Viscosity solutions of elliptic equations in R n : existence and uniqueness results

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1 Viscosity solutions of elliptic equations in R n : existence and uniqueness results Department of Mathematics, ITALY June 13, 2012 GNAMPA School DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS Serapo (Latina), June 11-15, 2012

2 References Results presented in this school are extracted from: G. Galise and A. Vitolo, Viscosity Solutions of Uniformly Elliptic Equations without Boundary and Growth Conditions at Infinity, International Journal of Differential Equations (2011).

3 Some previous results H. Brezis 1 u u s 1 u = f s > 1, f L 1 loc(r n )! solution (distributional sense) M. J. Esteban, P. Felmer, A. Quaas 2 F (D 2 u) u s 1 u = f s > 1, f L n loc(r n )! solution (L n -viscosity solution) 1 H.Brezis, Semilinear equations in R n without conditions at infinity, Appl. Math. Optim. 12 (1984), M.J.Esteban., P.L.Felmer and A.Quaas, Superlinear elliptic equations for fully nonlinear operators without growth restrictions for the data, Proc. Edinb. Math. Soc. 53 (2010),

4 Structure conditions (SC) Structure conditions Uniform estimates Existence F (x, u, Du, D 2 u) = f(x) in R n Assumptions on F : P λ,λ P + λ,λ (Y X) γ η ξ F (x, u, η, Y ) F (x, u, ξ, X) (Y X) + γ η ξ F (x, u, ξ, X) F (x, v, ξ, X) δ(u v) s if v < u, s > 1 F (x, 0, 0, 0) = 0 Example F (x, u, Du, D 2 u) = P + λ,λ (D2 u) + γ Du u s 1 u

5 Structure conditions (SC) Structure conditions Uniform estimates Existence Assumption on f: F (x, u, Du, D 2 u) = f(x) in R n f L p loc (Rn ) with p > p 0 = p 0(n, Λ/λ) (n/2, n). p 0 is the exponent such that the generalized maximum principle (GMP) holds true: GMP If f L p (Ω) with p > p 0 and u W 2,p loc (Ω) C(Ω) is an Lp -strong solution of the maximal equation then P + λ,λ(d 2 u) + γ Du f, max u max u + n Ω Ω Cd2 p f L p (Ω) (1) with d = diam(ω) and C a positive constant depending on n, λ, Λ, p, γd.

6 Structure conditions Uniform estimates Existence Lemma1 Let Ω be a domain of R n such that Ω R := Ω B R. Suppose that F satisfy structure conditions (SC) a.e. x Ω R. If u C(Ω R ) is an L p -viscosity solution (p > p 0 ) of the equation F (x, u, Du, D 2 u) f(x) with f L p (Ω R ), then for each r (0, R) we have sup Ω r u u + Ω + C 0(1 + R) µ/2 R µ (R 2 r 2 ) µ + C f Lp (Ω R ) (2) with µ = 2/(s 1), C 0 = C 0 (n, Λ, γ, s, δ) and C = C(n, p, λ, Λ, γr) are positive constants. Here u + Ω = sup u + if B R Ω B R Ω 0 if B R Ω.

7 Structure conditions Uniform estimates Existence Sketch of the proof: Osserman s barrier function Φ(x) = µ = 2/(s 1), C s 1 R C R R µ (R 2 x 2 ) µ, x < R = 2µδ 1 (Λ(n + 2(1 + µ)) + γr) (SC) F (x, Φ(x), DΦ(x), D 2 Φ(x)) 0 a.e. in Ω R w = u Φ is an L p -viscosity solution of P + λ,λ (D2 w) + γ Dw f(x) in Ω R {u > Φ} (GMP) conclusion.

8 Structure conditions Uniform estimates Existence Lemma2 Let Ω R, F and f as in Lemma1. If u C(Ω R ) is an L p -viscosity solution (p > p 0 ) of the equation then for each r (0, R) we have F (x, u, Du, D 2 u) = f(x), sup Ω r u u Ω + C 0(1 + R) µ/2 R µ (R 2 r 2 ) µ + C f Lp (Ω R ) (3) with C 0, C and u Ω = max(u + Ω, u Ω ) as defined in Lemma1.

9 Structure conditions Uniform estimates Existence Assumption: R > 0 ω R : R + R + such that ω R (t) 0 as t 0 + and F (x, v, ξ, X) F (x, u, ξ, X) ω R ( v u ) (4) a.e. in x for u + v + ξ + X R. (SC) =(SC)+(4) Theorem Let F : R n R R n S n R be measurable in x and satisfy the structure condition (SC) a.e. x R n for all (u, ξ, X) R R n S n. If f L p loc (Rn ), then equation F (x, u, Du, D 2 u) = f(x) has an L p -viscosity solution in R n for any p > p 0.

10 Structure conditions Uniform estimates Existence Sketch of the proof: f k C (R n ) such that lim k f k f L p (Ω) = 0 (4) solvability in the ball B 2 k of (DP) F = f k + continuous boundary condition Uniform estimates for h > k (SC) +C α - estimates sup u h C 0 + C f Lp (B 2 k+1 ) B 2 k u h Cα (B 2 k ) C 1 (1 + f Lp (B 2 k+1 )) Diagonal argument u hk u C(R n ) uniformly on every bounded domain Stability results conclusion.

11 Maximum Principle Uniqueness Maximum Principle Let δ > 0, s > 1 and Ω be a domain of R n. Suppose for a.e. x Ω that F (x, u, ξ, X) P + λ,λ (X) + γ ξ δ u s 1 u for all (u, ξ, X) R R n S n and u C(Ω) is an L p -viscosity solution (p > p 0 ) of the equation F (x, u, Du, D 2 u) 0 in Ω. If Ω = R n, then u 0 in R n. If Ω R n and u 0 on Ω, then u 0 in Ω.

12 Maximum Principle Uniqueness Mimum Principle Let δ > 0, s > 1 and Ω be a domain of R n. Suppose for a.e. x Ω F (x, v, ξ, X) P λ,λ (X) γ ξ δ v s 1 v for all (v, ξ, X) R R n S n and v C(Ω) an L p -viscosity solution (p > p 0 ) of the equation F (x, v, Dv, D 2 v) 0 in Ω. If Ω = R n, then v 0 in R n. If Ω R n and v 0 on Ω, then v 0 in Ω.

13 Maximum Principle Uniqueness F C(R n R R n S n ), f C(R n ) Theorem If F is indipendent of x and satisfies (SC) then the equation F (u, Du, D 2 u) = f in R n has a unique C-viscosity solution. Sketch of the proof: u, v solution, Ω = {u > v}. Jensen s approximations P + λ,λ (D2 (u v)) + γ D(u v) δ(u v) s 0 in Ω Maximum Principle u v...

14 Maximum Principle Uniqueness F C(R n R R n S n ), f C(R n ) Theorem Suppose that F satisies (SC) and that for all R > 0 there exist a constant K R > 0 and a function ω R : R + R + such that lim t 0 + ω R(t) = 0 and F (y, u, ξ, X) F (x, u, ξ, X) K R X y x + ω R ((1 + ξ ) y x ) (A2.1) as x, y R n, u ( R, R) and (ξ, X) R n S n. If p > p 0 and f M p := sup x R n f L p (B 1(x)) < +, (A2.2) then equation F (x, u, Du, D 2 u) = f has a unique C-viscosity solution. Sketch of the proof: u, v solutions, (A2.1)+(A2.2) u v satisfies a maximal equation...

15 Maximum Principle Uniqueness x F (x,,, ) measurable, f L p loc (Rn ), p > p 0 We suppose that for every R > 0 there exists c R > 0 such that P λ,λ (Y X) γ η ξ c R v u F (x, v, η, Y ) F (x, u, ξ, X) P + λ,λ (Y X) + γ η ξ + c R v u (5) for x R n and any R > 0, u, v ( R, R), ξ, η R n, X, Y S n (SC) =(SC)+(5) F (x, 0, 0, X) F (x 0, 0, 0, X) β F (x, x 0 ) := sup. X S n X X 0

16 Maximum Principle Uniqueness Theorem Suppose: (SC) holds true F (,,, X) convex sup r (0,r 0) ( 1/n β F (x, y) dy) n θ B r(x) for every x R n, with θ = θ(n, p, λ, Λ, r 0 ). Then the equation F (x, u, Du, D 2 u) = f(x) has a unique L p -strong solution u W 2,p loc (Rn ). Sketch of the proof: u, v solutions are L p -strong solution and by using (SC) we get a maximal equation for u v. We conclude from maximum principle.