Strongly nonlinear multiplicative inequalities involving nonlocal operators

Transcription

1 Strongly nonlinear multiplicative inequalities involving nonlocal operators Agnieszka Ka lamajska University of Warsaw Bȩdlewo, 3rd Conference on Nonlocal Operators and Partial Differential Equations, June, 2016

2 Outlines 1 Introduction 2 Inequalities without weight (Orlicz setting, together with Katarzyna Pietruska-Paluba) 3 Inequalities involving weight h( ) in dimension 1, with Lebesque measure, and applications (together with Jan Peszek) The special case Generalization to Orlicz spaces 4 n dimensional case (with Tomasz Choczewski) 5 Inequalities involving nonlocal operators (with Claudia Capogne and Alberto Fiorenza)

3 Introduction We are interested in inequality: f (x) p h(f (x))dx (a,b) ( p 1 ) p (a,b) and its generalizations. ( f (x)t h (f (x)) ) p h(f (x))dx, (1)

4 Assumptions: a < b +, p 2, f R, C0 f 0, 2,1 (a, b) R Wloc (a, b), sometimes we assume h : R R 0 is continuous (can be defined on R 0 provided f 0 ), T h ( ) is continuous, interpreted as the transformation of f : { H(λ) T h (λ) := h(λ) if h(λ) 0, 0 if h(λ) = 0, where H is primitive to h, Our motivations: Apply the new inequalities to singular PDEs.

5 Information about models where inequality applies Example models Thomas-Fermi model (1927, describes electric charge in isolated { neutral atom) y (t) = t 1 2 y(t) 3 2, t (0, ), y(0) = 0, lim t y(t) = 0. Emden-Fowler { problem (fluid dynamics): y + λq(x)y γ = 0, x (0, 1), γ > 0 y(0) = y(1) = 0, model of membrana and model of mikro-electro-mechanical system (MEMS), papers by Esposito and coauthors λf (x) u = in Ω R 2 (1 u) 2 0 u < 1 in Ω u = 0 on Ω models in cosmology, e.g. Makutuma model 1 u x 2 uq = 0, x R.

6 Unweighed simplest variant Theorem (Katarzyna Pietruska-Paluba and A.K, 2006) G( u )dx C G( u (2) u )dx, (2) where G is convex.

7 Inequalities involving weight, d=1 Theorem (Jan Peszek and A.K., 2011) (a,b) f (x) p h(f (x))dx C (a,b) under certain assumptions on h and f. ( f (x)t h (f (x)) ) p h(f (x))dx,

8 The special case The special case Theorem (Jan Peszek and A.K., 2011) Let 2 p <, θ R and f W 2,1 loc (R) be such that f has compact support. Assume additionally that at least one of the conditions is satisfied: 1 θ < 1 p, 2 θ > 1 p and f is nonnegative or (more generally) does not have isolated zeroes, 3 θ > 1 p and there exists ɛ such that for every r < R: (r,r) {x:0< f (x) <ɛ} ( ) f p f θ dx <.

9 The special case Then {x:f (x) 0} ( ) f p f θ dx ( ) p ( p 1 2 ff 1 θp {x:f (x) 0} f θ ) p dx. For θ = 1 2 and f 0 we retrieve Mazja s inequality know earlier.

10 Generalization to Orlicz spaces Generalization to Orlicz spaces We consider certain set of assumptions: (M) M : [0, ) [0, ) is (convex) differentiable N function, and M satisfies the condition: d M M(λ) λ M (λ) D M M(λ) λ for every λ > 0, (3) where D M d M 2. (h) h : (0, ) (0, ) is locally Lipschitz and H : (0, ) R is its locally absolutely continuous primitive and h H is well controlled by h 2 (precise formulation is omitted).

11 Generalization to Orlicz spaces Theorem (Jan Peszek and A.K, 2012) Assume that M satisfies (M), h : (0, ) (0, ) satisfies (h). Then any nonnegative f W 2,1 (R) such that f has compact support satisfies inequality M( f (x) h(f (x)))dx ( R ) C M f (x)t h (f (x)) h(f (x)) dx. R

12 Generalization to Orlicz spaces This inequality has been applied to second order capacitary estimates and isoperimetric inequalities.

13 Applications to the nonlinear eigenvalue problems Applications to the nonlinear ODEs I Consider the following O.D.E: { f (x) = g(x)τ(f (x)) a.e. in (a, b), f R (4) where a < b + and: τ : A R, A R is an interval, g L q (a, b), q [1, ], f W 2,1 loc ((a, b)), f (x) A, set R defines the boundary conditions (admitted to our inequalities).

14 Regularity We find function h( ) such that g(x) q f (x) q = = T τ(f (x)) h (f (x))f (x) 2q 2 h(f (x)), We apply: (a,b) ( 2q 1 ) 2q f (x) 2q h(f (x))dx (a,b) ( f (x)t h (f (x)) ) 2q h(f (x))dx.

15 Regularity We deduce that f 2q h(f ) C g q q. Let G = G τ be such transform of τ that (G(f )) 2q = f 2q h(f ) (G = h 1/(2q) ). Then G(f ) W 1,2q ((a, b)), so is λ-hölder continuous, where λ = 1 1 2q. we deduce the regularity and asymptotic behavior of solutions.

16 Asymptotic behavior Application (with Jan Peszek, generalization with Katarzyna Mazowiecka) Assumption: 1 q <, α q, κ = sign(α q ), 0 < b, g L q (0, b) and let f W 2,1 loc (0, b) and u 0 solves: f (x) = g(x)(f (x)) α a.e. on (0, b) and nonlinear boundary condition (mixed type): lim inf R b κ f (R) 2q 2 f (R)(f (R)) q(α+1)+1 lim sup κ f (r) 2q 2 f (r)(f (r)) q(α+1)+1 0. r 0

17 Asymptotic behavior Theorem (Jan Peszek, A.K., 2011) i) ii) b 0 b f (x) 2q f (x) q(α+1) dx C q g(x) q dx, sup { (f (x)) 1 α 2 (f (y)) 1 α 2 A q ( b 0 x y 1 1 2q ) 1 g(x) q 2q dx, 0 : x, y (0, b) iii) If α < 1 then lim r 0 f (r) =: f (0) exists and when f (0) = 0 ( b f (x) 1 α 2 A q x 1 1 2q 0 ) 1 g(x) q 2q dx. }

18 Asymptotic behavior Extensions were obtained with Katarzyna Mazowiecka.

19 Generalization to n-d (with Tomasz Choczewski) We obtained the analogue of multiplicative inequality having the form: f (x) p h(f (x))dx Ω ( ) p ( ) p p 1 f (x)th (f (x)) h(f (x))dx, (5) Ω and applications to the eigenvalue problems like: { f (x) = g(x)τ(f (x)) a.e. in Ω. f R (6)

20 we require information about second order vectorial Riesz transforms ( x j 1/2, papers by Tadeusz Iwaniec and coauthors) and best constants in Hardy inequalities. inequality applies to the model of elektrostatic micromechanical systems (MEMS), which is reduced to the following problem u = λf (x) (1 u) 2 w Ω u = 0 on Ω 0 < u < 1 in Ω where λ 0, f 0, u C 1 (Ω W 2,2 (Ω)), and Ω is open and bounded (papers by Esposito).

21 Weighted variants in 1-d (with Ignacy Lipka) C ( (a,b) (a,b) f (x) p h(f (x))ρ(x)dx ( f (x)t h (f (x)) ) p h(f (x))ρ(x)dx + (a,b) T h (f (x)) p h(f (x)) ρ (x) dx )

22 Inequalities involving nonlocal operators (with Claudia Capogne and Alberto Fiorenza) We obtain inequalities: M( f (x) h(f (x)))dx R ) A M (B p Mf (x)t h,p (f, x) h(f (x)) dx, R T h,p (f, x) := ( x Φp h(f (y))f ) (y) χ {f (y) 0} dy, f (x) 0 h(f (x)) p 1 0, f (x) = 0 and φ p (s) = s p 2 s. For p = 2 we have T h,p = T h, Mh is Hardy - Littlewood maximal function. The approach requires Simmonnenko and Boyed indeces.

23 Interpretation of the nonlinear transform For h 1 we have T h 1,p (f, x) = 1 p, p u = div ( u p 2 u ). In general T h,p (f, x) = 1 p (H(f )), Φ p (s) = s p 2 s. Φ p (h(f )) In particular T h,p (f, x) is nonlocal.

24 Example inequality C ( C R R R f (x) q (f (x)) α dx ( ) Mf q ( x p (x) f (y) p 1 (f (y)) α(p 1) dy ) q p f (x) αq p ( f (y) (f (y)) α) p 1 dy ) q p R (Mf (y)(f (y)) α) q p dy.

25 The tools Boyd indices and Simmonnenko indices: if the inequality holds with comvex function M then it holds with M 1 M. First step: M : [0, ) [0, ) is (convex) differentiable N function and M satisfies the condition: d M M(λ) λ where D M d M 2. M (λ) D M M(λ) λ for every λ > 0, (7) Second step: we substitute M with M 1 M but smaller lower Simmonnenko index d M. It leads to inequality with Boyd index i M := inf M1 Md M1 < 2.

26 C. Capogne, A. Fiorenza, A. Ka lamajska, Strongly nonlinear Gagliardo-Nirenberg inequality in Orlicz spaces and Boyd indices, preprint A. Ka lamajska, K. Mazowiecka, Some regularity results to the generalized Emden-Fowler equation under very weak assumptions, Math. Methods Appl. Sci. 38 (2015). A. Ka lamajska, J. Peszek, On some nonlinear extensions of the Gagliardo-Nirenberg inequality with applications to nonlinear eigenvalue problems, Asymptotic Analysis, Volume 77, Number 3-4 (2012). A. Ka lamajska, J. Peszek, On certain generalizations of the Gagliardo-Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities, Journal of Fixed Point Theory and Applications, Volume 13, Issue 1 (2013). A. Ka lamajska, K. Pietruska-Pa luba, Gagliardo-Nirenberg inequalities in Orlicz spaces, Indiana University Mathematics Journal 55(6) (2006).

27 Thank you!