HYPERCONTRACTIVITY, HOPF-LAX TYPE FORMULAS, ORNSTEIN-UHLENBECK OPERATORS (II)

. In this paper we study Hopf-Lax formulas, hypercontractivity, ultracontractivity, logarithmic Sobolev inequalities for a class of ﬁrst order Hamilton-Jacobi equations.


1.
Introduction. Inspired by [6], in this paper we study the problem 1) where α i , 1 ≤ i ≤ N , are nonnegative real numbers. More precisely, in Sections 2-5 we consider the case where α i > 0 for all i = 1, 2 . . . , N , and in Section 6 we generalize to the case where α i could vanish for some indeces i. If α i > 0 for all i = 1, 2 . . . , N , we find the formula u(x, t) = min y∈R N u 0 (y) + N j=1 α j 1 − e −2αj t (y j − e −αj t x j ) 2 }, solution, in the viscosity sense of (1.1). In the line of research of [1] we show hypercontractivity and ultracontractivity for the semigroup associated with our problem, and we obtain a class of logarithmic Sobolev inequalities. We also show extremal functions for our inequalities; these functions do not satisfy the global Lipschitz continuity assumption, which is used to establish the main properties of the semigroup. Neverthless it is possible to satisfy directly that they give solutions in a classical sense of the Cauchy problem.
Proof. We prove (2.5) The property (2.5) is proved. We prove (2.6). We have where a = max α j . We fix a compact set K ⊂ R N , then we can find a constant c 1 such that On the other hand, Then, we use the change of variable We have Finally, using (2.7) and (2.8), we get, taking C = max{c 1 , L u0 |x| + We also have that u is a Lipschitz continuous function in R N . (We do not give the proof, since it is very similar to [5], Lemma 2 p. 126 (see also [1]).
3. Hamilton-Jacobi equations and viscosity solutions. As a first step we show in formal way that the function is a solution of the Hamilton-Jacobi equations with initial data u 0 Now, since we apply Q αi t to the function as in the one-dimensional case, we can apply the result obtained in [1], to obtain Hence our claim follows. We refer to [4] for the notion of viscosity solution using test functions.
Proof. First we show that u is a subsolution. Thanks to the semigroup property for any x ∈ R N , t ∈ (0, +∞), s ∈ (0, t) and, for any y ∈ R N , Take a function φ ∈ C 1 (R N ) and assume that u − φ has a local maximum point in (x 0 , t 0 ). We fix a neighbourhood I 0 of (x 0 , t 0 ) such that ∀(x, t) ∈ I 0 , Then, the previous formula gives for (y, s) We set As a consequence, On the other hand, Then This means that ANTONIO AVANTAGGIATI AND PAOLA LORETI and this shows that u is a subsolution. Next we show that u is a supersolution. Take a function χ ∈ C 1 (R N ). Now assume that u − χ has a local minimum point in (x 0 , t 0 ). We have to prove that We argue by contradiction, and we assume that for all (x, t) in a neighbourhood I of (x 0 , t 0 ) and for some positive θ By assumption there exists a convex neighbourhood With the notation introduced above we have By (3.6), for any (x, t) ∈ I 0 , we have Taking s close to t 0 and x 1 the corresponding minimum such that the point (x 1 , s) ∈ I 0 , by the semigroup property (3.2), we have and we also have Hence Since, as σ describes [0, 1], (x(σ), t(σ)) describes the line segment with ending points (x 0 , t 0 ) and (x 1 , s), by the convexity of I 0 ∩I, the point (x(s), t(s)) ∈ I 0 ∩I ∀σ ∈ [0, 1]. Then, by (3.7), we have Since, for h small enough, I(h) > 0 we have a contradiction with the assumption that (x 0 , t 0 ) is a minimum point for u − χ. This shows that u is a supersolution and the proof is complete.
(4.6) can be written as We find (4.8) In order to have C(α, β, p, q, t) ≤ 1 for any t ∈ (0, +∞) we need If we can fix β such that and, for all the value of p such that p ≤ π α j j = 1, . . . , N we obtain C(α, β, p, t) ≤ 1 ∀t ∈ (0, +∞) (4.9) We have shown the following then Q α t is hypercontractive. More precisely, if (4.10) is satisfied we can fix β such that There are many different cases where (4.10) is satisfied. The simplest is when In this case we fix β = α and we have for all p such that 0 < p ≤ π α . Moreover, if α ≤ π the semigroup Q t in (4.11) is hypercontractive.

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ANTONIO AVANTAGGIATI AND PAOLA LORETI 4.2. Ultracontractivity. We now deal with the case of the norm L ∞ (R N ) of the function exp{Q α t u 0 }. We follow the arguments in [1], and we consider the norm L qe βt (R N ) in (4.7) and we pass to the limit as q → ∞. We recall that ultracontractivity for the nonlinear semigroup t → Q t means e Qtu L ∞ ≤ C e u L 1 . We have to estimate lim q→∞ C(α, β, p, q, t), using the explicit form of C(α, β, p, q, t), given by (4.8). We easily obtain applying to each term the one-dimensional computation (see [1]). Finally, we can state the following . Moreover, if all the constants α 1 , α 2 . . . , α N satisfy α i < π ∀i = 1, 2, . . . , N, and 1 ≤ p < π maxi{αi} , then for t large enough the semigroup Q t is ultracontractive. From (4.12), for p = 1, we observe that

The above inequality holds if and only if
The choice of q by (4.16) gives Indeed, we can do the computation as in the one-dimensional case (see [1]) to get equality. Then we may conclude stating the following

LSI. Given the semigroup t → Q α t defined by (4.1) we consider the function
with q a non descreasing C 1 function and u 0 smooth enough. From our previous computation in [1] we obtain the formula Taking into account that and that Q t u 0 (x) = u(x, t) is solution of the Hamilton-Jacobi equation (3.1) we have We recall the definition of entropy of a function h Then we select the function q(t). We take α 1 , α 2 , . . . , α N satisfying the condition (4.10). We fix β such that and we introduce the function F ⋆ (t) = e Qtu0 L pe βt (R N ) . In a similar way to the one-dimensional case we can state Lemma 5.1. We assume (2.1), (2.2), (4.10) and we take p satisfying then the function F ⋆ is non increasing in (0, 1 mini{2αi} log π p maxi{αi} ).
Proof. We take t 1 and t 2 ∈ (0, 1 mini{2αi} log π p maxi{αi} ) ( t 1 < t 2 ) and β satisfying (5.3).Then By (5.1) we have that and, by (5.2), Then, Finally we wish to give a conclusive result of (5.4) Theorem 5.2. In the same hypothesis of Lemma (5.1), for any function g ∈ H 1 (R N ) we have Proof. We apply (5.4) setting and we fix N positive numbers α 1 , . . . , α N with the conditions Then Then we take the limit as α i → π and t → 0 + we obtain (5.5) for e u0 . By density we obtain (5.5) in the general case. 6. Mixed model. In this Section we fix N = n + m and we represent the N - , and the function f defined in R N = R n × R m , which we represent with the nota- . In a similar way we shall use u(x, x ′ , t) = u(x 1 , . . . , x n , x ′ 1 , . . . , x ′ m , t), and we shall denote the gradient (D, D ′ ) with respect to the variables in R n × R m , with the position D = (∂ x1 , . . . , ∂ xn ) and in R N (6.1) A candidate function to be a solution is We need to define together with the semigroup Q α1 t , . . . , Q αN t we introduced above (from now denoted by Q α1 t,x1 , . . . , Q αn t,xn ) the semigroups Q α1 t,x1 , . . . , Q αn t,xn , Q 0 We observe that the one-dimensional semigroups Q α1 t,x1 , . . . , Q αn t,xn , Q 0 t,x ′ 1 , . . . , Q 0 t,x ′ m applied to functions of n + m variables x 1 , . . . , x n , x ′ 1 , . . . , x ′ m are pairwise permutable. Then our solution of (6.1) will be obtained by the change of variables In the following we shall use the notation It is not difficult to show that ∀s, t ∈ (0, +∞).
Indeed we can use the pairwise permutability of the one-dimensional semigroups, as we have already done in Section 2. In a similar way from the Lipschitz continuity of u 0 (x, x ′ ) we deduce the same property (with a different constant) for the function u(x, x ′ , t), and, also, the uniform convergence on compact subset to the initial datum as t → 0 + . The semigroup properties allow us to show tha u is viscosity solution of the Cauchy problem (6.1). Moreover, denoting by x ′ m the following holds Theorem 6.1. If u 0 ∈ Lip(R n × R m ), then for any compact subset K of R n × R m we have lim uniformly on K.
Now we study the hypercontractivity of Q (α,0) t (u 0 )(x, x ′ ). We fix β ≥ max{α i } and two positive numbers p and q such that p ≤ q and a = p q e −βt .
In order to apply the Brunn-Minkowski inequality, we introduce the functions It is easy to see that, In the above formula for the variable (x ′ 1 , . . . , x ′ m ) we used the vectorial notation. Using the substitution

ANTONIO AVANTAGGIATI AND PAOLA LORETI
Now, we fix γ 1 , . . . , γ n ,γ ′ 1 , . . . , γ ′ m in order to vanish y 2 j and y ′2 j . Hence we solve Hence, arguing as in Section 4 we find Furthermore, we obtain Taking into account that a = p qe βt , with an easy computation, we have where the constant C (α,0) is given by .

HOPF-LAX TYPE FORMULAS 541
Then we set This is the constant obtained in the Section 4 (N = n), assuming α i > 0 i = 1, . . . , n.
Let us examine the new factor 6.1. Optimality of the estimate. We discuss the optimality of our constant in the case α 1 = · · · = α n =: α ; α n+1 = · · · = α N = 0 and β = α We solve the problem and A B real, positive constants. 2 We find the solution Proposition 2. If A and B satisfy the assumptions 0 < B < A < α then there exists a unique t ⋆ ∈ (0, 1 2B ) solution of the equation Indeed to show the assert we draw the graphics of the functions 2Bt and A α (1 − e −2αt ) and its tangent at the origin and we take into account the assumptions. Then, we can state the Theorem Theorem 6.2. We take α 1 = · · · = α n ; α n+1 = · · · = α N = 0 and β = α. We consider real constants A and B such that 0 < B < A < α; the initial datum where t ⋆ is given by the unique positive solution of (6.7) and p ∈ (1, +∞). Then we have where the constant C ⋆ (α,0) is the constant C (α,0) given by (6.6) computed for the particular values of q and t * given by (6.8).
Proof. We take as in Section 4 qe αt = α α − A + Ae −2αt p p ∈ (1, +∞), then Arguing as in Section 4, we have On the other hand The above computation is valid for any positive t, while from now we fix t = t ⋆ (which exists and it is unique by the proposition (2)) such that .

HOPF-LAX TYPE FORMULAS 543
We set .
Here we have the constant .
We observe that this is the constant found by I. Gentil (see Theorem 2.1 [8]). Since q ⋆ = e αt⋆ q we also have which is exactly our factor in C (α,0) . This shows the optimality.