On semilinear elliptic equations with measure data

Transcription

1 On semilinear elliptic equations with measure data Andrzej Rozkosz (joint work with T. Klimsiak) Nicolaus Copernicus University (Toruń, Poland) Controlled Deterministic and Stochastic Systems Iasi, July 2 7, 2012 Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 1/28

2 Motivation Dirichlet problem in bounded domain D R d : Assume that u = f (, u) + µ in D, u D = 0. ( ) 1 f : D R R is a measurable function such that y f (x, y) is continuous, f (x, 0) = 0, y f (x, y) nonincreasing, For every c > 0, F c L 1 (D), where F c (x) = sup y c f (x, y), Ex. f (x, y) = y q 1 y for some q > 1. 2 µ - bounded smooth measure on D (Boccardo & Orsina: µ L 1 (D; dx) + H 1 (D), where H 1 (D) is the space dual to H0 1 (D); we may think that µ(dx) = g(x) dx, where g L 1 (D; dx)). Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 2/28

3 Motivation Comments. If g L 2 (D; dx) then by a solution of ( ) we mean a weak solution, i.e. u is a solution if u H0 1(D) and for v H1 0 (D), u v dx = f (x, u)v dx + v µ(dx) D D D = f (x, u)v dx + v(x)g(x) dx. D If g L 1 (D; dx) we cannot take v H0 1 (D) as test function (this is related to the fact that in general there is no solution of ( ) in H0 1(D)). We can take v C0 (D) as a test function. The space C0 (D) is, however, too small (the solution of equations of the form ( ) (with divergence form generator) in the sense of distributions may be not unique). D Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 3/28

4 Motivation Let Problem: to give proper definition of the solution to ( ) (mainly problem of uniqueness). To tackle the problem the so-called entropy solutions and renormalized solutions were introduced. T k u(x) = ( k) u(x) k. T - space of functions u : D R such that (a) u is finite a.e., (b) T k u H0 1 (D) for k > 0. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 4/28

5 Motivation Definition u is called an entropy solution of u = f (, u) + µ in D, u D = 0 ( ) if u T and for every v H0 1(D) L (D) and k > 0, u T k (u v) dx = f (x, u)t k (u v) dx D D + T k (u v) µ(dx). D Theorem (Bènilan, Boccardo, Brezis, Murat,... ) There exists a unique entropy solution of the problem ( ). Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 5/28

6 Motivation Comments. Both entropy and renormalized solutions are considered in the space T. The space T is not linear. Equations with non-radon measures µ. Equations with nonlocal operators (for instance fractional Laplacian). Probabilistic approach. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 6/28

7 Elliptic equations with measure data Suppose we are given E - locally compact, separable metric space, m - Borel measure on E such that supp[m] = E, (E, D[E]) - regular Dirichlet form on L 2 (E; m). Let A be the nonpositive self-adjoint operator corresponding to (E, D[E]), i.e. D(A) D[E], E(u, v) = ( Au, v), u D(A), v D[E]. We consider equations of the form Au = f (, u) + µ, where f : E R R is a measurable function and µ is a smooth measure on E (with respect to the capacity determined by the form (E, D[E])). Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 7/28

8 First examples 1 Let E = D R d, m(dx) = dx and E(u, v) = u v dx, D u, v D[E]. (a) D[E] = H0 1 (D), then A = with Dirichlet boundary conditions, (b) D[E] = H 1 (D), then A = with Neumann boundary conditions. 2 Let E = R d, m(dx) = dx and E(u, v) = û(x)ˆv(x) x α dx, R d D[E] = {u L 2 (R d ; dx) : û(x) 2 x α dx < } R d for some α (0, 2]. Then A = α/2. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 8/28

9 Probabilistic solutions (E, D[E]) Hunt process X = (Ω, X, F, (F t ), P x ) with life-time ζ. Ex. 1a: X - Wiener process, ζ = τ D, Ex. 1b: X - Wiener process, ζ =, Ex. 2: X - symmetric α- stable Lévy process. µ S (S - space of smooth measures on E) continuous additive functional A µ of X corresponding to µ in the sense of Revuz), i.e for every measurable bounded nonnegative f on E, Ex. If µ(dx) = g(x) dx, then 1 t lim t 0 t E m f (X s ) da µ s = f (x) µ(dx). 0 E A µ t = t ζ 0 g(x s ) ds, t 0. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 9/28

10 Probabilistic solutions Definition Let µ S be a smooth measure such that E x ζ 0 d Aµ t < for q.e. x E. We say that a quasi-continuous function u : E R is a probabilistic solution of the equation Au = f u + µ, ( ) where f u = f (, u), if E x ζ 0 f u(x t ) dt < and for q.e. x E. Ex. If µ(dx) = g(x) dx, then ζ ζ u(x) = E x f u (X t ) dt + E x 0 0 da µ t ζ ζ u(x) = E x f u (X t ) dt + E x g(x t ) dt. 0 Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 10/28 0

11 Probabilistic solutions, µ S M 0,b - space of all smooth signed measures on E with bounded total variation. Assumptions. (A1) f is measurable and y f (x, y) is continuous for each x E, (A2) (f (x, y 1 ) f (x, y 2 ))(y 1 y 2 ) 0 for y 1, y 2 R, x E, (A3) F c L 1 (E; m) for c > 0, where F c (x) = sup y c f (x, y), (A4) f (, 0) L 1 (E; m), µ M 0,b, (A3 ) for every c > 0 the function F c is locally quasi-l 1, i.e. t F c (X t ) belongs to L 1 loc (R +) P x -a.s. for q.e. x E, (A4 ) E x ζ 0 f (X t, 0) dt <, E x ζ 0 d Aµ t < for m-a.e. x E. Remark 1 (A3) implies (A3 ). 2 If (E, D[E]) is transient then (A4) implies (A4 ). Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 11/28

12 Probabilistic solutions, µ S D q - space of progressively measurable càdlàg processes Y such that E sup t 0 Y t q <. u FD q if the process t u(x t ) belongs to D q under P x for q.e. x E. u is of class (FD) if the process t u(x t ) is of class (D) (i.e. the family {u(x τ ), τ - finite stopping time} is uniformly integrable) under P x for q.e. x E. Theorem If µ S and f satisfies (A1), (A2), (A3 ), (A4 ) then there exists a unique solution u FD q, q (0, 1), of such that u is of class (FD). Au = f (, u) + µ Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 12/28

13 Probabilistic solutions, µ S Theorem (cont.) For q.e. x E there exists a unique pair of progressively measurable processes (Y x, M x ) such that Y x t T ζ T ζ = YT x ζ + f (X s, Ys x ) ds + da µ s t ζ t ζ T ζ dms x, t [0, T ], P x -a.s. t ζ for every T > 0, M x is a P x -martingale, Y x is of class (D), YT x ζ 0 if T. Moreover, for q.e. x E, It follows in particular that u(x t ) = Y x t, t 0, P x -a.s. ζ ζ u(x) = E x f (X s, u(x s )) ds + E x da µ s. 0 0 Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 13/28

14 Regularity of probabilistic solutions, µ M 0,b Assume that (E, D[E]) is transient. F e - extended Dirichlet space (u F e, if u < m-a.e. and there exists an E-Cauchy sequence {u n } D[E] such that u n u m-a.e.) S (0) 00 - class of measures µ S such that v(x) µ(dx) c E(v, v), v F e C 0 (E) E and Uµ is q.e. bounded, where Uµ F e is such that E(Uµ, v) = v(x) µ(dx), v F e C 0 (E). E Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 14/28

15 Regularity of probabilistic solutions, µ M 0,b Definition Assume that (E, D[E]) is transient and µ M 0,b. A quasicontinuous function u : E R is called a solution of in the sense of duality if ν, u = f u L 1 (E; m) and Au = f u + µ E u dν <, ν S (0) 00, ν, u = (f u, Uν) L 2 (E;m) + µ, Uν, ν S (0) 00. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 15/28

16 Regularity of probabilistic solutions, µ M 0,b Proposition Assume that (E, D[E]) is transient and µ M 0,b. If u is quasi-continuous and f u L 1 (E; m) then u is a probabilistic solution of Au = f u + µ ( ) iff u is a solution of ( ) in the sense of duality. Remark Assume that (E, D[E]) is transient and µ S (0) 0. If u is a probabilistic solution of ( ) and f u L 2 (E; m) then u F e and E(u, v) = (f u, v) L 2 (E;m) + v, µ, v F e, i.e. u is a weak solution of ( ). Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 16/28

17 Regularity of probabilistic solutions, µ M 0,b Proposition Assume that (E, D[E]) is transient, µ M 0,b and f (, 0) L 1 (E; m). If u is a probabilistic solution of ( ), then (i) f u L 1 (E; m) and f u L 1 (E;m) f (, 0) L 1 (E;m) + µ TV, (ii) T k (u) F e for every k > 0 and E(T k (u), T k (u)) k( f u L 1 (E;m) + µ TV ), (iii) The following condition of vanishing energy of the solution is satisfied: E(u1 {k u k+1}, u1 {k u k+1} ) f u (x) m(dx) + { u k} { u k} dµ. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 17/28

18 Regularity of probabilistic solutions, µ M 0,b Theorem Assume that (E, D[E]) is transient and µ, f satisfy (A1) (A4). Then there exists a unique probabilistic solution u of ( ) such that u is of class (FD) and u FD q for q (0, 1). Moreover, f u L 1 (E; m) and T k (u) F e for every k 0. Remark In general u / L 1 loc (E; m). Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 18/28

19 Regularity of probabilistic solutions, µ M 0,b Remark Let (E, D[E]) be a regular Dirichlet form and let g be a strictly positive bounded Borel function on E. Then the perturbed form (E g, D[E]), where E g (u, v) = E(u, v) + (u, v) L 2 (E;g dm) is a transient regular Dirichlet form on L 2 (E; m). The operator A g associated with (E g, D[E]) has the form A g = A + g, where A is associated with (E, D[E]). Therefore we may apply our regularity results to the equation Au + gu = f u + µ. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 19/28

20 Examples Let {ν t, t > 0} be a convolution semigroup of symmetric probability measures on R d and let ψ denote its characteristic symbol, i.e. ˆν t (x) = e i(x,y) ν t (dy) = e tψ(x). R d It is known that the Dirichlet form E(u, v) = û(x)ˆv(x)ψ(x) dx, R d where u, v D[E], D[E] = {u L 2 (R d ; dx) : û(x) 2 ψ(x) dx < } R d determined by {ν t, t > 0} is a regular Dirichlet form on L 2 (R d ; dx). If 1/ψ jest locally integrable on R d then (E, D[E]) is transient. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 20/28

21 Examples Proposition Assume that µ M 0,b, f satisfies (A1) (A4). If 1/ψ is locally integrable on R d then there exists a unique probabilistic solution of the problem ψ( )u = f (x, u) + µ, x R d. 1 (fractional Laplacian) ψ(x) = c x α for some α (0, 2], c > 0. The associated form is transient iff d > α. Observe that ψ( ) = c( 2 ) α/2 = c α/2. 2 (relativistic Schrödinger operator) ψ(x) = m 2 c 4 + c 2 x 2 mc 2. The associated form is transient if d 3. 3 (operator associated with Brownian motion with Bessel subordinator) ψ(x) = log((1 + x 2 ) + (1 + x 2 ) 2 1). Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 21/28

22 Examples Proposition Let D R d be an open bounded set. Assume that µ M 0,b, f satisfies (A1) (A4). If g : D R is a strictly positive bounded Borel function or g is nonnegative bounded Borel function and 1/ψ is locally integrable on D then there exists a unique probabilistic solution of the problem Remark u L 1 (D; dx). ψ( )u + gu = f (x, u) + µ, x D, u D c = 0. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 22/28

23 Examples D - open subset of R d, a : D R d R d - measurable function such that d a ij (x)ξ i ξ j 0, a ij (x) = a ji (x), x D, ξ R d. i,j=1 Consider the assumptions (H1) a ij L 2 loc (D), a ij x j L 2 loc (D), i, j = 1,..., d, (H2) There exists λ > 0 such that d i,j=1 a ij(x)ξ i ξ j λ ξ 2, x D, ξ R d. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 23/28

24 Examples Proposition Assume that µ M 0,b and f satisfies (A1) (A4) on D. If either (H1) or (H2) is satisfied and g : R R is a strictly positive bounded function or (H2) is satisfied, g is nonnegative and d 3 then there exists a unique probabilistic solution of the problem d i,j=1 (a ij (x) u ) + gu = f (x, u) + µ, u D = 0. x i x j Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 24/28

25 Examples Proposition Let D be a bounded domain in R with boundary of class C (locally given by a continuous function) and let a satisfy (H3). Assume that µ M 0,b, f satisfies (A1) (A4) on D and g is a strictly positive bounded Borel function on D. Then there exists a unique probabilistic solution of the problem d i,j=1 (a ij (x) u ) + gu = f (x, u) + µ in D, x i x j u n = 0 on D. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 25/28

26 Other possible applications Other interesting situations in which we encounter regular Dirichlet forms include Laplace-Beltrami operators on manifolds, quantam graphs, Hamiltonians with singular interactions, diffusions with Wentzell boundary conditions,... Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 26/28

27 Proofs - existence of probabilistic solutions 1 We prove the existence of a solution (Y n, Z n ) of the BSDE Y n t = n t n 1 [0,ζ] (s)f (s, Ys n ) ds + t n da µ s t dm n s, t [0, n], P x -a.s. (we use some ideas from the paper Ph. Briand, B. Delyon, Y. Hu, E. Pardoux, L. Stoica, L p solutions of BSDEs, SPA, 2003). 2 Letting n and using some a priori estimates on (Y n, Z n ) we show the existence of the pair (Y, Z) which for q.e. x E solves the BSDE Y t = t ζ t f (s, Y s ) ds+ t ζ t t ζ da µ s dm s, t 0, P x -a.s. t Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 27/28

28 Proofs - regularity of probabilistic solutions We know that for q.e. x E. To prove that ζ ζ u(x) = E x f u (X t ) dt + E x 0 0 da µ t E(T k (u), T k (u)) k( f u L 1 (E;m) + µ TV ) we consider a generalized nest {F n } such that 1 Fn f u m + 1 Fn µ S (0) 00 and we set Then ζ ζ u n (x) = E x 1 Fn f u (X t ) dt + E x 1 Fn (X t ) da µ t, x E 0 u n = U(1 Fn f + u 0 m + µ + ) U(1 Fn f u m + µ ) F e. We prove that u n u q.e. and (taking appropriate test functions) that ( ) holds true with u n in place of u on the left-hand side of the inequality. Andrzej Rozkosz, NCU (Toruń) Elliptic equations with measure data 28/28