A priori estimates of minimizers of an integral functional ... · A priori estimates of minimizers of an integral functional extending the variational Strong Maximum Principle Vladimir

A priori estimates of minimizers of an integral functionalextending the variational Strong Maximum Principle

Vladimir V. Goncharov, Telma J. Santos

CIMA, Departamento de Matematica, Universidade de Evora

27th IFIP TC7 Conference 2013 on System Modelling and OptimizationCote d’Azur, France

June, 29 - July, 3, 2015

Vladimir V. Goncharov, Telma J. Santos (CIMA, Departamento de Matematica, Universidade de Evora)A priori estimates of minimizers27th IFIP TC7 Conference 2013 on System Modelling and Optimization Cote d’Azur, France June, 29 - July, 3, 2015 1

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Outline

Inf (Sup)-convolution with a gauge function

Variational properties of the Inf (Sup)-convolution

Strong Maximum Principle. Traditional settings

Variational generalizations. From harmonic functions to convolutions

The problem with a linear perturbation

A priori estimates of minimizers near nonextremum points

”Cross-gap” extension of the Variational Strong Maximum Principle

Back to the limit case. Example

References


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This is the continuation of my talk given 2 years ago at the IFIPConference in Klagenfurt dedicated to varios properties of Inf-Convolutions

So, let me start with basic notations. Given a nonempty closed setC ⊂ Rn and a > 0 the Inf-convolution of a function θ : C → R with agauge function ρF 0(·) is defined as

u+θ,F (x) := inf

y∈Cθ (y) + a ρF 0 (x − y) (1)

Here F is a compact convex subset of Rn with 0 ∈ intF (a gauge), F 0

means the polar set for F and

ρF 0(ξ) := infλ > 0 : ξ ∈ λF 0

Similarly, the Sup-convolution is

u−θ,F (x) := supy∈Cθ (y)− a ρF 0 (y − x) (2)


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Under some supplementary condition on θ or/and C the Inf(Sup)-convolution above can be seen as a viscosity solution of aHamilton-Jacobi-Bellman equation, or as the Minimal Time necessary toachieve C by trajectories of some associated differential inclusion

We are interested instead in the ”minimal” (respectively, ”maximal”)property of this function among all the minimizers of some ”elliptic”Variational Problem


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Namely, given an open bounded connected region Ω ⊂ Rn containing Cand a lower semicontinuous convex function f : R+ → R+ ∪ +∞ withf (0) = 0 let us consider the integral functional

u(·) 7→∫Ω

f (ρF (∇u(x))) dx (3)

In order to formulate the first (comparison or extension) result we assumealso Ω to be convex and the boundary ∂F to be smooth

Denote bya := sup f −1(0)

Consider a closed (not necessarily convex) subset C ⊂ Ω and a functionθ(·) defined on C with a kind of slope condition:

θ(x)− θ(y) ≤ aρF 0(x − y), x , y ∈ C (4)


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Then the following result holds

Theorem A

The Inf-convolution u+θ,F (·) is a minimizer of the functional (3) on

u0(·) + W1,10 (Ω), which is maximal among all the minimizers with the

same boundary condition. In other words,

If u(·) is a continuous admissible minimizer of (3) on u0 (·) + W1,10 (Ω)

such that

(i) u (x) = u+θ,F (x) = θ (x) ∀x ∈ C

(ii) u (x) ≥ u+θ,F (x) ∀x ∈ Ω

thenu (x) ≡ u+

θ,F (x) , x ∈ Ω


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Similarly,

Theorem A’

The Sup-convolution u−θ,F (·) is a minimizer of the functional (3) on

u0(·) + W1,10 (Ω) for a given boundary function u0(·), which is minimal

among all the minimizers in the following sense:

If u(·) is a continuous admissible minimizer of (3) on u0 (·) + W1,10 (Ω)

such that

(i) u (x) = u−θ,F (x) = θ (x) ∀x ∈ C

(ii) u (x) ≤ u−θ,F (x) ∀x ∈ Ω

thenu (x) ≡ u−θ,F (x) , x ∈ Ω


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The results above can be considered as a natural extension of the StrongMaximum Principle for elliptic equations/variational problems

Let us follow the way from the classic setting for harmonic functions toTheorems above

The Strong Maximum Principle (SMP) for the harmonic functions(solutions of the Laplace equation ∆u = 0):

if a harmonic (in some open connected domain Ω ⊂ Rn) function attendsits minimum (maximum) in an interior point x0 ∈ Ω then it is constant


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Next, since ∆u = 0 is the Euler-Lagrange equation in the problem ofminimization of the functional

1/2

∫Ω

‖∇u(x)‖2 dx ,

one can formulate the Variational Strong Maximum Principle bysubstituting in the place of 1

2 t2 in the functional above any convex lower

semicontinuous function f : R+ → R+ ∪ +∞ with f (0) = 0

So we have the rotationally invariant functional∫Ω

f (‖∇u(x)‖) dx (5)


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Then, the Variational SMP can be formulated as follows

Given an open bounded connected region Ω ⊂ Rn there is no continuousnonconstant minimizer of the functional (5) on u0(·) + W1,1

0 (Ω) admitingits minimal (maximal) value in Ω,

or, equivalently,

if a continuous nonnegative (nonpositive) minimizer u(·) of (5) onu0(·) + W1,1

0 (Ω) touches zero at some point x0 ∈ Ω then u(x) ≡ 0


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The necessary and sufficient conditions guaranteeing the validity of theVariational SMP for (5) were proposed by A. Cellina in 2002:

(H1) ∂f ∗(0) = 0 (strict convexity at 0)

(H2) ∂f (0) = 0 (smoothness at 0)

Here f ∗ means the Legendre-Fenchel transform of f


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Variational generalizations

First we slightly extend the Cellina’s result to the functional with a generalgauge-type symmetry: ∫

Ω

f (ρF (∇u (x))) dx (6)

where F ⊂ Rn is a closed convex bounded set with 0 ∈ intF , and ρF (·) isthe Minkowski functional associated to F


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Then we assume that the hypothesis (H1) fails, i.e.,

∂f ∗(0) = f −1(0) = [−a, a]

for some a > 0 (the function f (·) is extended symmetrically to R)

In this case SMP in the traditional sense is obviously wrong, but it can beextended by considering in the place of the identical zero (or constant)another ”test” function, namely,

θ + a ρF 0(x − x0)

where θ ∈ R and x0 ∈ Ω

This is a particular case of the Inf-convolution when the set C = x0


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We have the following local result

Let Ω ⊂ Rn and a continuous minimizer u (·) of the functional (6) onu0(·) + W1,1

0 (Ω) be such that for some x0 ∈ Ω and δ > 0

u (x) ≥ u (x0) + aρF 0 (x − x0) ∀x ∈ x0 + δF 0 ⊂ Ω

Then

u (x) = u (x0) + aρF 0 (x − x0) ∀x ∈ x0 +δ

‖F‖ ‖F 0‖+ 1F 0

Here ‖F‖ and ‖F 0‖ are the radii of the spheres centred at the origincircumscribed around F and inscribed into F , respectively


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The previous result can be formulated globally whenever the gauge F hassmooth boundary (equivalently, ρF (ξ) is differentiable at each ξ 6= 0, or F 0

is rotund) and the region Ω is densely star-shaped w.r.t. x0 ∈ Ω (inparticular, Ω can be convex)

Ω ∈ Rn is said to be densely star-shaped w.r.t. x0 ∈ Ω if Ω ⊂ StarΩ(x0)where

StarΩ(x0) := x : λx + (1− λ)x0 ∈ Ω for all λ ∈ [0, 1]


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Namely, under these hypotheses

the inequality

u (x) ≥ u (x0) + aρF 0 (x − x0) ∀x ∈ Ω

implies that

u (x) = u (x0) + aρF 0 (x − x0) ∀x ∈ Ω

Observe that if a = 0 then one immediately obtains from above thetraditional Variational SMP (although under some suplementaryassumptions such as the star-sharpness of Ω)

Simple examples show that the latter assumption can not be avoided


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Considering in the place of a singleton x0 an arbitrary nonempty closedsubset C ⊂ Ω and in the place of a real number θ a Lipschitz continuousfunction θ(·) defined on C and satisfying the slope condition (4) thegeneralized Strong Maximum Principle admits form of an ExtremalExtention Principle (from a barrier inside Ω) as formulated above

But for this we need to assume the set Ω to be convex and F to havesmooth boundary


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Let us extend now the SMP in another way by considering the same”elliptic” integral functional but with linear perturbations:∫

Ω

[f (ρF (∇u(x))) + σu(x)] dx (7)

or ∫Ω

[f (ρF (∇u(x)))− σu(x)] dx (8)

with some real σ > 0

In the case f ρF (ξ) = 12‖ξ‖ the Euler-Lagrange equation for the problems

(7) and (8) is the Poison equation (with the constant non zero right-handside)

∆u(x) = ±σ


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We are interested in the following: how far the minimizers in (7) and (8)defer from the minimizers in the homogeneous case (with σ = 0) in theirextreme properties

In other words, we want to obtain some estimates of minimizers in aneighbourhood of their local minimum or maximum points that in thelimit case (σ = 0) would turn back the standart SMP (or its extensions)


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Indeed, such estimates can be made by using a class of a priori minimizersin the problems (7) and (8) found by A. Cellina in 2007

Namely A. Cellina proved that the functions

ω+x0,k

(x) :=n

σf ∗(σnρF 0(x − x0)

)+ k

(respectively,

ω−x0,k(x) := −n

σf ∗(σnρF 0(x0 − x)

)+ k),

x0 ∈ Ω, k ∈ R, minimize the functional (7) (respectively, (8)) on the classof Sobolev functions with the same boundary conditions

Here f ∗(·) means the Legendre-Fenchel conjugate of f (·) (symmetricallyextended to R)


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In order to be defined on whole Ω the following inclusions, naturally,should hold:

Ω ⊂ x0 ± b · nσF 0 (9)

where b := sup(domf ∗) (equivalently, b > 0 is such that f (·) is affine on[b,+∞[)

So, if Ω is enough small (or b > 0 is enough large, in particular, ifb = +∞, i.e., f (·) never becomes affine) then x0 satisfying the conditions(9) exist


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First, by using the solutions ω±x0,k(·) we obtain the following estimates of

minimizers near their nonextremum points

Let u(·) be a continuous minimizer in the problem (7), β > 0 be smallenough and x0 ∈ Ω satisfying the inclusion (9) be not a local maximumpoint of u(·) in the sense that

u(x0) < k − n

σf ∗(σnβ)

wherek := max

x∈x0+βF 0u(x)

Thenu(x) ≤ k − n

σ

[f ∗(σnβ)− f ∗

(σnρF 0(x − x0)

)]for all x ∈ x0 + βF 0 ⊂ Ω (i.e., ρF 0(x − x0) ≤ β)


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Similarly,

If u(·) is a continuous minimizer in (8), β > 0 is small enough and x0 ∈ Ωis not a local minimum of u(·) and, moreover,

u(x0) > k +n

σf ∗(σnβ)

wherek := min

x∈x0−βF 0u(x),

thenu(x) ≥ k +

n

σ

[f ∗(σnβ)− f ∗

(σnρF 0(x0 − x)

)]for all x ∈ x0 − βF 0 ⊂ Ω (ρF 0(x0 − x) ≤ β)


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”Cross-gap” extension of the Variational SMP

Finally, from the above estimates we deduce the following estimate

Let u(·) be a continuous minimizer in the problem (7). If x0 ∈ Ω is alocal maximum point of u(·), namely, there is δ > 0 withu(x0) ≥ u(x) for all x ∈ x0 − δF 0 ⊂ Ω. Then, in a slightly smallerneighbourhood of x0 (namely, in x0 − δ

1+‖F‖‖F 0‖F0) the inequality

u(x) ≥ u(x0)− n


)(10)

holds


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And, symmetrically,

If u(·) is a continuous minimizer in the problem (8) and x0 ∈ Ω is alocal minimum point of u(·) (with some δ > 0 we have u(x0) ≤ u(x)for all x ∈ x0 + δF 0 ⊂ Ω), then

u(x) ≤ u(x0) +n


)(11)

for all x ∈ x0 + δ1+‖F‖‖F 0‖F

0

Observe that the right-hand side of the inequality (10) (for a minimizer ofthe functional with the positive perturbation σu(x)) is exactly ω−x0.u(x0)(x)

(a Cellina’s minimizer of the functional with the negative perturbation−σu(x)) and vice versa


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So, close to extremum points of a continuous minimizer u(·) (maximum orminimum, respectively, depending on sign of the perturbation) we have agap between u(x0) and

u(x0) +n


),

oru(x0)− n


)containing the values of u(·)


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This ”cross-gap” shrinks to zero (i.e., to validity of the traditional SMP)as σ → 0± whenever the conjugate f ∗ is differentiable at the origin and(f ∗)′(0) = 0 (equivalently, f is stricly convex at the origin, orf −1(0) = 0)

If, instead, f −1(0) = [−a, a], a > 0, then f ∗ is affine (= ±at) near theorigin. Therefore, the ”cross-gap” is reduced to the interval between u(x0)and

u(x0)± aρF 0 (±(x − x0))

This, in turn, easily implies the (local) extreme properties of theconvolutions (with C = x0) what we talked about at the beginning


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Let p > 1 and q > 1 be its conjugate number, i.e., 1/p + 1/q = 1.Consider the function f (t) = tp

p , t ≥ 0, which satisfies all the standing

assumptions and whose conjugate is f ∗(t) = tq

q , t ≥ 0

Let us given a continuous minimizer u(·) in the problem

Minimize

∫Ω

[1

p‖∇u(x)‖p + σu(x)

]dx : u(·) ∈ u(·) + W1,1

0 (Ω)

Assuming that x0 ∈ Ω is a local maximum point of u(·) from the above

result we conclude that near x0 this minimizer satisfies a lower estimate

u(x0) ≥ u(x) ≥ u(x0)− 1

q

n

σ

(σn‖x − x0‖

)q= u(x0)−

(σn

)q−1‖x − x0‖q


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Symmetrically,

If u(·) is a continuous minimizer in

Minimize

∫Ω

[1

p‖∇u(x)‖p − σu(x)

]dx : u(·) ∈ u(·) + W1,1

0 (Ω)

,

and x0 ∈ Ω is a local minimum point of u(·) we have the following(upper) estimate, which holds in a neighbourhood of x0:

u(x0) ≤ u(x) ≤ u(x0) +(σn

)q−1‖x − x0‖q


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References

A. Cellina, On the Strong Maximum Principle, Proc. Amer. Math.Soc. 130 (2002), 413-418

A. Cellina, Uniqueness and comparison results for functionalsdepending on u and ∇u, SIAM J. on Control and Optim. 46 (2007),711-716

V. V. Goncharov, T. J. Santos, Local estimates for minimizers ofsome convex integral functional of the gradient and the StrongMaximum Principle, Set-Valued and Var. Anal. 19 (2011), 179-202


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MERCI BEAUCOUP DE VOTRE ATTENTION


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Documents

A priori estimates of minimizers of an integral functional ... · A priori estimates of minimizers of an integral functional extending the variational Strong Maximum Principle Vladimir