On semilinear elliptic equations with measure data

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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