Semilinear obstacle problem with measure data and generalized reflected BSDE

Transcription

1 Semilinear obtacle problem with meaure data and generalized reflected BSDE Andrzej Rozkoz (joint work with T. Klimiak) Nicolau Copernicu Univerity (Toruń, Poland) 6th International Conference on Stochatic Analyi and It Application Bȩdlewo, September 10 14, 2012 Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 1/19

2 Parabolic obtacle problem Let D R d (d 2) be a bounded domain, D T = [0, T ] D. We conider the problem u t + Au = f (, u) µ on {h 1 < u < h 2 }, h 1 u h 2, u(t, ) = ϕ, u(t, ) D = 0, t (0, T ), where A i a uniformly elliptic operator of the form and A = d ( (a ij (t, x) ) x j x i i,j=1 f : D T R R, ϕ : D R - meaurable function atifying ome condition (to be pecified later on), µ i a bounded mooth meaure on D T, h 1, h 2 : D T R are meaurable function uch that h 1 h 2 a.e.. Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 2/19 ( )

3 Parabolic obtacle problem Problem: olve ( ) for meaurable barrier h 1, h 2, ϕ L 1 (D), f atifying the monotonicity condition in u (for intance f (x, u) = u q 1 u for ome q > 1) and mild integrability condition, µ M 0,b (M 0,b - pace of all mooth igned meaure on D T with bounded total variation (for intance µ(dt dx) = g dt dx for ome g L 1 (D T )). Known reult for irregular barrier: M. Pierre (1979, 1980) - linear cae (f = f (x)) with L 2 data (i.e. ϕ, f, g L 2 ). T. Klimiak (2012) - y f (x, y) atifie the Lipchitz condition and the linear growth condition, L 2 data (i.e. ϕ, f (, 0), g L 2 ). Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 3/19

4 Cauchy-Dirichlet problem Let f u = f (, u). We firt conider the problem { u t + Au = f u µ on (0, T ) D, u(t, ) = ϕ, u(t, ) D = 0, t (0, T ). ( ) Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 4/19

5 Cauchy-Dirichlet problem X = {(X, P,x )} - Markov family aociated with A. Propoition Any µ M + 0,b admit a unique poitive AF Aµ of X uch that for q.e. (, x) D T, T E,x f (t, X t ) da µ,t = T for every poitive Borel function f : D T R. D f (t, y)p(, x, t, y) dµ(t, y) Here p i the tranition denity for X (weak fundamental olution for A). Ex. If µ(dt, dx) = g(t, x) dt dx for ome g L 1 (D T ) then A µ,t = t g(θ, X θ ) dθ. Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 5/19

6 Cauchy-Dirichlet problem Aumption. (H1) µ M 0,b (D T ), (H2) ϕ L 1 (D), (H3) f : [0, T ] D R R i a meaurable function uch that (a) y f (t, x, y) i continuou for q.e. (t, x) D T. (b) There exit µ R uch that (f (t, x, y) f (t, x, y ))(y y ) µ y y 2 for every y, y R and a.e. (t, x) D T. (c) f (,, 0) L 1 (D T ), (d) c>0 (t, x) up y c f (t, x, y) L 1 (D T ). Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 6/19

7 Cauchy-Dirichlet problem - main reult Let Theorem (T. Klimiak, A.R.) ξ = inf{t : X t / D}. Aume (H1) (H3). Then there exit a unique renormalized olution u of ( ). Moreover, ξ T u(, x) = E,x {1 {ξ>t }ϕ(x T ) + for q.e. (, x) D T. Remark u i an entropy olution of ( ). ξ T } + da µ,t f u (t, X t ) dt Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 7/19

8 Renormalized olution Theorem (Droniou, Porretta & Prignet) Each meaure µ M 0,b (D T ) admit a decompoition of the form µ = g t + div(g) + f, where g L 2 (0, T ; H 1 0 (D)), G = (G 1,..., G d ) L 2 (D T ) d, f L 1 (D T ). Remark. The above decompoition mean that T η dµ = g, η D T 0 t dt (G, η) L 2 + f η dm 1 D T for every η W(D T ) (m 1 denote the Lebegue meaure on D T ), where W = {η L 2 (0, T ; H 1 0 (D)) : η t L2 (0, T ; H 1 (D)). Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 8/19

9 Renormalized olution Definition (Droniou, Porretta & Prignet) A meaurable u : D T R i a renormalized olution of ( ) if (a) f u L 1 (D T ), (b) For ome decompoition (g, G, f ) of µ, u g L (0, T ; L 2 (D)), T k (u g) L 2 (0, T ; H 1 (D)) and u dm 1 = 0, lim n + {n u g n+1} (c) For any S W 2, (R) with compact upport, t (S(u g)) + div(a us (u g)) S (u g)a u (u g) = S (u g)f div(gs (u g)) + GS (u g) (u g) in the ene of ditribution, (d) T k (u g)(t ) = T k (ϕ) in L 2 (D) for all k 0. Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 9/19

10 Parabolic obtacle problem Definition We ay that a pair (u, ν) coniting of a meaurable function u : D T R and a meaure ν on D T i a olution of ( ) if (a) f u L 1 (D T ), ν M 0,b (D T ), h 1 u h 2, m 1 -a.e., (b) For q.e. (, x) D T, ξ T u(, x) = E,x {1 {ξ>t }ϕ(x T ) + ξ T } + d(a µ,t + A ν,t), f u (t, X t ) dt i.e. u i a renormalized olution of the problem { u t + Au = f u µ ν on (0, T ) D, u(t, ) = ϕ, u(t, ) D = 0, t (0, T ). Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 10/19

11 Parabolic obtacle problem We ay that u FD if the proce [, T ] u(t, X t ) i càdlàg under P,x for q.e. (, x) D T. Definition (continued) (c) For every h 1, h 2 FD uch that h 1 h 1 u h 2 h 2, m 1 -a.e. we have ξ T (u(t, X t ) h 1 (t, X t )) da ν+,t = 0, P,x -a.. and ξ T (h 2 (t, X t ) u(t, X t )) da ν,t = 0, P,x -a.. for q.e. (, x) D T (Here h i (t, X t) = lim <t, t h i (, X )). Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 11/19

12 Parabolic obtacle problem Comment. 1 If h 1, h 2 are quai-continuou and h 1 (T, ) ϕ h 2 (T, ), m-a.e. then condition (c) ay that (u h 1 ) dν + = (h 2 u) dν = 0. D T D T 2 In the linear cae with L 2 data condition (c) coincide with the condition introduced by M. Pierre (1979). Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 12/19

13 Parabolic obtacle problem Additional aumption. (H4) There exit a renormalized olution v of the problem { v t + Av = λ, v(t, ) = ψ, v(t, ) D = 0, t (0, T ) with ome λ M 0,b (D T ) and meaurable ψ atifying ψ ϕ uch that f v L 1 (D T ) and h 1 v h 2, m 1 -a.e. on D T. Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 13/19

14 Parabolic obtacle problem - main reult Theorem (T. Klimiak, A.R.) Aume (H1) (H4). (i) There exit a unique olution (u, ν) of ( ). Moreover, (a) u FD and E,x T up u(t, X t ) q <, E,x ( u(t, X t ) 2 dt) q/2 < t T for q (0, 1), (b) T k u L 2 (0, T ; H0 1(D)) for k > 0, where T ku = ( k) u k, (c) u L q (0, T ; W 1,q 0 (D)) for q < d+2 d+1. (ii) If h 1, h 2 are quai-continuou and h 1 (T, ) ϕ h 2 (T, ), m-a.e. then u i quai-continuou. Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 14/19

15 Parabolic obtacle problem - main reult Theorem (continued) (iii) Let u n be a renormalized olution of the problem { un t + A t u n = f un µ n(u n h 1 ) + n(u n h 2 ), u n (T, ) = ϕ, u n (t, ) D = 0, t (0, T ). Then u n u q.e. on D T and u n u in meaure m 1, (iv) If h 2 = + then ν n ν weakly, where ν n = n(u n h 1 ). Similar tatement in cae h 1 =. In the general cae more complicated formulation. Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 15/19

16 Sketch of proof (for ϕ = 0) (i) We prove exitence of a unique olution (Y, Z, L) of RBSDE Y t = τ + t t τ f (X t, Y t ) dt + τ t dl t dt + τ da µ t τ t τ τ t dm, P,x -a.. for q.e. (, x) D T and how that L = A ν for ome ν M 0,b. (ii) Taking t = and integrating with repect to P,x we conclude that u defined by u(, x) = Y, P,x -a.. atifie the nonlinear Fenynman-Kac formula u(, x) = E,x { ξ T (iii) We how that f u L 1 (D T ). ξ T } f u (t, X t ) dt + d(a µ,t + A ν,t). Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 16/19

17 Sketch of proof (for ϕ = 0) (iv) We know that ξ T u(, x) = E,x da γ,t, γ = f u m + µ + ν. We chooe a generalized net {F n } (i.e. F n F n+1, cap(k \ F n ) 0, γ (D T \ n=1 F n) = 0) uch that 1 Fn γ M 0,b W, where W i the pace dual to the pace W = {η L 2 (0, T ; H 1 0 (D)) : η t L2 (0, T ; H 1 (D)). and we define u n (, x) = E,x ξ T A 1 Fn γ,t. Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 17/19

18 Sketch of proof (for ϕ = 0) (iv) Then (L 2 theory of linear equation) u n i a weak olution of the problem { un t + Au n = 1 Fn γ on (0, T ) D, u n (T, ) = 0, u n (t, ) D = 0, t (0, T ), and hence u n i a renormalized olution. (v) Since 1 Fn γ (f u m + µ + ν) TV, it follow that u n u q.e. in D T and u i a renormalized olution of ( ). Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 18/19

19 Reference T. Klimiak, Reflected BSDE and the obtacle problem for emilinear PDE in divergence form, Stochatic Proce. Appl. 122 (2012) T. Klimiak, BSDE with monotone generator and two irregular reflecting barrier. Bull. Sci. Math. (2012) DOI: /j.bulci T. Klimiak, Cauchy problem for emilinear parabolic equation with time-dependent obtacle: a BSDE approach, Potential Anal., to appear. A. Rozkoz and L. Słomińki, Stochatic repreentation of entropy olution of emilinear elliptic obtacle problem with meaure data, Electron. J. Probab. 17 (2012), no. 40, Andrzej Rozkoz, NCU (Toruń) Elliptic equation with meaure data 19/19