Elliptic Operators with Unbounded Coefficients

Transcription

1 Elliptic Operators with Unbounded Coefficients Federica Gregorio Universitá degli Studi di Salerno 8th June 2018 joint work with S.E. Boutiah, A. Rhandi, C. Tacelli

2 Motivation Consider the Stochastic Differential Equation { dx(t, x) = F(X(t, x))dt + Q(X(t, x))db(t), t > 0, X(0) = x R N, (SDE) B(t) is a standard Brownian motion and Q(x) L(R N ). Probabilistic model of the physical process of diffusion Model in mathematical finance Model in biology

3 Motivation Consider the Stochastic Differential Equation { dx(t, x) = F(X(t, x))dt + Q(X(t, x))db(t), t > 0, X(0) = x R N, u(t, x) := E(ϕ(X(t, x)), ϕ C b (R N ) Then u solves Kolomogorov/Fokker-Planck equation u t (t, x) = N i,j=1 u(0, x) = ϕ(x), x R N, a ij (x)d ij u(t, x) + F(x) u(t, x), t > 0, (FP) (a ij (x)) := 1 2 Q(x)Q(x) can be unbounded.

4 Examples The Black-Scholes equation: v t σ2 2 x 2 2 v v rx + rv = 0, 0 < t T, x 2 x v(0, x) = (x K ) +, x > 0. (BS) v(t, x) = u(t t, x) is the value of european type option on the asset price x at time t; σ is the stock volatility; r is the risk-free rate; 0 < x is the underlying asset; K is the prescribed price.

5 Set y = log x. Then, v(t, x) = x α 1 2 e (α+1)2 8 σ 2t w ( σ 2 2 ) t, log x, where α = 2r σ 2 and w solves the heat equation w t = 2 w y 2 t > 0, y R, w(0, y) = e α 1 2 y (e y K ) + y R.

6 Examples Ornstein-Uhlenbeck operator: { dx(t, x) = MX(t, x)dt + db(t), t > 0, X(0, x) = x R N, (OU) M = (m ij ) L(R N ). Set M t := t 0 esm e sm ds. Then u(t, x) : = E(f (X(t, x))) = (2π) N 2 (det Mt ) 1 2 R N e 1 2 M 1/2 t (e tm x y) 2 f (y) dy solves the Ornstein-Ulhenbeck equation u t (t, x) = 1 u(t, x) + Mx u(t, x), t > 0, 2 u(0, x) = f (x) x R N. (OU)

7 Semigroups in short Let X be a Banach space. A family (T (t)) t 0 of bounded operators on X is called a strongly continuous or C 0 -semigroup if T (0) = I and T (t + s) = T (t)t (s) for all t, s 0; t T (t)x X is continuous for every x X.

8 Semigroups in short The generator of a strongly continuous semigroup A is defined as D(A) := {x X : t T (t)x is differentiable on [0, )} Ax := d dt T (t)x t=0 = lim t t 0 1 T (t) e ta (T (t)x x).

9 Semigroup approach to initial-boundary value problems Given X Banach space, A : D(A) X X with boundary conditions in D(A) { u (ACP 1 ) t (t) = Au(t) u(0) = u 0 (A, D(A)) generates a C 0 -semigroup (T (t)) t 0 on X (ACP 1 ) well posed with solution u(t) = T (t)u 0. Study qualitative properties: positivity, stability, regularity,...

10 The resolvent The Resolvent Operator is the operator R(λ, A) := (λ A) 1, λ ρ(a) := {λ C : λ A X is bijective} The semigroup is related to the Resolvent Operator T (t) is a C 0 -semigroup, T (t) e ωt then λ ρ(a) for Reλ > ω R(λ, A)f = 0 e λs T (s)f ds, R(λ, A) 1 Reλ ω.

11 Generation Theorems Hille, Yosida, (1948). Let (A, D(A)) closed, densely defined and λ ρ(a) for every λ > ω R(λ, A) 1 λ ω = A generates T (t) eωt Lumer, Phillips, (1961). Rg(λ A) = X, (λ A)f λ f, for all λ > 0 = A generates T (t) 1

12 Some trivial example Let ω R consider the operator A : R R, Ax = ωx d x(t) = Ax(t) = ωx(t) dt (1) x(0) = x 0 R(λ, A) = (λ A) 1 solves the equation λx ωx = y x = A closed, dense λ ρ(a) if λ > ω R(λ, A) = 1 λ ω R(λ, A)y = y λ ω H. Y. Theorem = and ρ(a) = C \ {ω} For t > 0, T (t) : R R Fixing x 0, T (t)x 0 = x(t) is a function in t and solves (1) y λ ω A generates T (t) e ωt

13 Some trivial example T (t)x = e ωt x Semigroup laws Generator Resolvent i) T (t + s)x = e ω(t+s) x = e ωt e ωs x = T (t)t (s)x ii) T (0)x = e µ0 x = x iii) lim t 0 T (t)x = x R(λ, A)y = Ax = lim t 0 T (t)x x t y λ ω = 0 = lim t 0 e ωt x x t e λs T (s)yds = 0 = ωx e λs e ωs yds

14 Kernel Representation If the coefficients of the differential operator have suitable regularity, the semigroup has a kernel representation. T (t)f (x) = k(t, x, y)f (y)dy. R N Consider the heat equation t u(t, x) = u(t, x), u(0, x) = f (x) 1 u(t, x) = T (t)f (x) = (4πt) N/2 e x y 2 4t f (y)dy In this case k(t, x, y) = 1 e x y 2 (4πt) N/2 4t The behavior of the semigroup depends on the behavior of the kernel The kernel is related to the eigenvalues and the ground state of the problem

15 Elliptic operators with unbounded coefficients We consider elliptic operators with unbounded coefficients of the form Au(x) = N i,j=1 a ij (x)d ij u(x) + N b i (x)d i u(x) + V (x)u(x). i=1 The realisation A of A in C b (R N ) with maximal domain D max (A) = {u C b (R N ) 1 p< Au = Au. W 2,p loc (RN ) : Au C b (R N )}

16 { t u(t, x) = Au(t, x) t > 0, x R N u(0, x) = f (x) x R N, (2) with f C b (R N ). To have solution we assume that for some α (0, 1), (1) a ij, b i, V C α loc (RN ), i, j = 1,..., N; (2) a ij = a ji and a(x)ξ, ξ = i,j a ij (x)ξ i ξ j k(x) ξ 2 x, ξ R N, k(x) > 0; (3) c 0 R s.t. V (x) c 0, x R N.

17 Consider the problem on bounded domains t u R (t, x) = Au R (t, x) t > 0, x B R u R (t, x) = 0 t > 0, x B R u R (0, x) = f (x) x B R, (3) with f C b (R N ). Then A is uniformly elliptic on compacts of R N and (3) admits unique classical solution u R (t, x) = T R (t)f (x), t 0, x B R with T R (t) analytic semigroup in C(B R ). The infinitesimal generator of (T R (t)) is (A, D R (A)), D R (A) = {u C 0 (B R ) 1<p< W 2,p (B R ), : Au C(B R )}.

18 Theorem 1 (i) (T R (t)) has the integral representation T R (t)f (x) = p R (t, x, y)f (y)dy, f C(B R ), t > 0, x B R B R with strictly positive kernel p R C((0, + ) B R B R ). (ii) T R (t) L(L p (B R )) per ogni t 0 e per ogni 1 < p < + ; (iii) T R (t) is contractive in C(B R ); (iv) for all fixed y B R, p R (,, y) C 1+ α 2,2+α ([s, t 0 ] B R ) for all 0 < s < t 0 and t p R (t, x, y) = Ap R (t, x, y), (t, x) (0, + ) B R.

19 f C(B R ) + (iv), gives u R C 1+ α 2,2+α ([s, t 0 ] B R ). Proposition 1 Let f C b (R N ) and t 0; then it exists T (t)f (x) = lim T R(t)f (x), x R N (4) R + and (T (t)) is a positive semigroup in C b (R N ).

20 Schauder interior estimates u R C 1+α/2,2+α ([ε,t ] B R 1 ) C u R L ((0,T ) B R ) Ce λ 0T f Theorem 2 Let f C b (R N ), then the function u(t, x) = T (t)f (x) belongs to C 1+ α 2,2+α loc ((0, + ) R N ) and it solves { u t (t, x) = Au(t, x) t > 0, x R N, u(0, x) = f (x) x R N.

21 The operator For c > 0, b R, α > 2 and β > α 2 we consider on L p (R N ) A := (1 + x α ) + b x α 2 x c x β. (5) Aim. Solvability of λu Au = f Properties of the maximal domains Generation of positive analytic semigroup

22 Related results b = c = 0 : Unbounded Diffusion: A = (1 + x α ) [G. Metafune, C. Spina, 10] α > 2, p > N N 2 b = 0 : Schrödinger-Type Operator: A = (1 + x α ) c x β [L. Lorenzi, A. Rhandi, 15] 0 α 2, β 0 [A. Canale, A. Rhandi, C.Tacelli, 16] α > 2, β > α 2 c = 0 : Unbounded Diffusion & Drift: A = (1 + x α ) + b x α 2 x [S. Fornaro, L. Lorenzi 07]: 0 α 2. [Metafune, Spina, Tacelli, 14] α > 2, b > 2 N p > N N 2+b Complete : A = x α + b x α 2 x x c x α 2 [G. Metafune, N. Okazawa, M. Sobajima, C. Spina, 16] N/p (s 1 + min{0, 2 α}, s 2 + max{0, 2 α}), c + s(n 2 + b s) = 0

23 Remark A := (1 + x α ) + b x α 2 x c x β. (1 + x α ) and b x α 1 x x are homogeneous at infinity. They have the same influence on the behaviour of A. ( possible dependence on coefficient b); x β with β > α 2 is super homogeneous. No critical exponent, but strong unboundedness with respect to diffusion and drift. A different approach is required; b x α 1 x x is not a small perturbation of (1 + x α ) c x β.

24 Solvability in C 0 (R N ) First consider the operator (A, D max (A)) on C b (R N ) where D max (A) = {u C b (R N ) 1 p< W 2,p loc (RN ) : Au C b (R N )}. It is known that we can associate to the parabolic problem { ut (t, x) = Au(t, x) x R N, t > 0, u(0, x) = f (x) x R N, f C b (R N ) (6) a semigroup of bounded operators (T min (t)) t 0 in C b (R N ) generated by A min = (A, ˆD), where ˆD D max.

25 Solvability in C 0 (R N ) The uniqueness relies on the existence of suitable Lyapunov function for A, i.e. 0 φ C 2 (R N ) : lim φ(x) = +, Aφ λφ 0, λ > 0. x Proposition 2 Assume that α > 2, β > α 2. Then φ = 1 + x γ function for A. : γ > 2 is Lypunov Proposition 3 T min (t) is generated by (A, D max (A)) C 0 (R N ), is compact, preserves C 0 (R N ).

26 Solvability of λu Au = f in L p The transformation v = u φ where φ = (1 + x α ) b α gives λu Au = f ( U) v = }{{} := H U = 1 4 φ φ f φ 1 + x α. φ φ + λ + c x β 1 + x α c x β 1 + x α For λ λ 0 we have 0 U L 1 loc then there exists G(x, y) such that v(x) = G(x, y) f (y)dy solves Hv = f R N u(x) = Lf (x) := G(x, y) φ(y) f (y)dy solves λu Au = f R N φ(x) 1 + y α We study the L p -boundedness of the operator L by estimates of G.

27 Green function estimates of U Since U(0) = λ > 0 and U behaves like x β α as x we have the following estimates C 1 (1 + x β α ) U C 2 (1 + x β α ) if β α, (7) 1 C x α β U C x α β if α 2 < β < α for some positive constants C 1, C 2, C 3, C 4. Z. Shen 95, gives estimate of G(x, y) if the potential belong to the reverse Holder class B q if q N 2.

28 Green function estimates of U f 0 is said to be in B q if C > 0 : ( 1 B B f q dx) 1/q C ( 1 B B ) fdx B R N. If β α 0 then x β α B. If N q < (β α) < 0 then x β α B q β > α 2 U B N 2 β α 2 U / B N 2

29 Green function estimates of U For every k > 0 there is some constant C(k) > 0 such that 1 m(x) G(x, y) := sup r>0 { r : C k (1 + m(x) x y ) k 1 r N 2 B(x,r) U(y)dy 1 1 x y N 2 }, x R N. Proposition 4 m(x) C(1 + x ) β α 2, β > α 2, C = C(α, β, N) Sketch of Proof. Observe that U CṼ.

30 Green function estimates of U Lemma 3 if β > α 2 G(x, y) C k x y k (1 + x ) β α 2 k 1 x y N 2 Finally we can prove the boundedness of L in L p (R N ). Theorem 4 C = C(λ): γ [0, β] and f L p (R N ) x γ Lf p C f p. For every f Cc (R N ) the function u solves λu Au = f

31 Theorem 5 Closedness & Invertibility of λ A p in D p,max (A) D p,max (A) = {u L p (R N ) W 2,p loc (RN ) : Au L p (R N )}. Assume that N > 2, α > 2 and β > α 2. For p (1, ) the following holds D p,max (A) = {u W 2,p (R N ) : Au L p (R N )}. The operator λ A p is closed and invertible. Moreover C = C(λ) > 0 : γ [0, β] and λ λ 0, we have γ u p C λu A p u p, u D p,max (A). The inverse of λ A p is a positive operator λ λ 0. Moreover, if f L p C 0 then (λ A p ) 1 f = (λ A) 1 f.

32 Weighted gradient and second derivative estimates D p (A) := {u W 2,p (R N ) : Vu, (1+ x α 1 ) u, (1+ x α )D 2 u L p (R N )} Lemma 6 C > 0: u D p (A) we have (1 + x α 1 ) u p C( A p u p + u p ), (1 + x α )D 2 u p C( A p u p + u p ). The space Cc (R N ) is dense in D p (A) endowed with the norm u Dp (A) := u p + Vu p + (1 + x α 1 ) u p + (1 + x α ) D 2 u p.

33 Generation of Analytic Semigroup Theorem 7 (A, D p (A)) generates a analytic semigroup in L p (R N ).

34 Some References [1] S. E. Boutiah, F. Gregorio, A. Rhandi, C. Tacelli: Elliptic operators with unbounded diffusion, drift and potential terms. J. Differential Equations 264 (2018), no. 3, [2] S. Fornaro, L. Lorenzi: Generation results for elliptic operators with unbounded diffusion coefficients in L p and C b -spaces, Discrete Contin. Dyn. Syst. 18 (2007), [3] L. Lorenzi, A. Rhandi: On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates. J. Evol. Equ. 15 (2015), no. 1, [4] G. Metafune, C. Spina: Elliptic operators with unbounded diffusion coefficients in L p spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. XI (2012),