Elliptic PDE with natural/critical growth in the gradient

Transcription

1 Elliptic PDE with natural/critical growth in the gradient September 15, 2015

2 Given an elliptic operator Lu = a ij (x) ij u + b i (x) i u + c(x)u, F(D 2 u, Du, u, x) Lu = div(a(x) u) + b i (x) i u + c(x)u, div(a(x, u, u)) consider Lu = g(x, u, u) Here the nonlinearity g grows in u like u 2.

3 Why natural or critical? Natural, since invariant w.r. to diffeomorphic changes of function u ψ(u), v(x) = ψ(u(x)).

4 Why natural or critical? Natural, since invariant w.r. to diffeomorphic changes of function u ψ(u), v(x) = ψ(u(x)). Since L in general form, also invariant w.r. to diffeomorphic changes of variable x Ψ(x), v(x) = u(ψ 1 (x))

5 Why natural or critical? Natural, since invariant w.r. to diffeomorphic changes of function u ψ(u), v(x) = ψ(u(x)). Since L in general form, also invariant w.r. to diffeomorphic changes of variable x Ψ(x), v(x) = u(ψ 1 (x)) Critical, since gradient term has same scaling as hessian term u(x) u t (x) = u(tx), t > 0.

6 L 0 u = c(x)u + <M(x) u, u > + h(x) in Ω u = g(x) on Ω L 0 u = a ij (x) ij u + b i (x) i u M is a bounded matrix, and the other coefficients have the necessary regularity to ensure satisfactory theory for the linear problem (M = 0).

7 L 0 u = c(x)u + <M(x) u, u > + h(x) in Ω u = g(x) on Ω L 0 u = a ij (x) ij u + b i (x) i u M is a bounded matrix, and the other coefficients have the necessary regularity to ensure satisfactory theory for the linear problem (M = 0). The size and sign of c(x) matter, both for the solvability and for the uniqueness of solutions.

8 Some history KAZDAN KRAMER 1975, various partial results BOCCARDO MURAT PUEL succeeded to obtain a full solvability result for the general class of divergence form operators, in the case when c(x) c 0 < 0 FERONE MURAT 2000 solvability for the case c 0 and observation that only possible if Mh small. BARLES-MURAT, B.-PORRETTA uniqueness. Related results by dall Aglio, Giachetti and Puel; Maderna, Pagani and Salsa; Grenon, Murat and Porretta; Abdellaoui, dall Aglio and Peral; Abdel Hamid and Bidaut-Véron; Boccardo, Gallouet, Murat...

9 B.S. ARMA Extension of the above to general class of (even fully nonlinear) NON-DIVERGENCE form equations, through maximum-principle type arguments. 2. And how about the case c > 0? solvability still holds for small c > 0, but in general not the uniqueness!

10 L 0 u = c(x)u + <M(x) u, u > + h(x) in Ω u = g(x) on Ω Theorem (ARMA 2010) 1 This problem has a unique solution if c(x) c 0 < 0 2 There exists δ 0 > 0 depending on λ, Λ, b n, diam(ω), such that if M 0 h + c + + M 0 c + (max Ω g) n δ 0 then there exists a solution. It is unique if c + = 0. 3 If c c 0 > 0, L 0 =, M = µ 0 I, g = h = 0, then there are at least two solutions.

11 { L0 u = c(x)u + <M(x) u, u > + h(x) in Ω u = 0 on Ω How about a more general non-uniqueness result? One might hope that whenever M 0 h + c + is small and we have a hypothesis which prevents M or c to be zero, say M 0 or M 0, and c 0, then we have at least two solutions.

12 Results for the model equation with L 0 =, M = µ(x)i, u = c(x)u + µ(x) u 2 + h(x)

13 Results for the model equation with L 0 =, M = µ(x)i, u = c(x)u + µ(x) u 2 + h(x) B.S. 2010: c(x) = c 0 > 0, µ(x) = µ 0 > 0, h = 0 exponential change and a Gidas-Spruck blow-up.

14 Results for the model equation with L 0 =, M = µ(x)i, u = c(x)u + µ(x) u 2 + h(x) B.S. 2010: c(x) = c 0 > 0, µ(x) = µ 0 > 0, h = 0 exponential change and a Gidas-Spruck blow-up. L. JEANJEAN B.S. 2014: µ(x) = µ 0 > 0 exponential change and a mountain-pass for a slow-growth nonlinearity.

15 Results for the model equation with L 0 =, M = µ(x)i, u = c(x)u + µ(x) u 2 + h(x) B.S. 2010: c(x) = c 0 > 0, µ(x) = µ 0 > 0, h = 0 exponential change and a Gidas-Spruck blow-up. L. JEANJEAN B.S. 2014: µ(x) = µ 0 > 0 exponential change and a mountain-pass for a slow-growth nonlinearity. D. ARCOYA, C. DE COSTER, L. JEANJEAN, K. TANAKA 2015; L. JEANJEAN, C. DE COSTER: 0 < µ 1 µ(x) µ 2 degree theory, Rabinowitz type bifurcation point of view, dependence of solutions on a parameter (c λc, λ R).

16 Results for the model equation with L 0 =, M = µ(x)i, u = c(x)u + µ(x) u 2 + h(x) B.S. 2010: c(x) = c 0 > 0, µ(x) = µ 0 > 0, h = 0 exponential change and a Gidas-Spruck blow-up. L. JEANJEAN B.S. 2014: µ(x) = µ 0 > 0 exponential change and a mountain-pass for a slow-growth nonlinearity. D. ARCOYA, C. DE COSTER, L. JEANJEAN, K. TANAKA 2015; L. JEANJEAN, C. DE COSTER: 0 < µ 1 µ(x) µ 2 degree theory, Rabinowitz type bifurcation point of view, dependence of solutions on a parameter (c λc, λ R). The decisive uniform a priori bound depends heavily on the variational structure of the Laplacian.

17 Can we relax the strict lower bound on µ(x)? { u = c(x)u + µ(x) u 2 + h(x) in Ω u = 0 on Ω

18 Can we relax the strict lower bound on µ(x)? { u = c(x)u + µ(x) u 2 + h(x) in Ω u = 0 on Ω The pictures suggest it might be hard to give a full answer.

19 Can we relax the strict lower bound on µ(x)? { u = c(x)u + µ(x) u 2 + h(x) in Ω u = 0 on Ω The pictures suggest it might be hard to give a full answer. SOUPLET: It is necessary that the intersection of supp(µ) and supp(c) contains a ball. If in addition n = 2; or n = 3 and power growth for µ and c close to Ω; or n = 3, 4, and µ(x) µ 1 > 0 on supp(c). then an uniform a priori bound holds. Uses the variational structure of the Laplacian and elliptic theory in weighted L p δ -spaces.

20 We report on another method for proving a priori bounds, which uses only fundamental results valid for all elliptic operators, namely quantitative strong maximum principle; weak Harnack inequality; local maximum principle.

21 We report on another method for proving a priori bounds, which uses only fundamental results valid for all elliptic operators, namely quantitative strong maximum principle; weak Harnack inequality; local maximum principle. For the problem at hand, this method permits to us to Prove a priori bounds for any uniformly elliptic operator L 0 ;

22 We report on another method for proving a priori bounds, which uses only fundamental results valid for all elliptic operators, namely quantitative strong maximum principle; weak Harnack inequality; local maximum principle. For the problem at hand, this method permits to us to Prove a priori bounds for any uniformly elliptic operator L 0 ; and do this, assuming that µ(x) µ 1 > 0 on supp(c), in any dimension.

23 L 0 u = c(x)u + <M(x) u, u > + h(x) in Ω u = 0 on Ω Theorem Let M(x) µ 1 I on supp(c), µ 1 > 0. If the problem with c = 0 has a solution, then it has at least two solutions for c 0 small, and the same bifurcation diagrams are valid.

24 Some observations If c < λ 1 then M 0, h 0 and u 0 is generic. u = c(x)u + µ(x) u 2 + h(x) ψ = c(x)ψ + h(x) Then v = u ψ is positive (MP) and satisfies v+ < ψ, v >= c(x)v + µ(x) v 2 + µ(x) ψ 2 modified elliptic operator.

25 Some observations If lack of a priori bound, then lack of a priori bound on supp(c). In O open in {c 0}, then in O both u sup O u and u 0 inf Ω u 0 are solutions of u = µ(x) u 2 + h(x) and they compare on the boundary, so by MP sup O u sup u + 2 u 0 O

26 Some observations So if lack of a priori bound, then for some x 0 Ω and some (half-)ball B around x 0 we have an unbounded in B sequence of solutions of and in B L 0 u = c(x)u + µ(x) u 2 + h(x) 0 < µ 1 µ(x) µ 2, c 0.

27 Some observations So if lack of a priori bound, then for some x 0 Ω and some (half-)ball B around x 0 we have an unbounded in B sequence of solutions of and in B L 0 u = c(x)u + µ(x) u 2 + h(x) 0 < µ 1 µ(x) µ 2, c 0. standard exponential change v i = µ 1 i (e µ i u 1) In B, for large u L 1 v 1 f 1 (x, v 1 ) c 0 c(x) v 1 log(v 1 ) L 2 v 2 f 2 (x, v 2 ) C 0 c(x) v 2 log(v 2 ) v 2 v A 1

28 In B, for large u L 1 v 1 c 0 c(x) v 1 log(v 1 ) + h 1 L 2 v 2 C 0 c(x) v 2 log(v 1 ) + h 2 v 2 v A 1 Theorem There exists a constant C depending on λ, Λ, b p, c, and a lower bound on B c, such that u C.

29 1. (QSMP) Lu 0, u > 0 in Ω = for each compact K Ω ( 1/ε0 inf u c ( Lu) 0) ε. K K 2. (WHI) Lu f, u > 0 in Ω = for each K K Ω ( ) 1/ε C(inf u + f n). K K u ε 3. (LMP) Lu f in Ω = for each K K Ω and p > 0 ( ( ) ) 1/p sup u C + f n K K u p

30 1. (QSMP) Lu 0, u > 0 in Ω = for each compact K Ω ( 1/ε0 inf u c ( Lu) 0) ε. K K 2. (WHI) Lu f, u > 0 in Ω = for each K K Ω ( ) 1/ε C(inf u + f n). K K u ε 3. (LMP) Lu f in Ω = for each K K Ω and p > 0 ( ( ) ) 1/p sup u C + f n K K u p Need boundary versions of these!!

31 1. (BQSMP) Lu 0, u > 0 in B + 2, u = 0 on {x n = 0} = ( ) u 1/ε0 inf c ( Lu) ε 0. B + x 1 n B 1 2. (BWHI) Lu f, u > 0 in B + 2, u = 0 on {x n = 0} = ( ( ) ) u ε 1/ε u C( inf + f n ). x n B + x 3/2 n B (BLMP) Lu f in B + 2, u = 0 on {x n = 0}, c L n+, = ( ) 1/p u sup C u p + f n B + x n B + 1 3/2

32 1. (BQSMP) Lu 0, u > 0 in B + 2, u = 0 on {x n = 0} = ( ) u 1/ε0 inf c ( Lu) ε 0. B + x 1 n B 1 2. (BWHI) Lu f, u > 0 in B + 2, u = 0 on {x n = 0} = ( ( ) ) u ε 1/ε u C( inf + f n ). x n B + x 3/2 n B (BLMP) Lu f in B + 2, u = 0 on {x n = 0}, c L n+, = ( ) 1/p u sup C u p + f n B + x n B + 1 3/2 Krylov original approach of writing a degenerate equation for u/x n.

33 L 1 v 1 c 0 c(x) v 1 log(v 1 ) + h 1 L 2 v 2 C 0 c(x) v 2 log(v 1 ) + h 2 v 2 v A 1 1 st ineq. + BQSMP = inf B + 1 v 1 v 1 c 0 inf x n B + x 1 n log ( inf B + 1 v 1 x n ) ( B + 1 c ε x 1+ε n ) So inf B + 1 v 1 C. Then 1 st ineq. + BWHI = x n ( ( B + 1 (v 1 ) ε ) 1/ε C B + 1 ( v1 x n ) ε ) 1/ε C. 2 nd ineq. + LMP + (log(v 1 )) n+1 Cv1 ε = ( ) A/ε ( ) A/ε sup v 2 (v 2 ) ε/a C (v 1 ) ε C. B + B B + 1