x^2: Same sign: global in time-solution. Opposite sign: Only local in time solution. Motivation for 'Method of Characteristic': Repetition, and continuation. ('Characteristics', 'the characteristic ODE's', 'characteristic curve'). Again reduces to: Global (in z) invertibility of the map X(t,z) = z + t DH(Dg(z)). Theorem (Hadamard-Caccioppoli(-Levy); no proof, see below): If F: R^n -> R^n is C^1, proper, and DF(z) is invertible for all z, then F is a _global_ C^1 diffeomorphism. Theorem (consequence): Global invertibility of maps depending on a parameter (and C^1 - dependence in the paramater of the inverse). Theorem (Local in time existence and uniqueness for the Cauchy problem): Let H, g: R^n -> R, both C^2, and let X(t,z) = z + t DH(Dg(z)). Assume there exists T > 0 so that (i) X(t,z) is proper (in z for fixed t in [0,T)) and (ii) det D_z X(t,z) non-zero for all z and t in [0,T). Then there exists unique classical C^2 (!) solution u=u(x,t) to the Cauchy problem in R^n x [0,T) of u_t + H(Du) = 0 with initial value g. (End of proof next time). Remark: Inverting same equation as when studied envelopes. For more on the general theory of the Method of Characteristics, see f.ex. [E] pp. 96-114 and [DB2] E. DiBenedetto, Partial Differential Equations (Second Edition), Springer (Cornerstones), 2010, pp. 225-230 and 267-274. For more on the Method of Characteristics for H-J eq's, see [E] pp. 113-114, [L] pp. 12-15(-20), [DB2] pp. 274-276, and [CS] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton—Jacobi Equations, and Optimal Control, Birkhäuser (Progress in Nonlinear Differential Equations and Their Applications), 2004 (see pp. 11-18). For proofs of the Hadamard-Caccioppoli(-Levy) Theorem, see [KP] S.G. Krantz and H.R. Parks, The Implicit Function Theorem, Modern Birkhäuser Classics, 2013 (see pp. 121-127), [AA] Ambrosetti and Arcoya, An Introduction to Nonlinear Functional Analysis and Elliptic Problems, Birkhäuser (Progress in Nonlinear Differential Equations and Their Applications), 2011 (see pp. 27-28), [AP] Ambrosetti and Prodi, A Primer on Nonlinear Analysis, CUP (Cambridge Studies in Advanced Mathematics), 1995 (see Chapter 3 (Theorem is p. 47)). 22 November: End of proof from last time: Theorem: Local in time existence and uniqueness for the Cauchy problem. (See above for precise formulation). Corollary: Let H in C^2, and assume g in C^2 with Dg and D^g both uniformly bounded on R^n. Then there exists unique classical C^2 (!) solution u=u(x,t) to the Cauchy problem of u_t + H(Du) = 0 with initial value g, for at least time T_0 = (M_1M_2)^{-1} where M_1 = sup_{z in R^n} ||D^2g(z)||, M_2 = sup_{|p|<=M_0} ||D^2H(p)||, where M_0 = sup_{z in R^n} |Dg(z)|. Remark: I.e., for any H in C^2, there is local-in-time existence of (unique) classical solution for any initial condition g with Dg and D^g uniformly bounded, for at least time T_0 (T_0 as above). Remark: The proof shows that the properness of the map X(t,z) = z + t DH(Dg(z)) follows as soon as DH(Dg(z)) is bounded on R^n (above ensured by 'Dg bounded'). As the example H(p)=p^2, g(x)=x^2 shows, the condition is not necessary for the Method of Characteristics to work. Corollary: Let H, g in C^2 be both convex, or both concave. Assume furthermore that Dg is uniformly bounded on R^n. Then there exists unique classical C^2 (!) solution u=u(x,t) to the Cauchy problem of u_t + H(Du) = 0 with initial value g, for all times. Remark: I.e., under the stated conditions on H and g, we have global-in-time existence of (unique) classical solution. Again, properness of the map X(t,z) = z + t DH(Dg(z)) would be enough; is again ensured by (unnecessary ?!?) condition 'Dg bounded'. Again, as the example H(p)=p^2, g(x)=x^2 shows, the condition is not necessary for having global-in-time existence via Method of Characteristics. [L] pp. 12-15(-20), [CS] pp. 11-18. (With changes!) 29 November: Re-cap: Method of characteristics (for u_t + H(Du) = 0) consists in (1) solving the characetristic eq for X (in this case, solvable globally in time), to get X(t;z) = z + t DH(Dg(z)), and (2) Inverting this equation for z (in this case, this may or may not be possible for all t). Potential problem: Two characteristics from two different points z_1 and z_2 may meet / cross at some time t*. Theorem: For H, g in C^2, let T* = sup{t>0 | I + tD^2H(Dg(z))D^2g(z) is invertible for all z in R^n}. If T' > T*, then there exists NO C^2 - solution u = u(x,t) in R^n x [0,T') to the Cauchy problem for u_t + H(Du) = 0 with initial value g. Discussion of Method of Characteristics for (more general) equation u_t + H(x,t,Du) = 0: The characteristic equations are exactly Hamilton's equation (with the Hamiltonian H). Discussion and remarks. Theorem: Let H, g in C^2, and let (X,P) be the solutions to Hamilton's equation with initial value (z,Dg(z)). Let U solve dU/dt = -H(X,t,P) + P D_pH(X,t,P) with initial value g(z). Assume there exists T > 0 such that (i) The maximal interval of existence of X,P,U contains [0,T) for all z in R^n, and (ii) The map X(t;z) is invertible in z with C^1 inverse Z(t;x) for all t in [0,T). Then there exists unique C^2 classical solution u = u(x,t) on R^n x [0,T) to the Cauchy pb for u_ t + H(x,t,Du) = 0 with initial condition g, given by u(x,t) = U(t,Z(t;x)). Discussion and remarks on Method of Characteristics for general 1st order equations, and higher order equations ('non-characteristic'). [CS] pp. 98-100(-104). For more on the general theory of the Method of Characteristics, see f.ex. [E] pp. 96-114 and [DB2] pp. 225-230 and 267-274. Chapter 2: Convex Analysis. Definition: Uniform continuity, modulus w. Theorem: f : A -> R is uniformly continuous iff it has a modulus of continuity w. Definition: Lipschitz continuity, Hölder continuity, locally Lipschitz/Hölder continuity (expressed via modulus of continuity). Theorem (Rademacher): A Lipschitz continuous function on an open set is differentiable a.e. (NO proof). Definition: Closed segment, convex set, strictly convex set. For a proof of Rademacher's Theorem, see [E] pp. 297-298, or [EG] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, Chapman and Hall/CRC, 2015, pp. 103-106 and 265-266. 30 November: Examples of convex/strictly convex sets (R^n, open/closed balls, closed segments, hyperplanes, closed halfspaces). Definition: For function f on convex set A: convex/strictly convex/uniformly convex with modulus w/strongly convex/semi-convex with a linear modulus/semi-convex with modulus w. Ditto for concave. Locally ditto. Examples: Affine functions, |x|, any norm, x^2, |x|^p, p>=1, exp(x). Remark: strongly convex implies uniform convex implies strictly convex implies convex implies semi-convex with linear modulus implies semi-convex. In particular: Any convex function is semi-convex. A function is convex _and_ concave iff it is affine. However, f can be both convex and _semi_concave. A function f is convex iff its epigraph epi(f) is a convex set. Definition: Support hyperplan (at point x) of a set A. Theorem: Any convex and closed set A has a support plane at every boundary point. (NO proof; consequence of Hahn-Banach (!), or see [Schn] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd Edition, CUP, 2013, p. 11. Consequences: Theorem: Let A be convex set, f : A -> R convex map. For all x in A there exists r in R^n such that f(y) >= f(x) + r (y-x) for all y in R^n. Discussion and remarks: The mapping y -> f(x) + r (y-x) determines the support hyperplane to f at x (i.e., to the graph of f at (x,f(x))). Consequence: Theorem: Jensen's Inequality. Another consequence: Theorem: A function is convex iff it is the supremum over a family of affine functions (if the sup is finite everywhere). Remarks: Proof shows that sup of convex functions is convex (if finite), and sup of semi-convex functions with same modulus is semi-convex with same modulus (if finite). Proposition: Any C^1 function on an open set is both locally semi-convex and locally semi-concave. [E] pp. 707-708; [CS] pp. 2-4, 29-38, and 273-277. 06 December: Proposition: For a C^2 - function f on an open and convex set A, relationships between positivity (as a matrix) of D^2 f and various degrees of convexity (and hence, between negativity of D^2 f and various degrees of concavity). Theorem: Any semi-concave function on a convex subset A of R^n is locally Lipschitz on the interior of A. Remarks: Also holds for concave, semi-convex, convex. In particular, these functions are all continuous on the interior of their domain of definition (but not necessarily everywhere - counter example). Also, combining with Rademacher, they are differentiable a.e. on the interior of their domain of definition. Remarks, comments, examples. Proposition: A function f on a convex open set A is semi-concave with a linear modulus iff it is continuous and mid-point semi-concave with a linear modulus. Remarks: Mid-point semi-concave with a linear modulus is often in the literature called 'semi-concave' (our general definition of semi-concave is different). Corollary: A function f on a convex set is concave iff it is continuous and mid-point concave. Theorem: Five equivalent characterisations of semi-concavity with linear modulus: (a) Definition of semi-concavity with linear modulus (with semi-concavity constant C). (b) x -> f(x) - C|x|^2/2 is concave. (c) f = f_1 + f_2 with f_1 concave, f_2 in C^2 with ||D^2 f|| <= C. (d) f is continuous and mid-point semi-concave with linear modulus. (e) f is inf over family of C^2 - functions f_i, with ||D^2 f_i|| <= C for all i. (Proof next time). [CS] pp. 2-4, 29-38, and 273-277. 09 December: Proof of five equivalent characterisations of semi-concavity with linear modulus (see above). Remark: Analogous result hold for strongly convex functions with constant m>0 ('m-strongly convex'): f is m-strongly convex iff x -> f(x) - m|x|^2/2 is convex. Theorem (Alexandroff/Alexandrov; NO proof): A convex function f on a convex open set A is 'twice differentiable a.e.': For a.e. x_0 there exist vector p and symmetric matrix B such that [ f(x) - f(x_0) - p(x-x_0) - <(x-x_0),B(x-x_0)> ] / |x-x_0|^{-2} -> 0 as x -> x_0. Definition: A function f : R^n -> R is superlinear iff f(x)/|x| -> infinity as |x| -> infinity. Definition: The (Fenchel-) Legendre Transform f* of a convex and superlinear function f. Lemma: The 'sup' in the definition of f* is attained (it is a 'max'). If it is attained at x* (non-unique), then f*(p) = p x* - f(x*). In particular, f* : R^n -> R (ie is _finite_). Proposition: Fenchel's Inequality / Young's Inequality: If f is convex and superlinear, and f* its Legendre transform, then f*(p) + f(x) >= x p for all x,p. Example: Legendre transform of x^2/2, of < x,Ax >/2 for A > 0 (positive definite matrix), and of |x|^r/r for r>1. Theorem: Let f : R^n -> R be convex and superlinear. Then f* is convex and superlinear (so that f** is well-defined), and f** = f. Remark: Recall definition of m-strongly convex and that, if f is C^2, is equivalent to D^2 f >= m. Analogously to the result on semi-concave with a linear modulus versus mid-point semi-concave with a linear modulus one gets: Corollary: A function f on a convex open set A is m-strongly convex iff it is continuous and mid-point m-strongly convex. Theorem: Assume f : R^n -> R is m-strongly convex. Then f* is well-defined and semi-concave with a linear modulus, with semi-concavity constant m^{-1}. (End of proof next time). [E] pp. 120-122 and formulae (36) and (37) p. 130 (for the last theorem); [CS] pp. 2-4, 29-38, 273-277, and 282-284. For a proof of Alexandroff's/Alexandrov's Theorem, see [EG] pp. 273-276. 13 December: End of proof from last time (see above). Remarks and outlook on Convex Analysis: Often done for functions into extended reals (some reasons: sup of convex functions then always convex; Fenchel-Legendre Transform then defined for _all_ funktions f; then f** <= f and f** = f iff f is convex and lower semi-continuous). Convex Analysis works also in infinite dimensional (topological) vector spaces V; 'x p' in def of Fenchel-Legendre transform replaced with the dual pairing between V and V'. Discussion of definition of Fenchel-Legendre transform wrt. second variable for L = L(x,v) (for L continuous, convex in v and uniformly (in x) superlinear in v). Here is a list of books on Convex Analysis (to be updated). Chapter 3: Hopf-Lax Formula. Motivation: (i) As seen (in Chapter 1), the Method of Characteristics 'rarely' produces _classical_ solution to Cauchy pb of H-J eq (even for H = H(p), when characteristics exist globally in time, they might cross). Discussion of 'generalised solutions'. (ii) Recall the n-parameter family v(x,t;y) = g(y) + (x-y)Dg(y) - tH(Dg(y)) of solutions of v_t + H(Dv) = 0 and how to build envelope: Critical points in y, and solve for y as C^1 - function of (x,t). More general 2n-parameter family w(x,t;y,z) = g(y) + (x-y) z - tH(z). Extremising in y gives z = Dg(y); inserting in w recovers v. Extremising via inf/sup (instead of finding critical points) gives a 'two-parameter envelope construction'. Choice: sup over z, then (!) inf over y (of w(x,t;y,z)) leads to the Hopf-Lax Formula: u(x,t) = inf_y sup_z w(x,t;y,z) = inf_y { g(y) + t H*((x-y)/t) } (if H is convex and superlinear). Will prove: u is Lipschitz; hence a.e. diff.; where diff., it solves the H-J eq. It has limit g(x) for t -> 0. (iii) In Classical Mechanics, the Lagrangian L = L(v) and the Hamiltonian H = H(p) are connected via L = H*, H = L* (= H**). In literature, often L is used (as notation). We will stick to H and H*. (iv) Another motivation for the Hopf-Lax Formula: Via Optimal Control and minimising the action plus initial cost. Lemma: For H convex and superlinear, g Lipschitz, let u be given by the Hopf-Lax Formula for t>0. Then u(x,t) = min_y { g(y) + t H*((x-y)/t) } (i.e., the inf is attained). Theorem: For H convex and superlinear, g Lipschitz, let u be given by the Hopf-Lax Formula for t>0 and let u(x,0) = g(x). Then u is Lipschitz on R^n x [0,infinity). (Proof next time). Corollary: For H convex and superlinear, g Lipschitz, let u be given by the Hopf-Lax Formula for t>0. Then lim_{t->0} u(x,t) = g(x), and u is differentiable a.e. in R^n x (0,infinity). [Ev] L.C. Evans, Envelopes and nonconvex Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 2014, 50, pp. 257-282 (see pp. 257-258). [CS] pp. 6-9; [E] pp 122-124 and 126-127. 14 December: Proof from last time (see above). Theorem (Semi-group property / Dynamical Programming Principle for Hopf-Lax): For H convex and superlinear, g Lipschitz, let u be given by the Hopf-Lax Formula for t>0. Then, for all s in (0,t), u(x,t) = inf_y { u(y,s) + (t-s) H*((x-y)/(t-s)) }. Remark: Writing u(.,t) = S_t(g), the theorem says that S_{t+s} = S_t ∘ S_s for t, s >= 0, hence, {S_t}_{t>=0} forms a semi-group. Theorem: For H convex and superlinear, g Lipschitz, let u be given by the Hopf-Lax Formula for t>0. Assume u is differentiable at some (x,t) in R^n x (0,infinity). Then u_t(x,t) + H(Du(x,t)) = 0. Summarising: Theorem: For H convex and superlinear, g Lipschitz, let u be given by the Hopf-Lax Formula for t>0 and let u(x,0) = g(x). Then (i) u: R^n x [0,infinity) -> R is Lipschitz, hence differentiable a.e. on R^n x (0,infinity). (ii) u(x,o) = g(x) for all x in R^n. (ii) If u is differentiable at (x,t) in R^n x (0,infinity), then u_t(x,t) + H(Du(x,t)) = 0. Ie. u is Lipschitz, solves the H-J eq _a.e._ and satisfies the initial value condition. Theorem (Continuity in the data for Hopf-Lax Formula): For H_1, H_2 convex and superlinear, g_1, g_2 Lipschitz, let u_1, u_2 be given by the corresponding Hopf-Lax Formula for t>0. Then, for t > 0, ||u_1(.,t)-u_2(.,t)||_{infinity} <= ||g_1-g_2||_{infinity} + t ||H_1-H_2||_{infinity}. [CS] pp. 9-11; [E] pp. 124-125 and 127-128. 20 December: Recall: Well-posedness a la Hadamard: Existence, uniqueness, continuity in data. Have: Lipschitz function (Hopf-Lax) solving H-J eq a.e., satisfying initial condition, and depending continuous on data. Examples: For H(p) = p^2/2 (1-dim), compute and draw characteristics (also crossings!) and compute Hopf-Lax, for initial condition x, x^2/2, -x^2/2, -|x|, |x| (Exercise!), 0 (!). Find: Several instances of NON-uniqueness (of Lipschitz function solving H-J eq a.e. and satisfying initial condition). Discussion. Theorem: For H convex and superlinear, g Lipschitz, assume furthermore: g is semi-concave with linear modulus, constant m>0. Let u be given by the Hopf-Lax Formula for t>0. Then, for all t>0, the map x -> u(x,t) is semi-concave with linear modulus, with constant m. Theorem: For H convex and superlinear, g Lipschitz, assume furthermore: H is m-strongly convex. Let u be given by the Hopf-Lax Formula for t>0. Then, for all t>0, the map x -> u(x,t) is semi-concave with linear modulus, with constant 1/(tm). Remark: The constant unbounded for t -> 0, reflecting that g may NOT be semi-concave (with linear modulus). Definition: u : R^n x [0,infinity) -> R is 'generalised solution in the sense of Kruzhkov' of the Cauchy problem for u_t + H(Du) iff it solves the equation a.e. in R^n x (0,infinity), satisfies the initial value condition, and for all t>0, the map x -> u(x,t) is semi-concave with linear modulus, with constant C(1+t^{-1}) for some C>0. Return to example with infinitely many Lipschitz-solutions: Most of them (!) NOT semi-concave: |x| is NOT semi-concave around 0. Definition: standard mollifier and mollification. [E] pp. 128-131, 713-714; [CS] pp. 8-9, 11, 18-22. 21 December: Proposition (Properties of mollification): Gives family {f^ε}_{ε>0} of C^infinity - functions, converging a.e. to f; converging uniformly on compacts if f is continuous; converging in L^p_loc if f in L^p_loc (p finite). Proposition (Further properties of mollification): For f Lipschitz, Df^{ε} = (Df)^{ε} and Df^{ε} converges a.e. to Df. If u(x,t) is semi-concave (in x, for t>0 fix) with a linear modulus with constant C(1+t^{-1}), then so is u^{ε} (mollified in x only). Remark/notation: Cone of determinacy, and its base. Proposition (Energy estimate): Given u and u~, two generalised solutions a la Kruzhkov of the Cauchy problem of u_t + H(Du) with H C^2 and convex, in a common cone of determinacy (with slope R given by max of |DH| over ball with radius given by the Lipschitz constants of u and u~), and assume u and u~ have the same constant of semi-concavity (in x) C(1+t^{-1}). Let E(s) be the energy given by the integral over the horizontal plane section of the cone at time/altitude s of a non-negative C^1 function of u - u~. Then E(σ) <= E(ε) exp( int_ε^σ nΛC(1+s^{-1})ds) for σ > ε. (Here, Λ is max of ||D^2H|| over ball with radius given by the Lipschitz constants of u and u~). Theorem (Uniqueness of generalised solutions a la Kruzhkov): For H C^2, convex, and superlinear, and g Lipschitz, there exists at most one generalised solutions a la Kruzhkov of the Cauchy problem for u_t + H(Du) with initial data g. [ Needed in proof: Lemma: For symmetric n x n matrices A, B with 0 <= A <= Λ and B <= k, tr(AB) <= n k Λ. Proposition: Grönwall's Inequality. ] Corollary: For H C^2, convex, and superlinear, and g Lipschitz, assume either (a) g is semi-concave or (b) H is strongly convex. Then the Hopf-Lax function given by H and g is the unique generalised solutions a la Kruzhkov of the Cauchy problem for u_t + H(Du) with initial data g. [E] pp. 716-718; 131-135; for Grönwall's Inequality, see pp. 711-712; [CS] pp. 22-25. Merry Christmas, Happy Hanukkah, & Happy Newyear! 
10 January: Chapter 4: Viscosity solutions. Recall: studied generalised solutions a la Kruzhkov for u_t + H(Du) = 0 and proved existence (Hopf-Lax) and uniqueness (via Energy estimate) under certain _conditions_ (on H and g). Aim this chapter: (a) Relax these conditions, (b) Do for more general equations ( u_t + H(x,Du) = 0). Method: Even more general concept of solution. Motivation (for later definition of 'viscosity solution'): Discussion of Maximum/Comparison Principle for harmonic, and discussion of sub-and superharmonic functions. Definition: A _continuous_ function u is subharmonic (in the viscosity sense) iff - Δv(x_0) <=0 whenever u-v has a local max at x_0 for a smooth function v. Analogously for 'superharmonic in the viscosity sense'. Generalises to 'elliptic 2nd order equations' - here enough to know: OK for H-J eqs. Definition: Of 'viscosity subsolution' and 'viscosity supersolution' and 'viscosity solution' of the equation u_t + H(t,x,Du) = 0 (for H continuous) in R^n x (0,T). Proposition (Consistency): Any classical solution u of u_t + H(t,x,Du) = 0 in R^n x (0,T) is a viscosity solution of u_t + H(t,x,Du) = 0 in R^n x (0,T). Proposition: Regular viscosity solutions are classical solutions. Needs: Lemma (Touching by C^1) - function: For u continuous, and differentiable at a point x_0, there exists C^1 - function v such that (i) u(x_0) = v(x_0), (ii) u - v has strict local max at x_0. (NO proof - see [E] pp. 586-587.) Theorem: For H convex and superlinear, g Lipschitz, let u be given by the Hopf-Lax Formula for t>0. Then u is a viscosity solution of u_t + H(Du) = 0 in R^n x (0,infinity). Remarks: (1) Under various conditions on H, one can prove uniqueness (via a Comparison Principle) of viscosity solutions of u_t + H(t,x,Du) = 0. In the case H = H(p), the Hopf-Lax function is then the unique viscosity solution under these conditions. (2) For various specific equations (more general than H = H(p)) , one can prove existence of viscosity solution either via (a) Perron's Method or (b) Optimal Control Theory. [E] pp. 581-588 and 603-604. For uniqueness of viscosity solutions (via 'doubling of variables' after Kruzhkov), see e.g. [E] pp. 588-592. For existence via Optimal Control Theory, see e.g. [E] pp. 592-603. For more on viscosity solutions and their properties, see the literature list, in particular: [L], [CS] (see above), and [CIL] M.G. Crandall, H. Ishii, and P.-L. Lions (1992), User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.). Vol. 27(1), pp. 1-67. [BCESS] M. Bardi, M.G. Crandall, L.C. Evans, H.M. Soner, and P.E. Souganidis, Viscosity solutions and applications, Springer-Verlag (1997). [B] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Springer-Verlag (1994). [BC-D] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Birkhäuser (1997). For more on _2nd_ order elliptic equations and viscosity solutions, see [BCESS] and [K] S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, 2nd edition (version: June 28, 2012). See also my previous classes, 'Viscosity Solutions for nonlinear PDEs I + II' (WS 2014/15 and SoSe2015). End of class/course/lecture! ----------------------------------- Letzte Änderung: 10 February 2017 (No more updates). Thomas Østergaard Sørensen |
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