Comparison Results for Semilinear Elliptic Equations in Equimeasurable Domains

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1 Comparison Results for Semilinear Elliptic Equations in Equimeasurable Domains François HAMEL Aix-Marseille University & Institut Universitaire de France In collaboration with Emmanuel RUSS Workshop on Geometric Aspects of Semilinear Elliptic and Parabolic Equations: Recent Advances and Future Perspectives Banff, May 2014

2 INTRODUCTION Semilinear elliptic equation { div(a(x) u) + H(x, u, u) = 0 in Ω u = 0 on Ω Bounded domain Ω R N of class C 2 (dimension N 2) Goal: to compare u with a (radially symmetric) solution v of { div(â(x) v) + Ĥ(x, v, v) = 0 in Ω v = 0 on Ω where Ω is the ball with the same measure as Ω ( Ω = Ω ) and  and Ĥ satisfy the same type of constraints as A and H

3 General assumptions A : Ω S n (R) is of class W 1, (Ω) and uniformly elliptic: A(x) Λ(x) Id with Λ L (Ω) and Λ(x) λ > 0 a.e. in Ω H : Ω R R N R is continuous and there exist 1 q 2 and three continuous functions a, b and f : Ω R R N R s.t. { H(x, s, p) a(x, s, p) p q + b(x, s, p) s f (x, s, p) b(x, s, p) 0 No bound on H from above The cases q = 1 and 1 < q 2 will be treated separately Existence and uniqueness results are different

4 Notions of solutions u Weak solutions: u H0 1(Ω), H(, u( ), u( )) L1 (Ω) and A(x) u ϕ + H(x, u, u) ϕ = 0 Ω Ω for all ϕ H0 1(Ω) L (Ω) If H(, u( ), u( )) L 2 (Ω), then test functions ϕ H0 1(Ω) Strong solutions u W (Ω) = W 2,p (Ω) 1 p<+

5 Distribution functions µ ϕ (t) = { x Ω, ϕ(x) > t } Schwarz symmetric decreasing rearrangement of ϕ: ϕ (x) = min { t R, µ ϕ (t) α n x n }, where α n = B 1 x Ω µ ϕ = µ ϕ (a) Function ϕ (b) Schwarz spherical rearrangement Goal: given a solution u in Ω, show that u v in Ω where v solves a comparable problem in Ω

6 LINEAR GROWTH IN THE GRADIENT Theorem 1 Assume q = 1: H(x, s, p) a(x, s, p) p + b(x, s, p) s f (x, s, p) Let u W (Ω) be a solution such that u > 0 in Ω and u 0 on Ω Then u v a.e. in Ω where v H0 1(Ω ) C(Ω ) is the unique weak solution of div( Λ(x) v) â(x) v f u (x) = 0 in Ω, v = 0 on Ω with some radially symmetric functions Λ, â L (Ω ) such that Λ, Λ 1 L1 (Ω ) = Λ 1 L1 (Ω) 0 < ess inf Ω Λ Λ(x) ess sup Ω 0 inf Ω R R a+ â(x) sup a + n Ω R R n and f u (x) is the Schwarz rearrangement of f u (y) = f (y, u(y), u(y))

7 Corollary For all functions a and f in L (Ω ) such that â a and f u f : u v a.e. in Ω where v H 1 0 (Ω ) C(Ω ) is the unique weak solution of div( Λ(x) v) a(x) v f (x) = 0 in Ω, v = 0 on Ω Proof. div( Λ(x) v) a(x) v f (x) = (â(x) a(x)) v + fu (x) f (x) 0 in the weak H 1 0 (Ω ) sense. Then, v v a.e. in Ω (weak maximum principle [Porretta]). Finally, u v a.e. in Ω.

8 Existence and uniqueness for the problem { div( Λ(x) v) â(x) v f u (x) = 0 in Ω v = 0 on Ω [Porretta] Remark: uniqueness = v is radially symmetric

9 Existence and uniqueness for the problem { div(a(x) u) + H(x, u, u) = 0 in Ω If u = 0 on Ω ω 1 (s s ) H(x, s, p) H(x, s, p) ω (s s ), ω > 0 H(x, s, p) H(x, s, p ) α(x) (1 + s 2/N ) p p α L r (Ω), r = N 2 /2 for N 3, r > 2 for N = 2, r = 2 for N = 1 = uniqueness If, additionally, H(x, s, p) β(x) (1 + s + p ), with β L t (Ω), t = N for N 3, t > 2 for N = 2, t = 2 for N = 1 = existence of a unique solution u H 1 0 (Ω) If, additionally, β L (Ω), then u W (Ω) If, additionally, H(x, 0, p) σ p and H(, 0, 0) 0, = u > 0 in Ω and u 0 on Ω

10 Some references in the literature Talenti (1976): A Id, Λ = 1, b 0, f L (Ω), div(a(x) u) + b(x)u = f (x) in Ω Then u v where v = f in Ω Talenti (1985): A Id, Λ = 1, α L (Ω, R N ), b 0, f L (Ω), div(a(x) u) + α(x) u + b(x)u = f (x) in Ω Then u v where v + α e r v = f in Ω and α = α L (Ω), e r (x) = x/ x = our first theorem

11 Further results: pointwise or integral comparisons between u and v in Ω or subdomains [Alvino, Trombetti, Lions, Matarasso], [Bandle, Marcus], [Cianchi], [Ferone, Messano], [Messano], [Trombetti, Vazquez] In most references: Λ is chosen as a constant λ and the second-order terms in Ω are λ v If one takes Λ = λ constant, then Λ = λ But, Λ can be chosen non-constant and, in general, Λ is not constant either

12 Quantitatively improved inequalities when Ω is not a ball Theorem 2 Same assumptions and notations as in Theorem 1, and Ω is not a ball There is η u > 0 such that Furthermore, if (1 + η u ) u v a.e. in Ω A W 1, (Ω) + Λ 1 L (Ω) + a L (Ω R R N ) + f L (Ω R R N ) M H(x, s, p) H(x, 0, 0) M ( s + p ) M H(x, 0, 0) 0, H(x, 0, 0) dx M 1 < 0 Ω then there is η = η(ω, N, M) > 0 independent of u, such that (1 + η) u v a.e. in Ω

13 AT MOST QUADRATIC GROWTH IN THE GRADIENT Theorem 3 Assume H(x, s, p) a(x, s, p) p q + b(x, s, p) s f (x, s, p) with 1 < q 2 and inf Ω R R N b > 0. Let u W (Ω) be a solution s.t. u > 0 in Ω and u 0 on Ω. Then u v a.e. in Ω where v H 1 0 (Ω ) C(Ω ) is the unique weak solution of div( Λ(x) v) â(x) v q + δ v f (x) = 0 in Ω, v = 0 on Ω with δ > 0 and some radially symmetric functions Λ, â, f L (Ω ) s.t. 0 < ess inf Λ Λ(x) ess sup Λ, Λ 1 L 1 (Ω ) = Λ 1 L 1 (Ω) ( ess inf Λ ) q 1 ( ess sup Λ ) 2q 2 0 inf a + â(x) sup a + ess sup Λ ess inf Λ inf f f (x) sup f, f = f u Ω Ω

14 Furthermore, for every ε > 0, there exists a radially symmetric function fε L (Ω ) such that and = µ fu µ fε (u v ε ) + L 2 (Ω ) ε where v ε H 1 0 (Ω ) C(Ω ) is the unique weak solution of div( Λ(x) v ε ) â(x) v ε q + δv ε f ε (x) = 0 in Ω, v ε = 0 on Ω

15 Corollary For all functions a â and f f and for all 0 < δ δ: u v a.e. in Ω where v H 1 0 (Ω ) C(Ω ) is the unique weak solution of div( Λ(x) v) a(x) v q + δ v f (x) = 0 in Ω, v = 0 on Ω Proof. div( Λ(x) v) a(x) v q + δ v f (x) = (â(x) a(x)) v q + (δ δ)v + f (x) f (x) 0 in the weak H0 1(Ω ) L (Ω ) sense. Then, v v a.e. in Ω (weak maximum principle [Barles, Murat]). Finally, u v a.e. in Ω.

16 Existence and uniqueness for the problem { div( Λ(x) v) â(x) v q + δ v f (x) = 0 in Ω v = 0 on Ω with v H0 1(Ω ) L (Ω ) Existence: [Boccardo, Murat, Puel] Uniqueness: [Barles, Murat] Remark: uniqueness = v is radially symmetric, and continuous in Ω

17 Existence and uniqueness for the problem { div(a(x) u) + H(x, u, u) = 0 in Ω u = 0 on Ω If H(x, s, p) = β(x, s, p) + H(x, s, p) β(x, s, p)s ν s 2, H(x, s, p) ρ + ϱ( s ) p 2 β(x, s, p) κ (γ(x) + s + p ), with ν, κ, ρ > 0, 0 γ L 2 (Ω) and ϱ : R + R + is increasing = existence of a solution u H0 1(Ω) L (Ω) [Boccardo, Murat, Puel] If H(x, s, p) M(1 + s r + p q ) with q < 1 + 2/N, M 0, r 0, r(n 2) < N + 2 = u W (Ω)

18 If s H and p H exist and { s H(x, s, p) σ > 0, p H(x, s, p) θ( s ) (1 + p ) H(x, s, 0) ϑ( s ) for some continuous nonnegative functions θ and ϑ = uniqueness of u H 1 0 (Ω) L (Ω), and u 0 if H(x, 0, 0) 0 [Barles, Murat] Further existence results, for the equation satisfied by v, when δ = 0 or even δ < 0, and when f is small in some spaces [Abdellaoui, Dall Aglio, Peral], [Ferone, Murat], [Ferone, Posteraro, Rakotoson], [Jeanjean, Sirakov], [Sirakov]

19 Existence or uniqueness do not always hold if δ 0 for general f [Abdellaoui, Dall Aglio, Peral], [Hamid, Bidaut-Véron], [Jeanjean, Sirakov], [Porretta], [Sirakov] Example [Porretta]: consider Set w = e v 1. Then v v 2 f (x) = 0 w f (x)(1 + w) = 0 Let λ 1 be the first eigenvalue of with Dirichlet b.c. If f (x) λ 1 (> 0), then v 0, whence w 0 and which is impossible. w = f (x)(1 + w) > λ 1 w

20 Some comparison results in the literature [Alvino, Lions, Trombetti] H = H(x, p) and A Id, Λ = 1, H(x, p) f (x) + κ p q with κ > 0, 0 f L (Ω) Then u v in Ω, where v H0 1(Ω ) L (Ω ) is any solution of v κ v q f (x) = 0 in Ω provided such a solution exists (it does if f L (Ω) is small enough) Further results: [Maderna, Pagani, Salsa], [Messano], [Pašić] [Ferone, Posteraro] A Id, Λ = 1, H(x, s, p) = div(f )+ H(x, s, p), H(x, s, p) f (x)+ p 2 with F (L r (Ω)) N, f L r/2 (Ω) and r > N Then u v in Ω, where v H0 1(Ω ) L (Ω ) is any solution of v v 2 div( F e r ) f (x) = 0 in Ω provided such a solution exists (it does if f and F are small enough)

21 [Tian, Li] H(x, s, p) f (x) + κ Λ(x) 2/q p q with κ > 0, 0 f L r (Ω) and r > N 2 /(3N 2) Then u v in Ω, where v H0 1(Ω ) L (Ω ) is any solution of div( Λ(x) v) κ Λ(x) 2/q v q f (x) = 0 in Ω provided such a solution v exists (it does if f is small enough)

22 Different assumptions Bounds on H vs. lower bound on H Assumption on the existence of v ( small coefficients) vs. assumption inf Ω R R N b > 0 and property δ > 0 (existence is automatically guaranteed, no smallness assumption) Problem with Ĥ(x, v) and f in Ω vs. Ĥ(x, v, v) and f, f ε in Ω

23 Particular case of our results Choose Λ = λ constant, a = α constant and f = γ constant Then u v in Ω, where v is the unique H 1 0 (Ω ) C(Ω ) solution of { λ v α v q + δ v = γ in Ω Even this particular case is new. v = 0 on Ω

24 Theorem 4 (Quantitatively improved inequalities when Ω is not a ball) Same assumptions and notations as in Theorem 3, and Ω is not a ball There is η u > 0 such that (1 + η u ) u v in Ω and ( ) (1 + η u ) u + v L ε 2 (Ω ) ε Furthermore, if q < 1 + 2/N, r 0, r(n 2) < N + 2 and A W 1, (Ω) + Λ 1 L (Ω) + a L (Ω R R N ) + f L (Ω R R N ) M b(x, s, p) M 1 H(x, s, p) H(x, 0, 0) M ( s r + p q ) H(x, s, p) M(1 + s r + p q ) M H(x, 0, 0) 0, H(x, 0, 0) dx M 1 < 0, Ω then there is η = η(ω, N, q, M, r) > 0 independent of u, such that (1 + η) u v in Ω and ( ) (1 + η) u + v L ε 2 (Ω ) ε

25 SOME ELEMENTS OF THE PROOFS div(a u) = H(x, u, u) Assume A Λ Id, A and Λ are C 1 (Ω), u is analytic in Ω. Then { Z = a [ 0, max Ω u ] }, y Ω, u(y) = a and u(y) = 0 is finite Denote Ω a = {y Ω, u(y) > a}, Σ a = {y Ω, u(y) = a} There holds Σ a = 0 for all 0 a max Ω u (c) Map a ρ(a) [0, R] such that Ω a = B ρ(a), is decreasing, continuous, one-to-one and onto. (d)

26 Non-critical values: Y = [ 0, max Ω u ] \Z, E = {x Ω, x ρ(y )} Symmetrized ellipticity function Λ, defined for all x E: u(y) 1 dσ(y) Σ ρ 1 ( x ) Λ(x) = > 0 Λ(y) 1 u(y) 1 dσ(y) Σ ρ 1 ( x ) co-area formula = Λ(x) 1 dx = Ω min Ω Λ Λ maxω Λ in Ω Λ(y) 1 dy Ω Symmetrized function û(x) = ũ( x ), radially symmetric, vanishing on Ω and such that, for a Z and x = ρ(a), Nα N x N 1 Λ(x) ũ ( x ) = div( Λ û)(z)dz = B ρ(a) div(a u)(y)dy < 0 Ω a The function û is positive in Ω, decreasing in x, it is of class W 1, (Ω ) H 1 0 (Ω ), C 2 (E Ω ) and C 1 (E {0})

27 Denote a u (y) = a(y, u(y), u(y)) Symmetrized function â defined for all x E: ( max a + u (y) Λ 1 (y) ) Λ(x) if q = 2, y Σ ρ 1 ( x ) 2 q â(x) = a u + (y) 2 2 q Λ(y) q 2 q u(y) 1 2 dσ ρ 1 ( x ) Σ ρ 1 ( x ) u(y) 1 Λ(x) q 2 dσ ρ 1 ( x ) Σ ρ 1 ( x ) (second case: 1 q < 2) min a u + Ω ( â(x) ) max a u + Ω ( ) maxω Λ q 1 min Ω Λ

28 Denote f u (y) = f (y, u(y), u(y)) Symmetrized function f defined for all x E: f (x) = Σ ρ 1 ( x ) f u (y) u(y) 1 dσ ρ 1 ( x ) Σ ρ 1 ( x ) u(y) 1 dσ ρ 1 ( x ) min Ω f u f (x) max f u Ω and f = Ω f u Ω

29 First key inequality: pointwise comparison between u and û For all x Ω with x = ρ(a) and for all y Σ a, û(x) u(y) = a = ρ 1 ( x ), that is u (x) û(x) (e) (f) Improved inequality when Ω is not a ball û(x) (1 + η) u(y), that is (1 + η) u (x) û(x) where η > 0 depends on Ω, N and on some bounds of u C 1,α (Ω)

30 Second key inequality: partial differential inequality For all x E Ω, there is y Σ ρ 1 ( x ) (that is, u(y) = ρ 1 ( x )) s.t. div( Λ û)(x) â(x) û(x) q f (x) div(a u)(y) a u (y) u(y) q f u (y) Improved inequality when m b = min Ω b u > 0, b u = b(, u( ), u( )) div( Λ û)(x) â(x) û(x) q + δ û(x) f (x) div(a u)(y) a u (y) u(y) q + b u (y) u(y) f u (y) where δ > 0 depends on m b, Ω, N and some bounds on some norms of u and the coefficients Results of independent interest. Different from Schwarz symmetrization.

31 End of the proof (with q = 1) div( Λ û)(x) â(x) û(x) f (x) div(a u)(y) a u (y) u(y) f u (y) div(a u)(y) + H(y, u(y), u(y)) = 0 and û is a weak H0 1(Ω ) subsolution of div( Λ û) â e r û f 0 in Ω Let w H 1 0 (Ω ) be the unique solution of div( Λ w) + â e r w f = 0 in Ω Maximum principle = û w in Ω = u û w in Ω

32 Sequence (g k ) k N s.t. g k f in L (Ω ) weak-* and µ gk Solutions z k H 1 0 (Ω ) of = µ fu div( Λ z k ) + â e r z k g k = 0 in Ω One has: z k w in H 1 0 (Ω ) weak and L 2 (Ω ) strong Solution z H 1 0 (Ω ) of div( Λ z) + â e r z f u = 0 in Ω One has: z k z (in particular: use of Hardy-Littlewood inequality) Solution v H 1 0 (Ω ) of div( Λ v) â v f u Maximum principle = z v = 0 in Ω = u w z k z v = conclusion: u v General case: data and eigenfunctions are not smooth enough Smooth approximations, uniform estimates,...: many technicalities Improved inequalities when Ω is not a ball