International Workshop on Topological and Variational Methods for ODEs Dedicated to Massimo Furi Professor Emeritus at the University of Florence Firenze, Dipartimento di Matematica e Informatica“U. Dini” June 3 – 4, 2014
! UNIVERSITÀ DEGLI STUDI DI TRIESTE Pierpaolo Omari Universit` a degli Studi di Trieste Dipartimento di Matematica e Geoscienze E-mail: omari@units.it An asymmetric Poincar´ e Inequality and Applications Joint work with Franco Obersnel and Sabrina Rivetti (UNITS) 2
Structure of this talk This talk is divided into two parts: 1. we introduce an asymmetric version of the Poincar´ e inequality in the space of bounded variation functions 2. based on this result, we study the existence of bounded variation solutions of a class of capillarity problems with possibly asymmetric perturbations. 3
POINCAR´ E INEQUALITIES The classical Poincar´ e-Wirtinger inequality Let Ω be a bounded domain in R N , with a Lipschitz boundary ∂ Ω . The classical Poincar´ e-Wirtinger inequality in BV (Ω) asserts that there exists a constant c > 0 such that every u ∈ BV (Ω) , with Ω u + dx � � � � Ω u − dx = 1 , if u � = 0 u dx = 0 i.e. r = , � Ω satisfies � � | u | dx ≤ | Du | . c Ω Ω Recall that u ∈ BV (Ω) if u ∈ L 1 (Ω) and its distributional gradient is a vector valued Radon measure with finite total variation � � � � v div w dx : w ∈ C 1 0 (Ω , R N ) and � w � L ∞ (Ω) ≤ 1 | Dv | := sup . Ω Ω 4
The Poincar´ e constant The largest constant c = c (Ω) for which the inequality � � | u | dx ≤ | Du | c Ω Ω holds is called the Poincar´ e constant and is variationally characterized by � � � � � | Dv | : v ∈ BV (Ω) , | v | dx = 1 c = inf v dx = 0 , . Ω Ω Ω Clearly, all minimizers, if any, yield the equality in the Poincar´ e inequality. 5
Why BV (Ω) instead of W 1 , 1 (Ω) ? Elementary examples show that � � � � � |∇ v | dx : v ∈ W 1 , 1 (Ω) , | v | dx = 1 inf v dx = 0 , Ω Ω Ω is not attained in W 1 , 1 (Ω) ; whereas, we have � � � � � | Dv | : v ∈ BV (Ω) , v dx = 0 , | v | dx = 1 inf Ω Ω Ω � � � � � = min | Dv | : v ∈ BV (Ω) , v dx = 0 , | v | dx = 1 Ω Ω Ω � � � � � |∇ v | dx : v ∈ W 1 , 1 (Ω) , | v | dx = 1 = inf v dx = 0 , . Ω Ω Ω 6
An asymmetric variant of the Poincar´ e inequality Our aim is to discuss the validity of an asymmetric counterpart of the Poincar´ e inequality, where u + and u − weigh differently, i.e. Ω u + dx � Ω u − dx � = 1 . r = � Namely, we show that for each r > 0 there exist constants µ > 0 and ν > 0 , with ν/µ = r , such that every u ∈ BV (Ω) , with Ω u + dx � � � u + dx − ν u − dx = 0 � � µ i.e. Ω u − dx = r , � Ω Ω satisfies � � � u + dx + ν u − dx ≤ | Du | . µ Ω Ω Ω 7
Variational characterization For each r > 0 we define µ and ν through the variational formulas � � � � v − dx = 0 , v + dx − r | Dv | : v ∈ BV (Ω) , µ = µ ( r, Ω) = inf Ω Ω Ω � � � v − dx = 1 v + dx + r Ω Ω and � � � � | Dv | : v ∈ BV (Ω) , r − 1 v − dx = 0 , v + dx − ν = ν ( r, Ω) = inf Ω Ω Ω � � � r − 1 v − dx = 1 v + dx + . Ω Ω Needless to say that in this way we find the best constants for which the inequality holds . 8
Minimum properties For each r > 0 , we have � � � � v + dx = 1 v − dx = 1 � | Dv | : v ∈ BV (Ω) , µ ( r ) = min 2 , 2 r Ω Ω Ω and � � � � v + dx = r v − dx = 1 � | Dv | : v ∈ BV (Ω) , ν ( r ) = min 2 , . 2 Ω Ω Ω Moreover, µ ( r ) , ν ( r ) > 0 and ν ( r ) = rµ ( r ) . 9
The curve C and its properties. We study the functions r �→ µ ( r ) r �→ ν ( r ) , and and the plane curve C = { ( µ ( r ) , ν ( r )) : r ∈ R + 0 } . Of course, by construction of C , the following holds: if ( µ, ν ) ∈ C , then every v ∈ BV (Ω) , with � � v + dx − ν v − dx = 0 , µ Ω Ω satisfies � � � v + dx + ν v − dx ≤ µ | Dv | . Ω Ω Ω 10
Symmetry For each r > 0 , we have µ ( r − 1 ) = ν ( r ) ; hence C is symmetric with respect to the diagonal. Continuity The function r �→ µ ( r ) (and hence r �→ ν ( r ) ) is both lower and upper semicontinuous (and thus continuous). Monotonicity The function r �→ µ ( r ) is strictly decreasing (and the function r �→ ν ( r ) is strictly increasing). Recall: � � � � v + dx = 1 v − dx = 1 � | Dv | : v ∈ BV (Ω) , µ ( r ) = min 2 , . 2 r Ω Ω Ω 11
Asymptotic behaviour as r → 0 + We have r → 0 + µ ( r ) = + ∞ r → + ∞ ν ( r ) = + ∞ ) . lim (and hence lim Asymptotic behaviour as r → + ∞ in dimension N ≥ 2 Assume N ≥ 2 . Then, we have r → + ∞ µ ( r ) = 0 lim (and hence r → 0 + ν ( r ) = 0) . lim Recall: � � � � v + dx = 1 v − dx = 1 � | Dv | : v ∈ BV (Ω) , µ ( r ) = min 2 , . 2 r Ω Ω Ω 12
The curve C in dimension N ≥ 2 13
Asymptotic behaviour as r → + ∞ in dimension N = 1 Assume N = 1 and let Ω = ]0 , T [ . Then, we have r → + ∞ µ ( r ) > 0 lim (and hence r → 0 + ν ( r ) > 0) . lim 14
The curve C in dimension N = 1 15
Asymptotic behaviour as r → + ∞ in dimension N = 1 Assume N = 1 and let Ω = ]0 , T [ . Then, we have r → + ∞ µ ( r ) > 0 lim (and hence r → 0 + ν ( r ) > 0) . lim This follows from � � � � v + dx = 1 v − dx = 1 � | Dv | : v ∈ BV (]0 , T [) , µ ( r ) = min 2 , . 2 r ]0 ,T [ ]0 ,T [ ]0 ,T [ and � v − ess inf v ≤ | Dv | , ∀ v ∈ BV (0 , T ) . ess sup ]0 ,T [ ]0 ,T [ ]0 ,T [ However we can deduce this fact from a more precise description of C in case N = 1 , which also provides the explicit value of the limit. 16
Explicit description of C in dimension N = 1 Assume N = 1 and let Ω = ]0 , T [ . Then, we have √ 0 : 1 √ µ + 1 � � ( µ, ν ) ∈ R + 0 × R + C = √ ν = 2 T . In particular, � � T , 2 2 ∈ C , T with 2 T the second eigenvalue c 2 of the Neumann 1 -Laplacian in ]0 , T [ as defined in [Chang, 2009], and 1 1 C is asymptotic to the lines µ = 2 T and ν = 2 T . 17
Moreover, for any given ( µ, ν ) ∈ C , a function u ∈ BV (0 , T ) satisfies � T � T � T � T � u + dx − ν u − dx = 0 u + dx + ν u − dx = | Du | µ and µ 0 0 0 0 ]0 ,T [ if and only if u is a positive multiple either of √ µ + √ ν √ ν 1 1 √ ν √ µ + √ νT, if 0 < x < T 2 µ ϕ ( x ) = √ µ + √ ν √ ν − 1 1 √ µ + √ νT ≤ x < T. if √ µ T 2 ν 18
The function ϕ 19
Moreover, for any given ( µ, ν ) ∈ C , a function u ∈ BV (0 , T ) satisfies � T � T � T � T � u + dx − ν u − dx = 0 u + dx + ν u − dx = µ and µ | Du | ]0 ,T [ 0 0 0 0 if and only if u is a positive multiple either of √ µ + √ ν √ ν 1 1 √ ν √ µ + √ νT, if 0 < x < T 2 µ ϕ ( x ) = √ µ + √ ν √ ν − 1 1 √ µ + √ νT ≤ x < T. if √ µ T 2 ν or of ϕ ( T − x ) . 20
Sketch of proof. The proof is based on a rearrangement technique: 1. we prove the validity of the asymmetric Poincar´ e inequality for decreasing functions √ whenever µ, ν ∈ R + √ µ + 1 1 0 satisfy √ ν ≥ 2 T 2. by exploiting some properties of decreasing rearrangements (area invariance, Polya-Szeg¨ o inequality), we extend the validity of the asymmetric Poincar´ e inequality to bounded variation functions 3. by using again the properties of decreasing rearrangements and the coarea formula, we characterize the functions yielding equality in the asymmetric Poincar´ e inequality √ 4. we show that if ρ, σ ∈ R + 1 1 2 T , then ρ = µ ( r ) , σ = ν ( r ) with r = σ 0 satisfy √ ρ + √ σ = ρ , i.e. ( ρ, σ ) ∈ C , and viceversa. 21
SOLVABILITY OF CAPILLARITY PROBLEMS We turn to the study of the capillarity-type problem � � 1 + |∇ u | 2 � − div ∇ u/ = f ( x, u ) in Ω , � 1 + |∇ u | 2 = κ ( x ) −∇ u · n/ on ∂ Ω . We are going to present some statements concerning non-existence, existence and multiplicity of solutions in the space of bounded variation functions. Our main aim is to study the case where the no convexity assumption is imposed on the associated action functional and solutions are not necessarily minimizers: this will be achieved by using the asymmetric variant of the Poincar´ e inequality we previously established and some tools of non-smooth critical point theory. Here for simplicity we will restrict ourselves to the discussion of the case of homogeneous conormal boundary conditions, i.e. κ = 0 . 22
Hereafter we assume that f : Ω × R → R satisfies the Carath´ (CAR) eodory conditions and there exist constants a > 0 and q ∈ ]1 , 1 ∗ [ and a function b ∈ L p (Ω) , (SGC) with p > N , such that | f ( x, s ) | ≤ a | s | q − 1 + b ( x ) for a.e. x ∈ Ω and every s ∈ R . � 1 ∗ = 1 ∗ = ∞ � N N ≥ 2 : N − 1 ; N = 1 : We set � s F ( x, s ) = f ( x, ξ ) dξ. 0 23
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